Quantum spin liquids: a review

The topological degeneracy is useful for formal theory and also numerical calculations, but is not very relevant to real systems. However, it is actually symptomatic of a more fundamental property of the topologically ordered state, which is that it supports exotic excitations known as anyons. The elementary excitations of the toric code appear rather simple: the minimal way to raise the energy of the Hamiltonian in equation (1) is to choose a single star or plaquette to be negative, i.e. ${{S}_{{{s}_{0}}}}=-1$ or ${{P}_{{{p}_{0}}}}=-1$, respectively (see figure 3). This clearly costs an excitation energy of $2{{K}^{\prime}}$ or 2K, respectively. We refer to these excitations as ‘electric’ (e) or ‘magnetic’ (m) particles, respectively. The interesting thing about these particles is that they cannot be created locally as individuals. For example, if one acts on the ground state with $\sigma _{i}^{z}$, this creates two e particles because the spin i is shared by two stars. One may also see this by noting that if one multiplies all the star operators, ${\prod}_{s}{{S}_{s}}=1$ identically, as each $\sigma _{i}^{x}$ operator appears twice in this product. To actually create an isolated e particle one needs to act on the ground state with a ‘string’ of adjacent $\sigma _{i}^{z}$ operators, which results in an e particle at each end of the string. Similarly, m particles are always created in pairs at the ends of analogous strings.

Zoom In Zoom Out Reset image size

The non-locality of the e and m particles imbues them with unusual ‘mutual statistics’—see figure 4. Consider a state in which an m particle is at the origin, and an e particle is far away from it. This is defined simply by the state in which S_s  =  1 and P_p  =  1 for all stars and plaquettes except those where the particles are located, where they take a negative sign. Now let us move the e particle in a closed path around the m particle. We move the e particle from one site to the next by acting with $\sigma _{i}^{z}$ where i labels the link connecting the two sites. By moving the particle one link at a time, we find that the final state is connected to the initial state by the action of a product of spins around this closed loop $\mathcal{L}$:

$$\begin{eqnarray}&&|{{\psi}_{\text{fin}}}\rangle =\underset{i\in \mathcal{L}}{\prod}\,\sigma _{i}^{z}|{{\psi}_{\text{init}}}\rangle.\end{eqnarray} \tag{ 5 }$$

Zoom In Zoom Out Reset image size

Now, we can use the discrete equivalent of Stokes’ theorem, to rewrite

$$\begin{eqnarray}&&\underset{i\in \mathcal{L}}{\prod}\,\sigma _{i}^{z}=\underset{p\in \mathcal{A}}{\prod}\,{{P}_{p}},\end{eqnarray} \tag{ 6 }$$

where $\mathcal{L}=\partial \mathcal{A}$, i.e. $\mathcal{A}$ is the area bounded by $\mathcal{L}$. Since the initial state has a magnetic particle inside $\mathcal{L}$, then exactly one P_p in this product is negative, and we find that

$$\begin{eqnarray}&&|{{\psi}_{\text{fin}}}\rangle =-|{{\psi}_{\text{init}}}\rangle.\end{eqnarray} \tag{ 7 }$$

Thus, the act of bringing the e particle around the m one has induced a π phase shift. We say that the e and m particles have non-trivial mutual statistics (specifically they are mutual ‘semions’). In addition to the e and m particles, one can also consider a composite ε particle which is just an e and m particle sitting next to one another. Unlike the e and m particles, which have trivial bosonic self-statistics, the ε particle is actually a fermion under self-exchange. This can be seen by similar considerations to those above.

The existence of these emergent anyons can be considered the defining feature of the toric code phase. Although we have discussed them based on the exactly soluble Hamiltonian in equation (1), they persist as excitations in the presence of arbitrary modifications to the Hamiltonian, provided those changes are not too large in magnitude. All the other properties of the phase can be understood based on these quasiparticles (see, e.g. [23]). For example, the topological degeneracy can be derived by considering the process of locally creating a quasiparticle pair, transporting one of the quasiparticles around a cycle of the torus, and then annihilating it with its partner.

The first phases with intrinsic topological order to be discovered were the Laughlin states in the fractional quantum Hall effect. Such fractional quantum Hall (FQH) states exist for both fermions (as in the much studied physical situation) and for bosons or spins. There is a family of such states, characterized by an integer k. In the kth state, there are k anyons, which can be thought of as carrying a ‘charge’ n/k and a ‘flux’ of n, where $n=1,\cdots,k$. The charge and flux are emergent, i.e. not real charge and flux, but they act similarly. Here, in the FQH context, k is related to the ‘filling fraction’ by $\nu =1/k$, and to the Hall conductance by ${{\sigma}_{xy}}=\nu {{e}^{2}}/h$. However, the fractional statistics of the anyons is, in general, more fundamental: one can start with a Laughlin state and add perturbations to the Hamiltonian which violate all the symmetries, including charge conservation, so that neither the filling fraction nor Hall conductance are well-defined. Nevertheless, the topological order of the state and the mutual statistics of the anyons are preserved. Thus, these Laughlin states can appear in much more general situations than in the FQH effect, including quantum spin systems in which there is no magnetic field, no charges, and there may be only discrete symmetries.

Specifically, bringing a charge q clockwise around a flux f results in a phase factor of ${{\text{e}}^{\text{i}2\pi qf}}$, so that taking an n anyon clockwise around an ${{n}^{\prime}}$ anyon results in a phase factor of ${{\text{e}}^{\text{i}2\pi n{{n}^{\prime}}/k}}$. Interchanging two identical anyons of type n is equivalent to half the rotation of one around the other, and so corresponds to a Berry phase of ${{\text{e}}^{\text{i}\pi {{n}^{2}}/k}}$. These unimodular numbers ${{\text{e}}^{\text{i}2\pi n{{n}^{\prime}}/k}}$ and ${{\text{e}}^{\text{i}\pi {{n}^{2}}/k}}$ describe the mutual and exchange statistics of the anyons. The fusion rule is simply that combining an n and an ${{n}^{\prime}}$ anyon results in an $n+{{n}^{\prime}}$ anyon (mod k). In the Laughlin sequence for fermions, k is an odd integer, and so we can see that fusing k primitive anyons with n  =  1 gives the n  =  k anyon whose exchange results in the factor angle ${{\text{e}}^{\text{i}\pi {{k}^{2}}/k}}=-1$. This has Fermi statistics, and can be identified with the bare electron. For bosons, k is even, and the same procedure of fusing k anyons gives a boson, corresponding to the bare boson. Indeed this correspondence tells us that when the microscopic constituents of the system are bosonic, as is the case for spin systems and hence QSLs, only the Laughlin states with k (even) can exist.

The Laughlin states are ‘chiral’: the definition of the anyonic statistics requires us to specify the clockwise sense of rotation of one anyon about another, and if we reversed this sense (or equivalently, preserved the sense, but complex conjugated all the phase factors) we would have a distinct and different topological state. Consequently, these states cannot be realized without breaking time-reversal symmetry. Moreover, and less obviously, these states must always have a gapless boundary. The gapless modes comprise the famous gapless edge states in the FQH effect. This is not obvious, but can be established entirely from the statistics of the anyons [29], and, like the anyonic statistics, the gapless edge modes exist for the Laughlin states even if all symmetries of the system are broken.

A perhaps full classification of topological phases exists in two dimensions. The problem in three dimensions is much less explored, and offers qualitative differences. An interesting example which has the most likelihood of experimental relevance is realized by the 3D toric code model—see figure 7. This is described exactly by equation (1) on, for example, the cubic lattice, where the stars S_s become 3D clusters of six bonds emerging from a site, and the plaquettes S_p remain as they are, but include all three orientations (xy, yz, and xz planes). It is solvable in the same way as the 2D toric code, and realizes a 3D topological phase. For example, with periodic boundary conditions, there are 2³  =  8 degenerate ground states, and no local operator can distinguish these states.

Zoom In Zoom Out Reset image size

Significant differences from two dimensions appear when we consider the excitations. We can again consider defect sites at which S_s  =  −1. These are point defects in three dimensions which may be called e particles. The magnetic excitations are, however, different. This is because one cannot create a single plaquette with P_p  =  −1. By construction, the product of ${\prod}_{p\in \mathcal{S}}{{P}_{p}}$ over all plaquettes forming a closed surface equals 1. This is the equivalent of a Gauss’ law integral and implies that the magnetic excitations form lines, which can be seen as piercing these plaquettes with P_p  =  −1 and which cannot terminate in the sample. Thus, the magnetic m defects form either closed loops or extended lines that cross the sample. With a finite amount of energy, only small loops can be created, which are topologically trivial from far away. Thus, one does not really have a non-trivial magnetic m particle in the spectrum of the 3D toric code phase. Instead of mutual statistics of particles, one has mutual statistics between the point e particles and the line m defects: if an e particle is transported around a closed loop which links with an m defect, a π phase shift is acquired.

Because of the line-like nature of the m defects, the 3D toric code phase is more stable than the 2D one. Long defect lines cost energy proportional to their length, which even at low non-zero temperature cannot be compensated by entropy. Hence, there is actually a finite-temperature phase transition from a low-temperature ‘deconfined’ phase to a high-temperature trivial one.

A naïve analysis consists of taking very small, but non-zero U, which we expect to introduce small fluctuations of B_p. Assuming they are small, we may then Taylor-expand the cosine term to obtain (up to a constant)

$$\begin{eqnarray}&&H\approx {{K}^{\prime}}\underset{a}{\sum}\,q_{a}^{2}+\frac{V}{2}\underset{\langle ab\rangle}{\sum}\,E_{ab}^{2}+\frac{K}{2}\underset{p}{\sum}\,B_{p}^{2}.\end{eqnarray} \tag{ 19 }$$

This is just the usual electromagnetic energy in terms of field strengths squared. When this expansion is valid, we say that the system is in the Coulomb phase, sometimes also known as the deconfined phase. It turns out that the naïve analysis is correct, and the expansion is valid, in the limit described ($V\ll K$) in three dimensions, but not in two dimensions. We will return to the 2D case later in section 3.5.

For now we concentrate on three dimensions. Since the expanded H is quadratic in E_ab and A_ab, it can be readily diagonalized by normal modes. We will not go into the details here, which are just the usual treatment of quantized electromagnetism, but on a lattice [43]. Since the normal modes with different momentum do not interact, the low-energy, long-wavelength physics is captured just by the continuum limit of equation (19):

$$\begin{eqnarray}&&H\approx {\int}^{}{{\text{d}}^{3}}x\,\left[\frac{\epsilon}{2}|\vec{E}{{|}^{2}}\,+\,\frac{1}{2\mu}|\vec{B}{{|}^{2}}\right].\end{eqnarray} \tag{ 20 }$$

This is just standard vacuum electromagnetism, with some effective dielectric constant and magnetic permeability μ. As a consequence, there are propagating linearly dispersing gapless ‘photon’ modes, with velocity ${{c}_{\text{eff}}}=\sqrt{\epsilon \mu}$, which exist with two transverse polarizations, just as in ordinary electromagnetism. Second, above an energy gap, non-zero charges can be present. For these charges, there is an additional contribution to the ground-state energy in the corresponding sector, which is finite but long-range: beyond the finite ${{K}^{\prime}}$ energy explicit in equation (19), one finds the usual long-range Coulomb potential, proportional to ${{q}_{a}}{{q}_{b}}/{{r}_{ab}}$, where r_ab is the distance between sites a and b.

A third property of the 3D Coulomb phase is not apparent from the naïve approximate Hamiltonian in equation (19). In addition to the electric charges, which are explicit here, implicitly in the original model of equation (13), magnetic charges are also possible. These are point defects in three dimensions, near which B_p is not small and so the expansion of the cosine is not possible, but where this is possible far away. Owing to the periodicity of the cosine, such a defect can act as a source of magnetic flux, owing to the $2\pi$-ambiguity of A and hence B_p. It is natural to define an unambiguous B_p to be the minimum of all its possible values, i.e. ${{B}_{p}}={{\left(\text{curl}\,A\right)}_{p}}-2\pi {{n}_{p}}$, with ${{n}_{p}}\in \mathbb{Z}$ chosen such that $|{{B}_{p}}|\leqslant \pi$. Then, if one sums B_p over a set of plaquettes surrounding the defect (the discrete analog of the surface integral), the result is a non-zero integer multiple of $2\pi$. This is possible because we have taken advantage of the $2\pi$ phase ambiguity, and is reflected in a divergence of n_p. This excitation is, for obvious reasons, called a ’magnetic monopole’. Within the Coulomb phase, the magnetic monopoles behave similarly to the electric charges: they are separated by a non-zero but finite energy gap (of order K rather than ${{K}^{\prime}}$) from the ground state, and experience a 1/r Coulomb potential with one another.

The Coulomb phase is not a topological phase, which must have a gap to all excitations. There is no ground-state degeneracy, nor is there an exponentially approximate one, on a torus or other multiply connected manifolds (there are low-energy states on a torus whose energy decays as a power law of system size associated with flux threading the torus [43]). The system does not support anyons, which are not well-defined in three dimensions anyway. However, the Coulomb phase does share the essential attribute that it supports excitations which are non-local, and cannot be created singly by any local operator.

The electric charge is the simplest such excitation. An isolated electric charge is defined, in the Kitaev-like limit where $V\to 0$, by a single site with non-zero q_a. Consider a general local operator, consisting of operators within some finite region R. It can be expanded in a sum of terms of the form

$$\begin{eqnarray}&&\mathcal{O}=F\left({{\left\{{{E}_{ab}}\right\}}_{\langle ab\rangle \in R}}\right)\times {{\text{e}}^{\text{i}\underset{{}}{{{{{\sum}^{}}_{\langle ab\rangle \in R}}}}\,{{m}_{ab}}{{A}_{ab}}}},\end{eqnarray} \tag{ 21 }$$

where m_ab must be integers to maintain periodicity. Acting on any given state, this operator shifts ${{E}_{ab}}\to {{E}_{ab}}+{{m}_{ab}}$. Notably m_ab  =  0 for bonds ab outside of R. Now, we can consider the total charge in a region $\hat{R}$ larger than R,

$$\begin{eqnarray}&&{{Q}_{{\hat{R}}}}=\underset{a\in \hat{R}}{\sum}\,{{\left(\text{div}\,E\right)}_{a}}=\underset{\langle ab\rangle \in \partial \hat{R}}{\sum}\,{{E}_{ab}},\end{eqnarray} \tag{ 22 }$$

where we rewrite it as a surface ‘integral’ using Gauss’ law. The change in this charge induced by $\mathcal{O}$ is then

$$\begin{eqnarray} &&\Delta {{Q}_{{\hat{R}}}}=\underset{\langle ab\rangle \in \partial \hat{R}}{\sum}\,{{m}_{ab}}=0.\end{eqnarray} \tag{ 23 }$$

This vanishes because the bonds m_ab in the final sum are outside the domain R acted on by $\mathcal{O}$. So local operators cannot create single charges, only neutral combinations such as dipoles. Nevertheless, in the Coulomb phase, these charges are elementary excitations. A different way to see this is that an electric charge can be sensed infinitely far away by its electric Coulomb force, or the electric-field lines emanating from it. Since a local operator cannot generate field lines far from the region it acts upon, it cannot create a non-neutral charge configuration. Likewise, magnetic monopoles cannot be created singly by local operators, since they too experience long-range Coulomb forces and are associated with magnetic-field lines which extend to infinity in the Coulomb phase. The existence of such non-loca,l but finite energy excitations is characteristic of a highly entangled phase.

We spent quite a bit of time describing Z₂ gauge theory and U(1) gauge theory. We emphasized that both possess, in appropriate dimensions, deconfined phases in which the gauge-charged particles ‘emerge’ as finite energy, but non-local excitations. However, there is another famous gauge theory which appears heavily in the condensed matter literature: Chern–Simons theory in two dimensions. Here, the Maxwell term is replaced or supplemented by a term with one fewer derivative. The U(1) Chern–Simons action is

$$\begin{eqnarray}&&{{S}_{\text{CS}}}={\int}^{}\text{d}t{{\text{d}}^{2}}x\,\left[\frac{k}{4\pi}{{\epsilon}^{\mu \nu \lambda}}{{A}_{\mu}}{{\partial}_{\nu}}{{A}_{\lambda}}-{{j}^{\mu}}{{A}_{\mu}}\right],\end{eqnarray} \tag{ 33 }$$

where ${{j}^{\mu}}(x,t)=\left(\,{{j}^{0}},\,\boldsymbol{j}\right)$ describes the space-time currents of quasiparticles,

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{j}^{0}}(x,t) & = & \underset{i}{\sum}\,{{n}_{i}}{{\delta}^{(2)}}\left(x-{{x}_{i}}(t)\right), \end{array}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \boldsymbol{j}(x,t) & = & \underset{i}{\sum}\,{{n}_{i}}\overset{\centerdot}{{\boldsymbol{x}_{{i}}}}\,{{\delta}^{(2)}}\left(x-{{x}_{i}}(t)\right). \end{array}\end{eqnarray} \tag{ 35 }$$

Here, n_i gives the ‘number’ of the minimal quasiparticles that one may consider moving together. We use the standard notation that lumps time in as a zeroth space-time component and index. The integer k is called the ‘level’ of the Chern–Simons theory, and the theory is often denoted by the symbol U(1)_k. Recall that the action for Maxwell electromagnetism involves two derivatives of the gauge field, so that the Chern–Simons action indeed has fewer derivatives and hence totally changes the low-energy properties of the gauge theory, dominating any Maxwell term if it is present.

Indeed, the presence of the Chern–Simons term opens a gap in the spectrum, which is of the order of the ratio of the Chern–Simons coefficient to the coefficient of the Maxwell term. In pure Chern–Simons theory, the gap is infinite, and the theory describes only global topological excitations. Consequently, the Chern–Simons gauge theory is known as a topological quantum field theory. In fact, the U(1)_k theory actually describes the Laughlin series of topologically ordered phases, discussed in section 2.3. The level of the Chern–Simons theory corresponds precisely to the notation used there.

To understand how this arises, note that if one integrates out A₀, one obtains a functional delta function, which imposes

$$\begin{eqnarray}&&\boldsymbol{\nabla} \times \boldsymbol{A}={{j}^{0}}(x,t)=\frac{2\pi}{k}\underset{i}{\sum}\,{{n}_{i}}{{\delta}^{(2)}}\left(x-{{x}_{i}}(t)\right).\end{eqnarray} \tag{ 36 }$$

Solving this equation completely determines $\boldsymbol{A}$ up to gauge transformations, so one can evaluate the remaining effective action S_eff in terms of the quasiparticle trajectories. The exponential, ${{\text{e}}^{\text{i}{{S}_{\text{eff}}}}}$ gives a quantum phase associated with the world lines of the particles. One can understand this phase physically as due to the Aharonov–Bohm phase a particle acquires on encircling the flux given in equation (36). One can see that the phase obtained on taking a particle i around particle j is

$$\begin{eqnarray}&&{{\theta}_{ij}}=\frac{2\pi}{k}{{n}_{i}}{{n}_{j}}.\end{eqnarray} \tag{ 37 }$$

The elementary act of exchanging two identical particles with charge ${{n}_{i}}={{n}_{j}}=1$ is equivalent to half the motion of taking one around the other, and so gives a phase $\theta =\pi /k$, so that the minimal quasiparticle is a fermion if k  =  1, but an anyon otherwise. Note that equation (37) corresponds precisely to the Laughlin series of quasiparticle states, discussed in section 2.3.1, with filling factor $\nu =1/k$.

The generalization of Chern–Simons theory to a multi-component form leads to many more states. The action is now

$$\begin{eqnarray}&&{{S}_{K}}={\int}^{}\text{d}t{{\text{d}}^{2}}x\,\left[\frac{1}{4\pi}{{\epsilon}_{\mu \nu \lambda}}\underset{I,J=1}{\overset{N}{\sum}}\,{{K}_{I,J}}A_{\mu}^{I}{{\partial}_{\nu}}\,A_{\lambda}^{J}-j_{I}^{\mu}A_{\mu}^{I}\right],\end{eqnarray} \tag{ 38 }$$

where K_I,J is a symmetric matrix with integer entries called the ‘K-matrix’.

It is believed that any abelian topological phase in two dimensions can be represented by equation (38). By a simple generalization of the analysis just discussed for a single component, one may obtain the physical meaning of the K-matrix theory. The K-matrix in equation (38) contains all the topological information of the quasiparticles of the phase, and thereby provides a compact description of them. K-matrices are $N\times N$ where N is the number of quasiparticle generators, which are represented by vectors $\boldsymbol{n}$ with a single non-zero entry, equal to one. All other quasiparticles in the theory can be obtained from them bringing these together, two at a time, an act known as ‘fusion’. A generic quasiparticle is described by an integer vector $\boldsymbol{n}=\left({{n}_{1}},\cdots,{{n}_{N}}\right)$, and fusing two quasiparticles gives a new quasiparticle whose vector is simply the sum of those of the two which were fused. By generalizing the manipulations outlined above for the single-component Chern–Simons theory, one obtains the anyonic statistics. The phase factor obtained by winding a quasiparticle described by the vector $\boldsymbol{n}$ around a quasiparticle described by the vector $\boldsymbol{m}$, which defines their mutual statistics, is ${{\text{e}}^{\text{i}\theta \left(\boldsymbol{m},\boldsymbol{n}\right)}}$, where

$$\begin{eqnarray}&&\theta \left(\boldsymbol{m},\boldsymbol{n}\right)=2\pi \boldsymbol{m}^{{T}}\centerdot {{K}^{-1}}\centerdot \boldsymbol{n}.\end{eqnarray} \tag{ 39 }$$

The self-statistics of a particle characterized by $\boldsymbol{n}$ is given by ${{\text{e}}^{\text{i}\theta \left(\boldsymbol{n}\right)}}$, where

$$\begin{eqnarray}&&\theta \left(\boldsymbol{n}\right)=\pi \boldsymbol{n}^{{T}}\centerdot {{K}^{-1}}\centerdot \boldsymbol{n}.\end{eqnarray} \tag{ 40 }$$

Note that several vectors may characterize particles of the same topological type, i.e. with identical statistics.

Some important properties of the K-matrix are the following. If and only if all the diagonal entries of K are even integers, then the microscopic degrees of freedom in the system are bosonic; otherwise, the microscopic degrees of freedom contain fermions. The total number of quasiparticle types is given by the absolute value of the determinant of K.

To understand this, we note that we can form special ‘basis’ particles defined by the vectors $\boldsymbol{b}^{{(i)}}=K\centerdot \boldsymbol{n}^{{(i)}}$, where $\boldsymbol{n}^{{(i)}}$ is one of the ‘generators’ with all elements equal to zero except for the ith one which is equal to one. It is straightforward to show that adding $\boldsymbol{b}^{{(i)}}$ to any vector $\boldsymbol{n}$ does not change its mutual statistics. Thus, the $\boldsymbol{b}^{{(i)}}$ describes particles which have trivial mutual statistics with all other particles. Since they have no non-local interactions amongst the other anyons, we should regard them as local particles, which can be considered as microscopic excitations. We can now consider the self-statistics of these microscopic basis particles. We obtain $\theta \left(\boldsymbol{b}^{{(i)}}\right)=\pi {{\left({{n}^{(i)}}\right)}^{T}}\centerdot K\centerdot {{n}^{(i)}}$, from which the condition on the diagonal entries follows.

To count the number of quasiparticle types, we note that we can regard any two quasiparticles which differ by fusing one with such a local basis excitation as topologically the same. Thus, we need to identify all the unique particles, i.e. integer vectors, which are not equivalent by such a fusion. Geometrically, the full set of integer vectors forms a N-dimensional hypercubic lattice. The $\boldsymbol{b}^{{(i)}}$ vectors form N ‘primitive lattice vectors’ for a superlattice. The number of unique quasiparticles is just the number of hypercubic lattice sites within a primitive unit cell defined by the $\boldsymbol{b}^{{(i)}}$: this is just the determinant of K.

The existence of chiral edge states—gapless 1D modes which propagate only in one direction along the boundary of the sample—follows rather directly from the Chern–Simons theory, as shown by Wen [53]. For the multi-component theory of equation (38), the number of such chiral models is the difference between the number of positive and negative eigenvalues of the K-matrix. This is not so easy to show from quasiparticle statistics, but follows from analysis of the Chern–Simons theory with a boundary [54].

The multi-component Chern–Simons theory captures the toric code example discussed in section 2.1. It has the K-matrix

$$\begin{eqnarray}{{K}_{\text{TC}}}=\left(\begin{array}{c} 0 & 2 \\ 2 & 0 \end{array}\right).\end{eqnarray} \tag{ 41 }$$

The e and m are particles are the generators of the fusion group for the toric code and are represented by $\boldsymbol{n}=(1,0)$ and $\boldsymbol{n}=(0,1)$, while the fermion ε is at (1,1) and the trivial particle at (0,0). We also see that this K-matrix is non-chiral, i.e. it has one positive and one negative eigenvalue, and so there are no protected chiral edge states in the toric code. The K-matrix theory has been used recently to discuss the role of symmetries in 2D topological phases [55, 56]. We discuss symmetries in QSLs in section 5.

The above Chern–Simons theory is a compact and powerful method to analyze abelian topological phases. It also can be extended to non-abelian groups, to discuss non-abelian topological phases. However, it is important to note that the Chern–Simons action contains no more and no less information than the mutual and self-statistics of anyons, as discussed in section 2.3. The only result which appears somewhat distinct from those statistics is the relation of edge states to bulk topological order. We believe that the universal aspects of this relationship also follow from the anyonic statistics, though this is more subtle [29, 57]. In our view, Chern–Simons theory is just one way to represent the universal properties—the fundamental data describing the anyons—of some particular topological phases. It is convenient for some purposes, but certainly not essential. This is quite different from usual U(1) Maxwell theory of section 3.1, where the electric-field fluctuations and the photon mode of the gauge theory itself are observable by local measurements.

There is a generalization of Chern–Simons theory known as BF (or background field) theory. In 2  +  1 dimensions it is equivalent to Chern–Simons theory so we do not discuss it. However, it can be formulated in three dimensions, and describes universal features of some 3d topological phases. The basic action of BF theory is

$$\begin{eqnarray}&&{{S}_{\text{BF}}}={\int}^{}\text{d}t{{\text{d}}^{3}}x\,\left[\frac{k}{2\pi}{{\epsilon}^{\mu \nu \sigma \lambda}}{{b}_{\mu \nu}}\,{{f}_{\sigma \lambda}}-{{j}^{\mu}}{{a}_{\mu}}-{{\tilde{{j}}\,}^{\mu \nu}}{{b}_{\mu \nu}}\right],\end{eqnarray} \tag{ 42 }$$

where

$$\begin{eqnarray}&&{{f}_{\mu \nu}}={{\partial}_{\mu}}{{a}_{\nu}}-{{\partial}_{\nu}}{{a}_{\mu}}.\end{eqnarray} \tag{ 43 }$$

Here, ${{f}^{\mu \nu}}$ is an electromagnetic-field-strength tensor, ${{a}_{\mu}}$ is a usual gauge field, and ${{b}_{\mu \nu}}$ is a two-form anti-symmetric gauge field. These fields are coupled to currents of generalized ‘quasiparticles’, but while ${{j}^{\mu}}$ describes the usual particle world lines, ${{\tilde{j}}^{\mu \nu}}$ describes worldsheets of line-like objects. This is in accordance with our discussion of line-like m excitations and particle-like e excitations in the 3D toric code. Indeed, the BF theory with k  =  2 reproduces the quasiparticle statistics of that model. As shown by Hansson et al [58], the BF theory also applies at long distances (compared to the magnetic penetration length) to a 3D superconductor with a full gap, when electromagnetic fluctuations are included. According to [58], a conventional superconductor is in the same phase, topologically, as the 3D toric code. In this correspondence, the m lines are simply the vortices of the superconductor, and the e particles are the electronic quasiparticles, and they indeed have the same mutual statistics as in the toric code. We are not aware of how or even if the BF theory distinguishes the statistics of electronic quasiparticles from the Bose statistics of the e particles in the toric code.

BF theory has been much less applied in condensed matter physics than Chern–Simons theory has. We refer the reader to the articles by Hansson et al [58] and one by Cho et al [59] for further discussion, or to the high-energy literature where there is much more work, but with which we are unfamiliar.

It may happen, for specific choices of $t_{ij}^{\alpha \beta}$ and $\Delta _{ij}^{\alpha \beta}$, that equation (48) has more than just the Z₂ parity symmetry. For example, if $\Delta =0$, H₀ is invariant under a general phase rotation of the fermions, ${{f}_{j\alpha}}\to {{\text{e}}^{\text{i}\chi}}{{f}_{j\alpha}}$. This contains the parity transformation and enlarges it, but like the parity symmetry of the previous section, it is unphysical, and just part of the gauge invariance introduced by the ansatz of equation (45). As the phase χ is arbitrary, this is called a ’U(1) state’. Thus, following the same reasoning, we should improve the mean field theory by promoting this to a gauge symmetry, and requiring physical states to be invariant under the associated gauge transformation. We let ${{H}_{0}}\to {{H}_{1}}$, in this case with

$$\begin{eqnarray}&&{{H}_{1}}=\underset{ij}{\sum}\,t_{ij}^{\alpha \beta}{{\text{e}}^{\text{i}{{A}_{ij}}}}f_{i\alpha}^{\dagger}{{f}_{j\beta}}+{{H}_{g}},\end{eqnarray} \tag{ 53 }$$

where ${{A}_{ij}}=-{{A}_{ji}}$ is a lattice gauge field, analogous to the vector potential. We require that the full theory is invariant under the gauge transformation ${{f}_{j\alpha}}\to {{\text{e}}^{\text{i}{{\chi}_{j}}}}{{f}_{j\alpha}}$, $f_{j\alpha}^{\dagger}\to {{\text{e}}^{-\text{i}{{\chi}_{j}}}}f_{j\alpha}^{\dagger}$, and ${{A}_{ij}}\to {{A}_{ij}}+{{\chi}_{i}}-{{\chi}_{j}}$. The gauge part of the Hamiltonian has the same form introduced in section 3.1, equation (13), and involves the ‘electric’ field ${{E}_{ij}}=-{{E}_{ji}}$ conjugate to the vector potential (see equation (12)). Because A_ij is a periodic variable (see equation (53)), the eigenvalues of E_ij are quantized to be integers (that is, generally, separated by integers). In the literature, the periodicity of A_ij is denoted by calling the latter a compact U(1) gauge field.

We must again express the condition that physical states are gauge-invariant. This means that the gauge generator Q_i, where the gauge transformation is enacted by ${{U}_{i}}={{\text{e}}^{\text{i}{{\chi}_{i}}{{Q}_{i}}}}$, commutes with H, i.e. is a constant in the physical manifold. The generator is

$$\begin{eqnarray}&&{{Q}_{i}}={{\left(\text{div}\,E\right)}_{i}}-f_{i\alpha}^{\dagger}\,{{f}_{i\alpha}}=-1.\end{eqnarray} \tag{ 54 }$$

where the particular constant of  −1 is chosen so that in the limit in which the electric field vanishes, the microscopic constraint is satisfied. This is the analog of the sign choice in equation (51), and corresponds to the condition of half-integral spin per site.

Equations (53), (13) and (54) give a prescription to translate the mean-field solution to a physical description of low-energy states in the U(1) case. This full Hamiltonian is what is called a compact U(1) gauge theory with (fermionic) matter fields. It corresponds most directly to the mean-field approximation in the limit of large K, in which case fluctuations of A_ij are suppressed. However, because A_ij is a continuous variable, we must be cautious: there is no gap for excitations which induce fluctuations of A_ij. Hence, it is not obvious that the singular case $K=\infty$, which in fact corresponds to the mean-field state, qualitatively describes the same physics as the generic situation $K<\infty$, when fluctuations are allowed. The question is whether the mean-field U(1) state is stable to gauge fluctuations.

First, we discuss a ‘simple’ case. For large K we may certainly make a first approximation to the problem by taking A_ij  =  0, which is a particular gauge satisfying $\text{curl}A=0$. Then we can diagonalize H₁ in equation (53), to obtain a mean-field fermion spectrum. In the simplest cases, this is gapped, and so it appears safe to integrate out the fermions. We expect that, in this case, we will obtain a pure U(1) gauge theory, i.e. a Hamiltonian of the form of equation (13), but with renormalized couplings, and the constraint in equation (54) replaced by one no longer involving fermions, which has the form given in equation (18). The fixed ‘background’ charges q_i might depend upon the site i, and should be determined from a more careful study. Technically we can derive equation (18) in particular cases systematically, if we can tune the mean-field band structure ($t_{ij}^{\alpha \beta}$) so that the fermion gap becomes very large. For example, on a cubic lattice, one may take a mean-field state of the form

$$\begin{eqnarray}&&t_{ij}^{\alpha \beta}={{\delta}^{\alpha \beta}}\left[u{{\delta}_{ij}}{{\epsilon}_{i}}-t{{\delta}_{|i-j|,1}}\right],\end{eqnarray} \tag{ 55 }$$

where ${{\epsilon}_{i}}=+1(-1)$ on the A (B) rock-salt sublattice of the cubic lattice. Equation (55) just describes fermions with nearest-neighbor spin-preserving hopping t and a staggered potential u. When u is large, the ground state simply has a two-sublattice structure with two fermions on each site of one sublattice, and zero fermions on each site of the other sublattice. Replacing $f_{i\alpha}^{\dagger}\,{{f}_{i\alpha}}=1-{{\epsilon}_{i}}$, we then obtain equation (18) with ${{q}_{i}}={{\epsilon}_{i}}$. This specific case is illustrative, but the specific form of the q_i is not particularly important for the stability arguments summarized below.

For the case of gapped spinons, therefore, we arrive at an effective pure compact U(1) gauge theory on the lattice, without fluctuating matter fields. This is just the problem formulated in section 3.1, equations (13), (12) and (18). As discussed there, the naïve large K limit of this theory is just Maxwell electromagnetism, which describes the familiar electromagnetic photon modes (with two polarizations) in free space in three dimensions. In two dimensions, it describes a similar photon with a single polarization (as the magnetic field can only be directed out of the plane). Because of the similarity to standard electromagnetism, the state described in this way is known as a Coulomb phase, or sometimes a photon phase. This Coulomb phase is, as discussed in section 3.5, stable in three dimensions, but not in two. This means that for this class of mean-field states in two dimensions, the parton mean field is qualitatively wrong at low energies and long distances.

An interesting twist on the stability issue in two dimensions is the situation in which the spinons are gapless. Integrating out these gapless modes is a dangerous and uncontrolled procedure, and so the analysis of the pure gauge theory is not applicable. In general, there are two conflicting tendencies. First, the spinons themselves (if they are bosons), or some collection of them (e.g. a spinon-anti-spinon pair) might condense, disrupting the state. This seems more likely in the gapless case than the gapped one since the spinons or collections of them can be generated with very small energy. Conversely, the monopole events, which involve fluctuations of magnetic flux, should be suppressed by the zero-point fluctuations of the spinons, which carry the dual electric gauge charge: due to the commutation relation in equation (12), fluctuations of electric fields suppress fluctuations of A_ij and hence magnetic flux.

The theoretical challenge is that the theory, even without monopoles (i.e. the compact theory) of gauge fields coupled to gapless spinons, is generally complicated and not easy to solve, and there are many possible cases involving different ‘band structures’ of the spinons. Consequently, this problem is still largely unresolved. We summarize the major known results here:

• Dirac fermions: Neglecting monopoles, theorists have heavily studied the problem of quantum electrodynamics in three space-time dimensions, or QED₃, described by the low-energy action,
  $$\begin{eqnarray}\begin{array}{*{35}{l}} {} & {} & S={\int}^{}\text{d}\tau \,{{\text{d}}^{2}}x\left[\underset{i=1}{\overset{N}{\sum}}\,{{{\bar{\psi}}}_{i}}\left({{\partial}_{\tau}}-i{{a}_{0}}-i\left(\vec{\nabla}-i\vec{a}\right)\centerdot \vec{\sigma}\right){{\psi}_{i}} \right. \\ {} & {} & \left. \,\,\,\,\,\,\,\,\,\,\,\,\,+\,\frac{1}{4g}{{\left({{\partial}_{\mu}}{{a}_{\nu}}-{{\partial}_{\nu}}{{a}_{\mu}}\right)}^{2}}\right], \end{array}\end{eqnarray} \tag{ 56 }$$
  where ${{\psi}_{i}}$ is an N-component vector of two-component spinors (spinor indices suppressed), and ${{a}_{0}},\vec{a}$ are the time and spatial components of a U(1) gauge field. It is known that, for sufficiently large N, i.e. N  >  N_c with some finite N_c, this theory is in a conformally invariant phase with power-law correlations, described by some non-trivial RG fixed point. This is known as an ‘algebraic spin liquid’ [64] or ‘Dirac spin liquid’ [65] phase. For sufficiently large N, moreover, the critical phase is stable to monopoles [66]. It is believed that, for small N, the critical phase of QED₃ becomes unstable. The precise nature of this instability, the critical N for which this occurs, and the dependence of this instability upon microscopic details, are not well understood. We do not know if a critical Dirac spin-liquid state is stable in any physically relevant situation (for example, N  =  4 for the proposed Dirac spin liquid on the kagomé lattice [65]).
• Fermi surface: Another situation of interest is the case in which the spinons form a partially filled band with a Fermi surface (or multiple Fermi surfaces). This is known as a ‘spinon Fermi-surface state’ [67–69]. Following the heuristic arguments above, since this case has more low-energy ‘electric’ excitations (spinons) than the Dirac fermion case, it is expected to be more stable to monopoles than the latter situation. However, many other instabilities are possible involving pairing, particle–hole condensates, etc. Many theories postulate that, like the Dirac case, this situation supports a gapless power-law regime, known as a ‘strange metal’. The properties of such a strange metal itself are not well understood, and the subject of much current study [70]. As for the above case, we do not know definitively whether a spinon Fermi-surface state is stable in any physically relevant context, but such states have been suggested to be relevant to organic triangular lattice materials—see section 7.3.3.
• Critical bosons: Unlike fermions, bosons are never generically gapless, because negative-energy single-particle modes are inherently unstable to condensation. Thus, for generic spin-liquid states described by the bosonic parton construction, one obtains only gapped spinons. However, one may contemplate the situation in which a system is tuned to a quantum phase transition point at which the gap of bosonic spinons vanishes. It is possible for a U(1) gauge theory to be stabilized in two dimensions at this point. This phenomena is known as ‘deconfined quantum criticality’ [71], and has been argued to describe the phase transition between Néel and valence-bond solid (VBS) orders on the square lattice.

If one is interested in the universal properties of the QSL phase, the gauge theory (e.g. equation (61)) contains a great deal of extraneous information and far more degrees of freedom than necessary. It is also tied to a particular representation, i.e. the parton construction. A more minimal and also more physical question related to symmetry is to ask about the quantum numbers of the excitations. For example, if the QSL preserved spin-rotation symmetry, we may ask about the SU(2) spin of the spinons. Likewise, we can ask how the excitations transform under space-group operations.

We make first a general remark about quantum numbers of excitations in general. In a many-body system in the thermodynamic limit, excitations are usually expected to be local. If the system has a gap, then they can be considered as ‘particles’, and even in the gapless case often a quasiparticle description holds. When we talk about the quantum numbers of the excitation we really mean this relative to the ground state. What does ‘relative’ mean? We should think of a quasiparticle as created by some operator. When we talk about the quantum numbers of the quasiparticle, we really refer to the transformation properties of this operator. So it is better to think of quasiparticle quantum numbers as referring to transformation properties of operators rather than of states. This distinction arises already in very simple non-entangled systems. Consider, for example, an ordered ferromagnet, composed of spin-1/2 spins. The ground state itself has some large spin, which might be half-integer if the total number of spins is odd. Due to the broken symmetry, we can only discuss the quantum number of total spin along the magnetization axis—call it S^z—since rotation symmetry about this axis is unbroken. The operator which creates such an excitation is $\mathcal{O}=S_{i}^{-}$, if the ground state has S^z  =  N/2. Note that $\mathcal{O}$ creates integer spin $\Delta {{S}^{z}}=1$, that is, under a U(1) transformation, $U={{\text{e}}^{\text{i}\theta {{S}^{z}}}}$, ${{U}^{\dagger}}\mathcal{O}U={{\text{e}}^{\text{i}\theta}}$, with a unit coefficient in the exponential. Even though the spins themselves are half-integer, the excitations have integer spin. More formally, extending to full spin-rotation symmetry, the spin operators themselves are vectors, and transform under SO(3) rather than SU(2). From any finite combination of them, we can build only representations of SO(3). So in conventional, short-range entangled phases, a spin-1/2 quasiparticle, which is not a single-valued representation of SO(3), cannot arise.

In this sense, the spin-1/2 spinons of a QSL are ‘fractional’ quasiparticles. Crudely speaking, they are created by spinors ${{f}_{i\alpha}}$ or $f_{i\alpha}^{\dagger}$. This is allowed because physical operators always involve an even number of fermion operators. More generally, the non-local quasiparticles of a QSL may form multivalued representations of physical symmetries. They must, of course, satisfy a constraint that physical states, which are ‘neutral’ collections of such quasiparticles, must decompose into ordinary representations.

All this assumes that we can define quasiparticles as sharp excitations, which is the case in gapped, topological QSLs, but may not be the case in other QSLs. Nevertheless, we proceed with this assumption for now. The representations of physical symmetries acting on individual quasiparticles may be ‘projective’. This means that if there are two symmetry operations g and h, and their composition gh, then the actions U_a of these operations on a quasiparticle state a obey

$$\begin{eqnarray}&&{{U}_{a}}(g){{U}_{a}}(h)=\omega (g,h){{U}_{a}}(gh),\end{eqnarray} \tag{ 63 }$$

where $\omega (g,h)$ can be a non-trivial unimodular number (exponential of a phase). The algebra of the U_a can be considered a projective representation of the symmetry group. This is another PSG, associated now to a quasiparticle rather than a parton mean field theory. A priori, more complex behavior might yet occur, e.g. if there are some internal degrees of freedom of the quasiparticle, then a priori these ‘flavors’ may change under a transformation, and U_a(g) might be a matrix in this space. More generally, one type of quasiparticle could transform into another under a symmetry. When the group operations are violated by the phases $\omega \ne 1$, then we might say that the symmetry is fractionalized. For the case of a spin-1/2 spinon, this is the case because in SO(3), we can consider, for example, g  =  h to be a spin rotation by π about some axis, so that gh is the identity. However, for a spin-1/2 particle, $\omega =-1$ for a $2\pi$ spin rotation, so SU(2) is indeed realized projectively for spinons.

Other symmetries may also be realized projectively for quasiparticles of QSLs. For example, in the next section, we will discuss an instance of projectively realized translational symmetry.

Kitaev in [17] (see also [23] for a summary) introduced a simple and appealing model of spin-1/2 spins with anisotropic exchange on a honeycomb lattice:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} H={{J}_{x}}\underset{\langle ij\rangle \in x}{\sum}\,\sigma _{i}^{x}\sigma _{j}^{x}+{{J}_{y}}\underset{\langle ij\rangle \in y}{\sum}\,\sigma _{i}^{y}\sigma _{j}^{y}+{{J}_{z}}\underset{\langle ij\rangle \in z}{\sum}\,\sigma _{i}^{z}\sigma _{j}^{z}, \end{array}\end{eqnarray} \tag{ 70 }$$

where the three different directions of bonds are labeled by $\mu =x,y,z$ (see e.g. figure 10), and $\sigma _{i}^{\mu}$ are Pauli matrices as usual. Kitaev showed that equation (70) is, remarkably, exactly soluble by a fermionic parton construction. That is, a version of the parton mean-field method discussed in section 4.1 is actually exact for this model.

Zoom In Zoom Out Reset image size

Let us define the plaquette operator, ${{W}_{p}}=\sigma _{1}^{x}\sigma _{2}^{y}\sigma _{3}^{z}\sigma _{4}^{x}\sigma _{5}^{y}\sigma _{6}^{z}$. The latter commutes with the Hamiltonian, [W_p, H]  =  0, for all plaquettes, i.e. each ${{W}_{p}}=\pm 1$ is a constant of the motion. This is a local Z₂ gauge symmetry of H which is physical: states with different eigenvalues of W_p are allowed and W_p is measurable in principle. Since there is one W_p per unit cell, the subspace of states with all W_p fixed has only one two-level system per unit cell. We can bring this out by introducing Majorana partons. Defining ${{\sigma}^{\mu}}=\text{i}c{{c}_{\mu}}$ where $c,{{c}_{\mu}}$ are Majorana fermion operators, ${{c}^{\dagger}}=c$, we obtain

$$\begin{eqnarray}&&H=\frac{i}{4}\underset{\langle ij\rangle}{\sum}\,{{\hat{A}}_{ij}}{{c}_{i}}{{c}_{j}},\end{eqnarray} \tag{ 71 }$$

where ${{\hat{A}}_{ij}}=2{{J}_{{{\gamma}_{ij}}}}{{\hat{u}}_{ij}}$, with ${{\hat{u}}_{ij}}=\text{i}c_{i}^{{{\gamma}_{ij}}}c_{j}^{{{\gamma}_{ij}}}$, where ${{\gamma}_{ij}}=\mu$ if $\langle ij\rangle \in \mu$, following the convention defined below equation (70). This is an exact faithful rewriting of the Hamiltonian if we keep only states such that

$$\begin{eqnarray}&&{{D}_{i}}|\psi \rangle =\,|\psi \rangle,\quad \text{with}\quad {{D}_{i}}=c_{i}^{x}c_{i}^{y}c_{i}^{z}{{c}_{i}}.\end{eqnarray} \tag{ 72 }$$

For example, we can enforce equation (72) by acting on an arbitrary state with the projector $P={\prod}_{i}\frac{1+{{D}_{i}}}{2}$. At this point, we note that D_i in equation (72) is the generator of a Z₂ gauge symmetry, which is artificial and induced by the Majorana representation.

Now, because in equation (71), each $c_{i}^{{{\gamma}_{ij}}}$ appears only once, $\left[{{\hat{u}}_{ij}},{{\hat{u}}_{kl}}\right]=0$, and so $\left[{{\hat{u}}_{ij}},H\right]=0$, so that we may treat H as a free (c_i) fermion operator by working in each of the eigenvalue sectors where ${{\hat{u}}_{ij}}=\pm 1$. Recognizing that the plaquette operator may be rewritten in the physical subspace as:

$$\begin{eqnarray}&&{{W}_{p}}=\underset{\langle ij\rangle \in \partial p}{\prod}\,{{\hat{u}}_{ij}},\end{eqnarray} \tag{ 73 }$$

we identify $\hat{u}$ as a gauge potential, and W_p as the associated ${{\mathbb{Z}}_{2}}$ flux operator, as seen by the c fermions.

The ground state should have no vortices [87], and therefore, in the ground state, $\forall p$ W_p  =  1. An acceptable choice of {u_ij} satisfying this constraint is u_ij  =  1 if i belongs to the A sublattice, and u_ij  =  −1 otherwise. Diagonalizing equation (71), we find the following dispersion for c fermion excitations: ${{\epsilon}_{\mathbf{k}}}=\pm 2|{{J}_{x}}{{\text{e}}^{\text{i}\mathbf{k}\centerdot {{\mathbf{n}}_{1}}}}+{{J}_{y}}{{\text{e}}^{\text{i}\mathbf{k}\centerdot {{\mathbf{n}}_{2}}}}+{{J}_{z}}|$, where ${{\mathbf{n}}_{1,2}}=\left(\pm \sqrt{3}/2,3/2\right)$ (for the choice of honeycomb lattice where z bonds are vertical). When ${{J}_{x}}\leqslant {{J}_{y}}+{{J}_{z}}$ and ${{J}_{y}}\leqslant {{J}_{z}}+{{J}_{x}}$ and ${{J}_{z}}\leqslant {{J}_{x}}+{{J}_{y}}$, the fermion spectrum contains two accidental zero-energy Dirac points which lie along the k_y  =  0 line. This allows us to distinguish between two types of phases in the phase diagram: one gapless, and one gapped.

Let us now connect this to the discussion of the toric code and Z₂ gauge theory/topological order in sections 2 and 3.2. The presence of the Z₂-valued W_p operators indicates the presence of an emergent Z₂ gauge structure. A ‘defect’ plaquette with W_p  =  −1 represents a non-local excitation which we would like to identify with one of the anyons of the topological phase. Strictly speaking, this need only be possible for the gapped phase, but we can (following the considerations in, for example, section 4) regard the gapless phase as a Z₂ gauge theory with gapless matter. Following the ‘flux’ notation, it is natural to regard the W_p  =  −1 particle as either an m or e anyon. If we, in fact, track how the fluxes should be described if we make the system anisotropic and enter a gapped phase, we see that some of the fluxes must be identified as m and others as e particles, in alternating rows of plaquettes. To understand this, we can consider the loop operator which moves a flux around a closed loop. Imagine, for example, taking a flux around the filled plaquette in figure 10. To move it from one plaquette to its neighbor, we should change the sign of one link ${{\hat{u}}_{ij}}$, which is shared by the two plaquettes in question. For example, a z link with ${{\hat{u}}_{ij}}=\text{i}c_{i}^{z}c_{j}^{z}$ anticommutes with $\sigma _{i}^{z}$, so $\sigma _{i}^{z}$ performs the action of moving the flux. To move it around the entire shaded plaquette, we must perform six such steps, with appropriate spin operators. One can readily see that the operator which moves a flux around the central loop is exactly W_p! Thus, we see that if the central plaquette contains a flux W_p  =  −1, then moving a second flux in a loop around this central one leads to a π phase shift. This is precisely the mutual statistics expected between e and m particles. It holds already in the isotropic limit.

What is clear is the the c fermion is unambiguously a fermion, so should be identified with the ε particle. As expected, the ε particles sense the m or e particles as a π flux. It is interesting to then think of the decomposition of the spin operator into the fundamental anyons. The formula $\sigma _{i}^{\mu}=\text{i}{{c}_{i}}{{c}_{i\mu}}$ implies that a ε particle is created (by c_i). The ${{c}_{i\mu}}$ operator, in fact, creates a pair of fluxes on the two plaquettes sharing site i and separated by the type μ bond. Following the above discussion, these should be identified as an e and an m particle. So we see that a spin operator creates one of the anyons each; symbolically,

$$\begin{eqnarray}&&\sigma \sim {\mathsf {e}}\,{\mathsf {m}}\,\varepsilon.\end{eqnarray} \tag{ 74 }$$

This is rather natural since e and m fuse to ε and two ε particles fuse to the identity, i.e. this combination is gauge-neutral, and contains zero net electric and magnetic charge.

The identification with the topological order of the toric code is complete in the gapped phase, and indeed one can actually map the honeycomb model to the toric code in the limit in which one of the couplings is large. In the gapless state, the m and e statistics are ill-defined, since moving the vortex induces dynamics in the low-energy fermions: adiabatic transport is not possible. However, interestingly, upon introducing a time-reversal-breaking perturbation, the Dirac cones become gapped, and one observes that a Majorana fermion binds to each m and e particle. The resulting statistics of these composite excitations is non-abelian, and the system also displays robust chiral modes at the edge. This can be shown explicitly by introducing, for example, time-reversal-breaking second nearest-neighbor interactions, but it is expected that all phases obtained by other time-reversal-breaking perturbations should be smoothly connected to one another.

The gapless phase has become of particular interest because of the (surprising) relevance of the model in equation (70) with ${{J}_{x}}={{J}_{y}}={{J}_{z}}$ to honeycomb materials with strong spin orbit coupling and superexchange through 90°-bond paths. We will return to this in detail in section 7.3.1. This has led to considerable interest in computing physical properties from Kitaev’s solution. Some properties are peculiar: for example, the dynamic spin-structure factor exhibits a gap, despite the presence of gapless Majorana fermions [88]. This feature turns out to be particular to the soluble model, and the false gap is filled in if Kitaev’s model is perturbed by symmetry-allowed interactions [89]. Signatures of Majorana fermions in Raman scattering may be more robust [90].

We would also like to point out the existence of a great many generalizations of Kitaev’s soluble honeycomb model to other lattices with a similar local structure [91–94]. This includes a variety of 3D models, which feature ${{\mathbb{Z}}_{2}}$ QSLs with an intriguing zoology of fermionic band structures [95]. Some 3D ‘hyperhoneycomb’ structured lithium iridates actually exist and have been studied experimentally [96, 97].

A wide set of highly entangled phases have been constructed using a method which might be considered ‘reverse engineering’. The idea is to find a Hamiltonian H for which a desired state minimizes the energy. The strategy is to construct H as a sum of terms, each of which is a positive coefficient multiplying a positive semi-definite local operator, i.e. whose eigenvalues include zero and are otherwise positive:

$$\begin{eqnarray}&&H=\underset{a}{\sum}\,{{c}_{a}}{{P}_{a}},\end{eqnarray} \tag{ 75 }$$

where P_a are the operators with semi-definite eigenvalues, and c_a  >  0. Typically these operators are chosen as projectors, i.e. to have all eigenvalues zero or one. In this situation, the energy is bounded below by zero, and so any state $|\psi \rangle$ which is annihilated by H, which requires ${{P}_{a}}|\psi \rangle =0$, is a ground state. When this is accomplished, the Hamiltonian may be said to be ‘frustration-free’ [98–100]. The uniqueness of these states, and their stability, must be established separately, and on an individual basis for each such model.

We see that the toric code model belongs to this class. Some other iconic models can be written in the same way—for example, the Heisenberg ferromagnet has this form, if the P_a are taken to be projectors onto the spin-0 sector on each bond. A wide class of models of this type are the ‘string-net’ ones of Levin and Wen, which realize a large set of highly-entangled phases. In general, however, they are even more contrived than the toric code Hamiltonian.

One popular family of models in this class are quantum dimer Hamiltonians [101] tuned to so-called ‘Rokhsar–Kivelson points’ [6]. In these models, the Hilbert space consists not of physical spins but of nearest-neighbor dimer coverings of some lattice. Resorting to quantum-dimer models is motivated by Anderson’s RVB picture of QSLs, in which the states are described as superpositions of singlet S  =  0 pairings of individual spins, e.g. $|\psi \rangle =\frac{1}{\sqrt{2}}\left(|\uparrow \downarrow \rangle \,-\,|\downarrow \uparrow \rangle \right)$ on each bond for S  =  1/2 spins. Correspondingly, in the quantum dimer model, the spins are replaced by a Hilbert space spanned by a basis of states consisting of all dimer coverings: colorings of bonds, representing the singlets, in which every lattice site is covered once and only once. The Hamiltonian contains terms which are diagonal and off-diagonal in this basis. For example, a well studied dimer model on the triangular lattice is

Here, the dark/thick bonds represent links which are ‘occupied’ by dimers, i.e. have dimer number  =  1, while pale/thin bonds are unoccupied links with dimer number  =  0, and Hilbert space is defined as a direct product over bonds which have either 0 or 1 dimer. We included the two standard couplings: J is an ‘off-diagonal’ term which flips the dimer configuration on an elementary plaquette, while V is a ‘diagonal’ term which penalizes plaquettes with ‘flippable’ dimers.

In general, these two terms compete. Large positive V favors states with no flippable plaquettes. Such states are annihilated by J, which on its own can achieve, on a single plaquette, an energy of  −J, by forming a superposition of flippable states. In between, a compromise can occur, as a superposition of states with both flippable and unflippable plaquettes. Such a maximal superposition state has the best chance of realizing a QSL.

On the triangular lattice, one indeed finds a QSL ground state for a region where $J\lesssim V$ [15]. This can be understood from a method introduced by Rokhsar and Kivelson. Then, we note that for Hamiltonians of this type, when J  =  V, it becomes a sum of projectors:

where

The projector Hamiltonian is known as a ‘Rohksar–Kivelson’ (RK) point in the dimer literature. Similar RK points occur in deformations of the XXZ model with Ising frustration. These models will be discussed in section 6.2. Following the general discussion above, any state which is annihilated by the projector is an exact ground state of H_RK. This includes both unflippable states and massive superpositions which are actually Z₂ QSL states in the toric code universality class.

To construct a highly entangled ground state $| \Psi \rangle$, we can do the following. We simply add all configurations of dimers, in the occupation basis, and give them exactly equal and positive weight in the superposition. Consider the action of the operator for a single plaquette on this state. Singling out this plaquette, we can write the state as

where $|{{ \Psi }_{u}}\rangle$ is a state in which the given plaquette is not flippable, and $| \Phi \rangle$ is some state defined on the space of bonds outside the given plaquette. The form of the term in parenthesis follows from the choice of equal weights which defined $| \Psi \rangle$. Now, if we act on this wavefunction with the first term in equation (79) is annihilated because that plaquette is unflippable. The second term is annihilated because the state in parenthesis is orthogonal to in equation (93). The same argument applies for all plaquettes, so the full Hamiltonian annihilates $| \Psi \rangle$.

In general, the superposition in $| \Psi \rangle$ contains a few too many states: for example, it includes non-flippable states which are zero-energy eigenstates on their own. We can correct this by a simple procedure. We start with some base state containing a maximal number of flippable plaquettes, for example with aligned columns of occupied bonds. Then, we construct a ‘tree’ of additional states by successively and repeatedly acting with the flip term on each plaquette, and keeping all the states (in the occupation basis) which we accumulate in this way. Once we have (for a finite lattice) obtained all the states we can find in this way, we just superimpose them with equal weight. This is, again, an exact zero-energy eigenstate at the RK point.

Does the answer depend upon the starting state? Yes, but in an interesting way. Consider, In particular, the case with periodic boundary conditions, i.e. a torus, with an even number of links in both directions. Now, draw a straight line cutting through a set of bonds that encircles the torus in one of the directions. We can count the number of occupied dimers crossing this line. The parity of this dimer number is the same for any two states connected by the flip term. Hence, we obtain two different (orthogonal) states if we superimpose configurations in which this is even or odd. We can do the same for a line in the other direction around the torus, so the total number of ground states is $2\times 2=4$. This is precisely the expected fourfold degeneracy of the toric code on a torus. One can verify that the other universal ground state properties associated with Z₂ topological order hold for these wavefunctions: dimer correlations are exponentially decaying; no local observables can distinguish the four degenerate states, and they have the proper topological entanglement entropy [102].

At the RK point, these toric code states are degenerate with other, less flippable, states. However, if we perturb away from the RK point to $V=J-\epsilon <J$, then these less flippable states are disfavored, and the toric code states emerge as the true unique ground states. In this way, one can argue that equation (76) supports a Z₂ QSL state. If we perturb in the opposite direction, to $V=J+\epsilon >J$, then the unflippable states dominate, and we lose the superposition entirely. Thus, we see that the soluble RK point lies at a phase transition between the topological phase and a trivial one. This is not unique to the triangular lattice quantum dimer model: many projector models lie at such phase transitions.

While the above construction of ground states works for simple quantum-dimer models at RK points very generally, this does not always yield a Z₂ QSL. For bipartite lattices, a much larger degeneracy of RK wavefunctions on the torus exists. It turns out that in this situation, the dimer model itself, even away from the RK point, can be mapped to a U(1) compact gauge theory without matter fields. The idea is to define an electric-field variable E_ab on link ab which is equal to the dimer number if a is on sublattice A and b is on sublattice B, and equal to minus the dimer number if the sublattices are reversed. Since on a bipartite lattice, all links connect one site of each sublattice, this completely transcribes the dimer states into electric-field states. The dimer constraint that one dimer overlaps each site then implies that $\text{divE}$ is fixed to  +1 (−1) on the A (B) sublattice. This is exactly the Gauss’ law constraint of a U(1) gauge theory, which is compact since ${{E}_{ab}}=0,\pm 1$ is an integer. The full quantum-dimer model in this case is a compact U(1) gauge theory with some fixed background charges. From the general arguments in section 3.5, we see that this model does not support a deconfined phase in two dimensions: the Coulomb phase is unstable. In fact, the RK point for such lattices is finely tuned to a critical point where something like the Coulomb phase exists, but it becomes unstable following the general arguments as soon as it is perturbed. This was first understood rather generally in unpublished work by Henley, and is reviewed in [103]. In three dimensions, since the compact U(1) gauge theory can support a Coulomb phase, one can argue that the perturbed quantum-dimer model on a bipartite lattice also contains a U(1) QSL phase.

Despite the strong RVB motivation, it is hard to quantitatively connect quantum-dimer models to physical systems. Indeed, the projection of wavefunctions to the nearest-neighbor sector is ad hoc. Moreover, the physical nearest-neighbor singlet product basis states are not orthogonal, and orthogonalizing them is quite non-trivial [6, 104, 105]. Finally, as alluded to above, dimer models, where the Hilbert space is restricted to dimer coverings, are incomplete: they do not, for example, describe excitations with non-zero spin.

Certain models of this type are known to host QSL states. One example is the case of the kagomé lattice with equal first-, second-, and third-neighbor Ising terms: more precisely V_ij  =  V when i and j are on the same hexagon, while V_ij  =  0 otherwise. This model was first studied by Balents, Fisher, and Girvin in [106], for the case in which all the XY exchanges on the analogous bonds are taken as equal (to J). It has been dubbed the BFG model, though the latter condition has been relaxed in subsequent studies. When $V\gg J$, the low-energy sector consists of all classical Ising states with a total spin $S_{\text{hex}}^{z}=0$ on each hexagon. This is a highly degenerate set of states, with an extensive entropy, but which is highly constrained. Indeed, upon including the XY exchange perturbatively using degenerate perturbation theory, the resulting effective Hamiltonian within that degenerate manifold has the structure of a Z₂ gauge theory [106]. Through analytical arguments and numerical simulations [106–109], the ground state in this limit has, indeed, been shown to be in the deconfined phase, i.e. it realizes a Z₂ QSL, with the characteristic excitations like those of the toric code. Note that, in the strong Ising limit, $V\gg J$, the sign of J is actually not important, and a QSL also occurs for antiferromagnetic J. However, the latter case is much less accessible to numerical approaches.

Similar methods to those used for the BFG model above can be applied to the 3D pyrochlore lattice. Here, nearest-neighbor exchange is sufficient to obtain a QSL in the large V limit. In this case, the low-energy states are those with $S_{\text{tet}}^{z}=0$ on each tetrahedron of the pyrochlore lattice. Because these tetrahedra themselves form a bipartite diamond lattice structure (if one connects the centers of tetrahedra), the effective model constrained to the low-energy subspace ends up having the structure of a U(1) gauge theory rather than a Z₂ one [43]. The ground state of this effective Hamiltonian is believed to be in the deconfined Coulomb phase.

This model is of much more practical interest than the BFG one, in that it is (in its local axes version) potentially realizable in actual rare-earth magnets, known as quantum-spin-ice materials—see section 7.3.2. In this context, it is conventional to write the Hamiltonian as

$$\begin{eqnarray}&&H={{J}_{zz}}\underset{\langle ij\rangle}{\sum}\,{\mathsf {S}}_{i}^{z}{\mathsf {S}}_{j}^{z}-{{J}_{\pm }}\underset{\langle ij\rangle}{\sum}\,\left({\mathsf {S}}_{i}^{+}{\mathsf {S}}_{j}^{-}+{\mathsf {S}}_{i}^{-}{\mathsf {S}}_{j}^{+}\right),\end{eqnarray} \tag{ 81 }$$

where i,j are nearest-neighbor sites on the pyrochlore lattice. This is the model equation (80) at ${{J}_{zz}}={{V}_{\langle ij\rangle}}$ and ${{J}_{\pm }}={{J}_{\langle ij\rangle}}/2$, and all other further-neighbor couplings are set to zero. This model was shown to host, for J_zz  >  0, $|{{J}_{\pm }}|/{{J}_{zz}}\ll 1$ and for either sign of ${{J}_{\pm }}$, a U(1) quantum spin liquid best described by a form of compact quantum electrodynamics on the lattice. The first term, in J_zz, is the Hamiltonian of nearest-neighbor classical spin ice. Because, up to a constant, the sum over nearest-neighbor bonds can be split into a sum over tetrahedra t of $\frac{1}{2}{{\left({\sum}_{i\in t}{\mathsf {S}}_{i}^{z}\right)}^{2}}$, the ground-state manifold of the J_zz term alone is extensively degenerate—it is the famous two-in/two-out manifold where two spins point ‘in’ and two spins point ‘out’ in each tetrahedron. Degenerate perturbation theory in ${{J}_{\pm }}/{{J}_{zz}}$ on this manifold has, as its first non-trivial term, the ‘ring exchange’ Hamiltonian

where $K=\frac{12J_{\pm }^{3}}{J_{zz}^{2}}$. Next, we use the fact that each pyrochlore site is shared by two tetrahedra with centers $\mathbf{r}$ and ${{\mathbf{r}}^{\prime}}^{{}}$, which reside on a diamond lattice. Moreover, the sites $\mathbf{r}$ and ${{\mathbf{r}}^{\prime}}^{{}}$ reside, one each, on the two bipartite sublattices of this diamond lattice. This allows one to define

$$\begin{eqnarray}&&{\mathsf {S}}_{i}^{z}={{\epsilon}_{\mathbf{r}}}{{E}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}},\quad \quad {\mathsf {S}}_{i}^{+}={{\text{e}}^{\text{i}{{\epsilon}_{\mathbf{r}}}{{A}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}}}},\end{eqnarray} \tag{ 83 }$$

where we define ${{\epsilon}_{\mathbf{r}}}=+1$ on one diamond sublattice and  −1 on the other. Here, E and A are canonically conjugate (see equation (12)) integer-spaced and $2\pi$-periodic variables, respectively, as introduced in section 3.1 on U(1) gauge theory. The staggering factor is useful because only the pair $\mathbf{r}$ and ${{\mathbf{r}}^{\prime}}$, and not which diamond site is which, is determined for a fixed i. In particular, equation (83) is well-defined with this inclusion if the E and A variables are vector-like, such that ${{E}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}}=-{{E}_{{{\mathbf{r}}^{\prime}}\mathbf{r}}}$ and ${{A}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}}=-{{A}_{{{\mathbf{r}}^{\prime}}\mathbf{r}}}$. Now, enforcing the constraint $S_{i}^{z}=\pm 1/2$ with a term $\frac{U}{2}\left(E_{\mathbf{r}{{\mathbf{r}}^{\prime}}}^{2}-1/4\right)$, U  >  0 and large, one obtains

i.e. the Hamiltonian of compact quantum electrodynamics, and exactly of the form discussed in section 3.1. (Here, , i.e. the lattice curl of A, as defined in equation (15) for .) Not only is the Hamiltonian now of the form of a compact U(1) gauge theory, but also we obtain, as in section 3.1, Gauss’ law, $\text{div}E=0$.

The rewriting in terms of rotors (A and E are mathematically analogous to the angle θ and angular momentum ${{L}_{z}}=-\text{i}{{\partial}_{\theta}}$, respectively, describing rigid body motion of a planar rotator, or rotor) is helpful for intuition and some formal manipulations, but we emphasize that this is entirely equivalent to the ring model in equation (82), which already has the compact U(1) gauge invariance. The model, in either form, is non-trivial, however, since there is no small parameter (e.g. in equation (84) we must take $U\to \infty$ to accurately represent the original spin system—this limit is non-trivial because E is half-integer). However, Monte Carlo calculations indicate that the small U Coulomb phase of the rotor Hamiltonian of equation (84) actually extends all the way to the infinite U limit [110, 111]. Therefore, we conclude that, at small transverse coupling, the XXZ model on the pyrochlore lattice realizes a 3D U(1) QSL.

The above treatment is based on perturbation theory in ${{J}_{\pm }}$. In section 7.3.2, we will discuss how to extend the rewriting equation (83) to include matter fields and therefore obtain a more complete gauge theory valid for all values of the exchange.

Many other models of ‘Ising frustration’ of this type have been studied. Moessner and Sondhi particularly focused on models in which quantum fluctuations are induced by a transverse field rather than XY exchange. This violates the LSM conditions and thus trivial states are possible and highly entangled states are less likely. Indeed, to our knowledge, no non-trivially entangled states have been found in such models. XXZ models on various other lattices and with other forms of V_ij have also been studied—for example with nearest-neighbor terms on the triangular and kagomé lattices. Apart from the cases discussed in the previous sections, these models, though degenerate when J_ij  =  0, order strongly at T  =  0 when an infinitesimal J_ij is introduced, realizing conventional short-range entangled gapped states. It should be noted that transverse field Ising models in 2  +  1 dimensions are generally dual, by Kramers–Wannier duality, to Ising gauge theories [14]—a formal link that has often proven useful. For example, the fully frustrated Ising model in a transverse field on the honeycomb lattice is dual to an Ising lattice gauge theory on the triangular lattice, and thereby related to the triangular lattice quantum-dimer model. It is important to emphasize, however, that because the duality transformation is non-local, what is local in one representation (e.g. the Ising model) becomes non-local in another (e.g. the dimer model/gauge theory). Hence, this duality does not imply any topological order, entanglement, or non-local excitations in the transverse field Ising model.

Instead, sharp predictions of QSL phases in SU(2) symmetric Heisenberg models come from numerics. Typically if the exchange couplings are not frustrated, an ordered phase emerges, so the focus is on highly frustrated situations. In these cases, direct QMC is prohibited by a strong sign problem. Instead, one turns to less scalable methods. It is not our purpose here to describe different computational methods in any depth, but only to discuss some of the most used techniques, and their advantages and disadvantages.

The most unbiased method is exact diagonalization, for which the only bias is the choice of finite cluster. However, only very small systems can be solved in this way (the current record is 48 spins [124]), so it may not be clear how to extrapolate to the thermodynamic limit. Still largely unbiased is the DMRG, which can obtain numerically exact ground-state properties (as well as properties of some low-energy states) in many cases for systems with a strip geometry, for which the DMRG is best suited by its intrinsically 1D formulation. In many cases the length of the strip can be made very large, but expanding the width is prohibitive, and limited to (at best) an order of 10–12 sites. This is still a significant enlargement compared to exact diagonalization in terms of system size. DMRG does have a slight bias toward low-entanglement states, but since the difference between QSLs and competing states typically comes from a subdominant correction to the area-law term, this is probably not a severe limitation, and it is one that can be diagnosed and overcome.

In recent years, DMRG has come to be regarded as a member of a larger family of techniques which rely upon the entanglement structure of ground states. Quantum information theory has emphasized that the area law obeyed by nearly all ground states puts strong constraints upon the wavefunctions of these states, which can be captured well by tensor network forms, as discussed in section 2.5. In fact, the DMRG can be viewed as an algorithm to find the optimal MPS for a given Hamiltonian, which has led to various improvements in the DMRG taking explicit advantage of the MPS representation. This allows DMRG to directly address various entanglement-related properties. For example, calculation of the entanglement entropy and, indeed, the full spectrum of the reduced density matrix is straightforward by DMRG (though extracting the universal terms in entanglement entropy may be challenging due to finite size effects). Furthermore, the MPS form allows a conceptual extension of a wavefunction to infinite size, by defining the wavefunction for arbitrary size through a translationally invariant tensor. This allows one to much more directly address topological properties and, for example, explicitly construct all the degenerate states on the torus, and to calculate the full exchange statistics of anyons in a topological phase [125]. Other numerical approaches based on higher dimensional tensors, and which may thus be more intrinsically suited to two or three dimensions than DMRG, are actively being developed. However, results so far are limited, and are not yet clearly competitive with more established methods.

There are a variety of techniques which incorporate a higher degree of bias, in exchange for various benefits. We have already discussed the use of variational wavefunctions based on Gutzwiller-projected fermionic partons. Calculations using these wavefunctions can be carried out by a Monte Carlo procedure quite efficiently. The results can also be improved by introducing additional variational parameters, for example, by multiplying the Gutzwiller-projected wavefunction by additional factors which do not introduce additional signs into the calculation. Various improved techniques have been explored by several groups, and certainly yield, as expected from the variational principle, lowered estimates of the ground-state energy, which compare well with other approaches. It is somewhat less clear how accurate these approaches are for correlations and other physical properties, as compared to the true ground state. In general there is potential here for variational methods to mislead, since the variational principle, strictly speaking, implies that the variational state is close to the ground state when its total energy, which scales with the system volume, is close to the exact ground-state energy. This means that a small error in energy density can lead to a qualitatively wrong wavefunction if the system size is large [126]. Whether this is actually a problem in practice is unclear. We will mention results from Gutzwiller-projected variational wavefunctions for some specific models below.

Other techniques are designed to describe particular phases. These methods include high-temperature series expansions and series expansions around particular ordered phases, and coupled cluster methods. These techniques require a guess or knowledge of the phase the system is in, but with that assumption can provide accurate calculations of many quantities such as ground-state and excitation energies, order parameters, etc. A high-temperature series is often a useful technique for quantitatively understanding thermodynamics of a model, which can be compared directly with experiment, as a means of refining the model description. Series expansions generally do not describe QSL states of Heisenberg-type models (though they have been successfully applied to special models such as the perturbed toric code).

There are further methods which share some similarities with high-temperature expansions, but are based on Abrikosov fermions and a perturbative or diagrammatic treatment of effective spin interactions. One such technique is the functional renormalization group, which has been applied to kagomé and other systems [127, 128]. A relatively new technique is diagrammatic quantum Monte Carlo [129, 130] which, like a high-temperature series, is limited to high-to-intermediate temperatures, but gives access to details of real-time correlation functions—a recent application reveals a spin-ice regime for the nearest-neighbor Heisenberg antiferromagnet on the pyrochlore.

Having outlined the methods of analysis, we now summarize the state of understanding of QSLs in several iconic frustrated Heisenberg models.

The spin-1/2 Heisenberg model on the kagomé lattice has been one of the most intensively studied problems in frustrated quantum magnetism. Historically, most of the focus, naturally, has been on the minimal model with a single antiferromagnetic exchange J₁ on nearest-neighbor bonds, while recently the effects of second- and third-neighbor exchange, J₂ and J₃, have been studied.

We first focus on the nearest-neighbor model, which has a tumultuous past and remains controversial today. Classically, it has an extremely large degeneracy of continuously connected ground states, within which local rotations of small numbers of spins are possible [131]. This would be expected to encourage large quantum fluctuations and invalidate spin-wave theory for small spin S. Very early on, this model was suggested to have a non-magnetic ground state [132], which might be a QSL [123] or a VBS. Subsequently, exact diagonalization studies on up to 27 spins supported a gapless QSL ground state [133], and countless theorists studied the problem in innumerable ways. In 2007, a Gutzwiller-projected fermion variational calculation proposed a gapless U(1) Dirac QSL state [65], while series expansions argued strongly for a VBS state [134]. The VBS prediction stood as the leading numerical candidate for several years, until a pioneering DMRG study conclusively demonstrated that VBS order vanishes, and strongly suggested a gapped QSL state [135]. Soon other DMRG simulations supported the gapped QSL, both by improved estimates of gaps [136] and calculations of the topological contribution to the entanglement entropy on cylinders, which was found to be equal to $\ln 2$ within small numerical errors [24] (note that in the latter paper, a small ${{J}_{2}}=0.1{{J}_{1}}$ was included, which further stabilized the QSL state and improved numerical convergence). This is the expected universal value for a toric code phase, and together with the other DMRG results point strongly toward a gapped Z₂ QSL ground state. Gapped Z₂ QSLs on the kagomé lattice can be described by partons. Though variational wavefunctions based on Gutzwiller-projected fermion wavefunctions [137] seem not to yield good energies for Z₂ states [138], the energy of a Gutzwiller-projected bosonic wavefunction was competitive [139]. The difficulty of working with such wavefunctions limited the size of the latter calculations, however. Taken together with the DMRG, these works point toward a specific ‘symmetry-enriched’ Z₂ QSL state, i.e. with a specific PSG for all the toric code anyons, for the kagomé lattice. Rather remarkably, the same QSL phase can be described in both the bosonic and fermionic formulations. It is known as the Q1  =  Q2 Schwinger boson state [123] or the ${{Z}_{2}}[0,\pi]\beta$ fermionic mean-field state [137]. While the matching of the PSGs for both parton constructions is strong evidence for their equivalence [140–142], there is even stronger evidence: the states have been explicitly matched in a certain limit, i.e. their wavefunctions become identical [143].

Though a fairly complete picture of a putative Z₂ QSL is emerging, improvements on the Dirac QSL wavefunction also provide competitive variational energies, so that a gapless U(1) QSL state continues to be advocated [138, 144, 145]. It should be emphasized that the intrinsic stability of the U(1) Dirac QSL state has not been established, i.e. we do not know whether this state actually exists as a phase in any 2D system without fine tuning of parameters. Nevertheless, the variational results raise valid concerns. Hence, the presence of a gapped Z₂ QSL in the nearest-neighbor kagomé system and even ${{J}_{1}}-{{J}_{2}}$ model remains an open issue.

Recent simulations including third-neighbor exchange (here, J₃ is defined as the exchange between sites on opposite corners of a hexagon of the lattice) have uncovered a second and less controversial QSL on the kagomé lattice. When $0.3<{{J}_{2}}\approx {{J}_{3}}<0.7$, DMRG finds a chiral QSL state [146–148]. This realizes an old idea of Kalmeyer and Laughlin [4], who suggested a QSL state could be described by the standard mapping of spin-1/2 spins to hard-core bosons (i.e. S^z  =  −1/2 is an empty site, and S^z  =  +1/2 is a site occupied by a boson), and constructing a bosonic FQH state at filling factor $\nu =1/2$. This is a chiral topological phase, of the type described in section 2.3.1. It not only has anyonic excitations and topological order, but (as discussed in section 5.5) breaks time-reversal symmetry. Consequently, it displays a non-zero order parameter: the scalar spin chirality, $\langle {{\vec{S}}_{i}}\centerdot {{\vec{S}}_{j}}\times {{\vec{S}}_{k}}\rangle \ne 0$, for certain sets of three nearby spins. In this case, the DMRG has fully verified the expected anyons in this state, and their statistics [147, 148], and there is no controversy with respect to variational wavefunctions: these calculations find the same state [149]. In parallel work, it has been unambiguously demonstrated that energetically favoring certain patterns of the spin chirality stabilizes a chiral QSL [150].

The frustrated J₁-J₂ Heisenberg model on the square lattice is another long-studied model, which was considered for possible relevance to the cuprates, and more recently for the iron pnictides and their relatives. Classically, it undergoes a transition from the usual Néel state with ordering wavevector $(\pi,\pi )$ to a striped antiferromagnet with ordering at $(\pi,0)$ or $(0,\pi )$, with increasing ${{J}_{2}}/{{J}_{1}}$ (both couplings are antiferromagnetic). It is believed that near the classical degeneracy point ${{J}_{2}}={{J}_{1}}/2$, a non-magnetic ground state occurs [151–158].

The nature of this ground state is controversial, with the most popular contender being a VBS with either a structure of columns of singlet bonds or an array of square-singlet plaquettes. However, a QSL state has also been proposed in this region. The most recent works seem to favor a plaquette VBS in the region $0.5<{{J}_{2}}/{{J}_{1}}<0.6$, and leave open the possibility of an intermediate QSL phase for $0.4\gtrsim {{J}_{2}}/{{J}_{1}}<0.5$. Alternatively, there may be a direct transition from the Néel to VBS phase at some point in this interval, which might realize the interesting proposal of deconfined quantum criticality [71] (see also section 4.4.3).

Anderson first proposed the RVB QSL state for the nearest-neighbor spin-1/2 Heisenberg model on the triangular lattice. This proved to be incorrect: this model has long-range magnetic order [159–161], consisting of spins oriented at 120 degrees to one another in a coplanar three-sublattice pattern. This led the search for QSLs to move away from the triangular-lattice Heisenberg model (though other types of models on the triangular lattice were heavily pursued).

Nevertheless, recent DMRG studies have revisited the problem, including second-neighbor exchange J₂. These studies found a gapped spin-liquid region lying at $0.08\leqslant {{J}_{2}}/{{J}_{1}}\leqslant 0.16$ [162, 163]. Evidence was not provided through the entanglement entropy signature, but via the more conventional detection of exponentially decaying spin and dimer correlations, which give evidence for the absence of such long-range order. A gap to the singlet and triplet spin excitations, as well as a twofold ground-state degeneracy for a cylinder geometry may point to a ${{\mathbb{Z}}_{2}}$ quantum-spin liquid. Some indication of spin chirality was also observed [163], however, so there may be some competition between a chiral spin liquid and a Z₂ state. Reminiscent of the situation in the kagomé Heisenberg model (section 6.4.2), variational wavefunction calculations support a gapless QSL state instead of either of these gapped states [164].

The pyrochlore lattice, which is formed of corner-sharing tetrahedra, is the 3D archetype of a frustrated lattice. The nearest-neighbor model is clearly extremely frustrated, and like its 2D relatives, for S  =  1/2 Heisenberg spins lacks any small parameter with which to attack it analytically. Being 3D, it does not lend itself to numerical simulations well, so computational results are extremely minimal. Exact diagonalization on very small clusters of up to 12 spins was performed years ago [165] pointing to a quantum-spin-liquid phase as evidenced by exponentially decaying spin-spin correlation functions.

Even biased calculations are limited. We found only two published works proposing fermionic parton wavefunctions for the pyrochlore lattice [166, 167], but these works assumed a U(1) structure and did not carry out a full analysis of even all possible symmetric QSL states, much less symmetry-breaking states. Several papers study the pyrochlore by breaking inversion symmetry and introducing two distinct exchange interactions J and ${{J}^{\prime}}$, within each of the two types of tetrahedra, treating J exactly, but ${{J}^{\prime}}$ perturbatively or approximately [168, 169]. This approach results in a VBS state. It is unclear whether this approach implies anything for true centrosymmetric pyrochlores, but so-called ‘breathing pyrochlores’ do exist with symmetry appropriate for the $J-{{J}^{\prime}}$ model. The XXZ model, discussed in section 6.2.2, is tractable in the Ising limit, but it seems unlikely that the QSL found there extends to the isotropic point.

Given the ubiquity of simple ordered states, it is natural to approach the problem of finding QSLs in terms of destabilizing such states. Clearly, frustration is, therefore, an essential feature. In two dimensions, nearest-neighbor interactions on the kagomé (corner-sharing triangles) and triangular (‘face’-sharing triangles) lattices in three dimensions, the pyrochlore (corner-sharing tetrahedra) and FCC (edge-sharing tetrahedra) lattices are common examples of geometrically frustrated lattices. But, frustration may also be introduced by further neighbor couplings (e.g. while the diamond lattice is bipartite, second neighbors form two FCC lattices), or by competing interactions, for example, when interactions are anisotropic, or are higher-order in spins, or by introducing, e.g. a magnetic field.

There are many reviews of the physics of frustration [21, 22, 188]. Mostly, these take a classical or semi-classical perspective, which has limited impact on the QSL problem. For our purposes, frustration is merely a guide to help select systems which might be reasonable candidates for QSLs.

Another feature commonly advocated to be important for the realization of QSLs is a low spin. This is based on the familiar notion that in the large S limit, spin-S spins become classical which, in turn, is based on the fact that the commutator of two spin operators, if normalized by their ‘length’ S, is of the order 1/S. It is indeed true for Heisenberg models and, more generally, models with bilinear exchange interactions. However, this conclusion is not general, and must be considered a bit more carefully in practice. A better way to think about the classicality of a spin system is in terms of localization in Hilbert space. A classical state can be considered as the fully polarized ($S_{i}^{z}=S$) state with a spin quantization axis chosen at each site to coincide with the classical orientation of the spin vector. The single-spin operator $\vec{S}$ becomes classical for large S in the sense that it generates transitions only between states with S^z differing by $\pm 1$, and for a Hamiltonian which involves only individual $S_{i}^{\mu}$ operators (and not products of multiple operators at a single site), the low-energy states become localized near the fully polarized one, because it requires a high-order action of the Hamiltonian to connect to states far from this one. However, this conclusion rests on the bilinear exchange form of the Hamiltonian.

Indeed, larger spin systems can be highly quantum under certain circumstances. A simple way this occurs is if there are strong single-ion terms that split the spin degeneracy. This can lead to a doublet ground state of the single site problem, which then acts like an effective spin-1/2, if exchange is small compared to the splitting to the next multiplet. This is, in fact, typical for f electron spins in the lanthanide sequence. A second, less common, way that larger spins can be quantum is through multipolar spin interactions. This means couplings involving a product of many $S_{i}^{\mu}$ operators on a single site. Such operators do not necessarily become classical for large S. Indeed, if we allow all possible operators acting on a spin-S spin, it becomes simply a N  =  2S  +  1-level system, and we could choose a Hamiltonian which has SU(N) symmetry (see section 4.1) that is more quantum than any SU(2) spin model. Multipolar interactions escape classicality precisely by connecting all the N states directly, thereby avoiding localization near any particular spin projection. The microscopic physics of multipolar interactions is complex, but may arise out of orbitally dependent super-exchange, and spin–orbit coupling [178, 189, 190].

A different strategy to find liquid ground states of spins is to look at systems close to the more common sort of quantum liquid: the metallic state. It is not unreasonable to think that a QSL is a natural intermediate state between a metal and an ordered antiferromagnet. This has already been discussed theoretically in section 6.6. Certainly proximity to a Mott metal-insulator transition leads to new sources of quantum fluctuations, e.g. virtual excitations across the Mott gap. The proximity to the Mott transition is an often-cited justification for the spin-liquid-like behavior observed in the organics (see section 7.3.3).

Neutron Scattering is often considered the method of choice to study magnetic properties. It gives access to the spin-spin correlation function $\mathcal{S}\left(\omega,\mathbf{k}\right)\sim \langle S_{-\mathbf{k},-\omega}^{\mu}S_{\mathbf{k},\omega}^{\nu}\rangle$ of the system, where ω and $\mathbf{k}$ are the energy and momentum transfers, respectively. First, consider the elastic ($\omega =0$) signal. Long-range magnetic order manifests as elastic scattering at magnetic Bragg peaks, whose location in momentum space gives information on the type of order. In principle, any magnetic intensity at $\omega =0$ at T  =  0 implies a non-zero ground-state expectation value of the spin operator, since using the spectral representation $\mathcal{S}\left(0,\mathbf{k}\right)\propto \langle 0|S_{-\mathbf{k}}^{\mu}|0\rangle \langle 0|S_{\mathbf{k}}^{\nu}|0\rangle$. This takes the form of a resolution-limited Bragg peak if the ground-state moments form an ordered pattern, but may be a smooth function of momentum if the static moments are random. A ground state with no static moments whatsoever should really have zero elastic intensity. In practice, one cannot measure the zero of energy with absolute precision, so that one observes ‘quasi-’elastic weight, i.e. the integral of $\mathcal{S}\left(\omega,\mathbf{k}\right)$ over some narrow frequency range. This can be non-zero without any static moments, if there are low enough energy excitations with non-zero matrix elements to the ground state. This might arise in a QSL due to multi-spinon continua, collective modes like the photon of the three-dimension U(1) Coulomb phase, etc. For these types of excitations, for most QSL phases, the quasi-elastic weight would be expected to be localized in only some regions of the Brillouin zone, and show some characteristic sharp momentum dependence. Quasi-elastic weight may also be due to quasi-static moments created by impurities, which is likely the origin if this weight depends very smoothly on momentum.

The inelastic signal, for ω on the scale of the exchange J, reveals excitations of the system. The spectral representation implies that $\mathcal{S}\left(\omega,\mathbf{k}\right)$ receives contributions only from excitations with energy ω and net momentum $\mathbf{k}$, for which a single spin operator has a non-zero matrix element with the ground state (at T  =  0). The notable feature of QSLs is that their quasiparticle-like excitations are non-local, which means that individual quasiparticles do not have such non-zero matrix elements. Rather, single spin operators acting on the ground state can create the non-local quasiparticles only in pairs. This leads to an intrinsically smooth signal, where at each momentum transfer $\mathbf{k}$, the intensity is spread out over a continuum of frequencies ω, only constrained by the condition that ${{\mathbf{k}}_{1}}+{{\mathbf{k}}_{2}}=\,\mathbf{k}$. Under ideal circumstances, then, some QSLs should exhibit an entirely continuum spectrum. This is in sharp contrast to short-range entangled states such as conventional ordered magnets, or VBS states, in which the elementary excitations are essentially spin-flips, and a single-spin operator creates an individual quasiparticle. In the latter case, this quasiparticle has a definite dispersion ${{\epsilon}_{qp}}\left(\mathbf{k}\right)$, and contributes a resolution-limited peak to the inelastic signal.

While an entirely continuum spectrum is possible for a QSL, it is not the only possibility. Even if the elementary quasiparticles are non-local, a pair of them may form a bound state, which may be a local object and can contribute a sharp peak to $\mathcal{S}\left(\omega,\mathbf{k}\right)$. The presence or absence of such a bound state is a non-universal feature which can depend upon the position within a single QSL phase. Some QSLs, in addition to non-local quasiparticles, have intrinsic emergent but local excitations. This is the case for the U(1) QSL in a 3D Coulomb phase (section 3.6), which will be discussed in the quantum-spin-ice context in section 7.3.2. This is an attractive feature of 3D U(1) QSLs, since it provides an unambiguous smoking-gun measurement than can, in principle, be made using a well-established method.

While standard x-ray scattering is mainly used to measure atomic structure, resonant x-ray scattering is being increasingly applied as a probe of magnetism, often substituting for neutron scattering’s traditional role. This is an active and rapidly changing experimental field, where capabilities have grown dramatically in the past decade. The technique typically relies on probing a resonant atomic transition, and detecting small changes relative to the large energy of this transition. Consequently, the energy resolution of resonant x-ray scattering is worse than neutrons: in current experiments, the energy width is of the order of a few tens of meV, while neutrons easily achieve 0.1 meV and a few μeV resolution is possible with specialized techniques. Conversely, the x-rays have excellent momentum resolution, often better than neutrons. X-rays require much smaller samples due to high resonant-scattering efficiency and high-intensity sources. They are also preferable in some materials, such as iridates, which are highly absorbent to neutrons.

Due to the complexity of resonant processes, x-ray scattering probes more than just the spin-structure factor $\mathcal{S}\left(\omega,\mathbf{k}\right)$. In addition, it can measure other types of excitations which are created not by single spin operators acting on the ground state, but by more complex (but unfortunately still local) operators. The literature on the theory of these measurements in resonant inelastic x-ray scattering is still developing, and we refer the reader to [191, 192].

The most fundamental thermodynamic measurement, specific heat c_v, plays several roles in the study of quantum magnetism. First, it is the most unbiased detector of phase transitions. Any phase transition at T  >  0 is accompanied by a singularity in c_v(T), which is usually detectable. Second, it provides a means to determine entropy, using integration based on the standard thermodynamic relations. One of the simplest tests of the validity of a model spin Hamiltonian is whether it correctly captures the spin entropy released from high to low temperature, which should equal $\ln (2S+1)$ per site for effective spin-S spins. A classical spin-liquid regime can be revealed by a plateau in the entropy, which is associated with an intermediate temperature region of low specific heat.

The most interesting, yet perhaps most subtle, use of specific heat is to gauge the number of low-energy excitations in a quantum magnet. In a gapped state, the low-temperature magnetic specific heat is exponentially suppressed and obeys an Arrhenius law. In a gapless state, it is a power of temperature which reveals the density of states of low-energy excitations. For an ordered d-dimensional Heisenberg antiferromagnet, for example, we expect ${{c}_{v}}\sim A{{T}^{d}}$. Various power laws are predicted for gapless QSL states, including 3D U(1) Coulomb QSLs [193], and various 2D QSL states with gapless fermionic spinons [65, 181, 186].

There can be complications. In general, magnetic specific heat is not separately measurable, and some subtraction procedure is usually performed to remove the lattice contribution. Usually, this is most reliable at low temperature, where, for a gapless QSL, the magnetic contribution may be significantly larger than the lattice one. Another difficulty is disorder, which may affect the density of states of some QSLs, changing the intrinsic behavior or inducing entirely new low-energy excitations.

Specific heat is powerful because of its generality, but this is also its Achilles heel. It is sensitive to any excitation, and not only excitations of spins. In complex materials, especially at low temperatures, many other effects can contribute, such as molecular tunneling excitations and nuclear levels. So care needs to be taken to verify that a ‘magnetic’ specific heat is, in fact, due to magnetic degrees of freedom.

Uniform DC magnetic susceptibility χ is often the first measure of a new magnetic material. Like specific heat, it serves several purposes. It typically shows a peak or cusp at a phase transition, so can be a check for magnetic or other ordering. The presence of a Curie behavior $\chi (T)\sim A/T$ at temperatures which are high compared to the exchange interactions is the simplest indication that a material contains localized spins at all! The coefficient A determines the magnetic moment of the spins. The correction to the Curie behavior is indicative of the strength of the interactions. Typically, the susceptibility is fitted to the form $\chi (T)=A/\left(T-{{ \Theta }_{\text{CW}}}\right)$, where ${{ \Theta }_{\text{CW}}}$ is the Curie–Weiss temperature. If this formula is applied when $T\gg {{T}_{\text{CW}}}$, it becomes $\chi (T)\sim A/T\left(1+{{ \Theta }_{\text{CW}}}/T+\cdots \right)$, so that ${{ \Theta }_{\text{CW}}}$ can be viewed as the first correction to the Curie law for independent moments in a high-temperature expansion. This correction can be directly calculated for any exchange model, which allows one to extract some information on the exchange interactions. For example, for the standard Heisenberg antiferromagnetic Hamiltonian of equation (85) with spin-S spins, one obtains

$$\begin{eqnarray}&&{{ \Theta }_{\text{CW}}}=-\left(\underset{j}{\sum}\,{{J}_{ij}}\right)\frac{S(S+1)}{3{{k}_{\text{B}}}},\end{eqnarray} \tag{ 91 }$$

assuming all sites i are equivalent. Note that, typically, ${{ \Theta }_{\text{CW}}}<0$ for antiferromagnets. From a single measurement of $\chi (T)$, one may determine T_c for a magnetic ordering transition and the Curie–Weiss temperature, which allows one to construct the dimensionless ‘frustration parameter’ $f=\,|{{ \Theta }_{\text{CW}}}|/{{T}_{c}}$. When f is large, it indicates that the system avoids ordering even when spins are highly correlated, which may suggest incipient QSL behavior, though it can have other explanations.

The Curie–Weiss analysis is only applicable when $T\gg |{{ \Theta }_{\text{CW}}}|$, a regime which may not be achievable. Consequently, it may be better to fit experimental data to a higher-order expansion or other approximate form of susceptibility determined somehow for a model Hamiltonian. Beyond just the scale of interactions, one can sometimes learn, from single-crystal measurements, about more details from the full tensorial structure of the susceptibility: anisotropic g-factors, anisotropic exchange, or both. A word of caution: fitting the susceptibility may be dangerous as it is usually relatively smooth and featureless, and so cannot reveal more than one or two combinations of parameters in the Hamiltonian. Nevertheless, these methods are essential to get some insight into the scale at least, and sometimes some details of the microscopics of a material.

As with specific heat, susceptibility at low temperature may also be related to the low-energy excitations of a material. This seems natural since one may obtain susceptibility from thermodynamics: $\chi =-{{\partial}^{2}}F/\partial {{H}^{2}}{{|}_{H=0}}$. In a more detailed sense, however, this depends upon the physics of the magnetic system. In general the spin susceptibility arises from coupling of the external field to the magnetic moment operator. In many models, such as Heisenberg ones, the magnetic moment is proportional to the total spin, and the total spin is conserved by Heisenberg interactions. This implies that the field couples to a conserved quantity, and we can construct simultaneous eigenstates of the field term and the zero-field Hamiltonian. In this situation, the eigenstates of the system do not evolve with the field, only their energies do. So the susceptibility just reflects a change of occupation of eigenstates, and can be expressed in terms of spin-dependent density of states. In this situation, the low-temperature susceptibility is directly related to the density of excitations: it is large when there are many gapless excitations, and small when there are few or none. For example, within a Heisenberg description a U(1) spinon Fermi-surface QSL state is expected, on these grounds, to have a susceptibility which tends to a constant at T  =  0 (similar to Pauli paramagnetism in a metal), while a gapped Z₂ QSL has an exponentially small susceptibility. These conclusions are, however, incorrect if the Hamiltonian does not have spin-rotation symmetry around the field axis. When this symmetry is absent, the magnetic susceptibility tends to a constant at T  =  0, irrespective of the presence or absence of low-energy excitations. The non-zero susceptibility in this case arises not from a repopulation of levels, but from the evolution of the ground-state wavefunction with applied field.

Nuclear magnetic resonance (NMR) uses nuclear levels of atoms in a sample as a probe of their local environment—see [194] for a review in the frustrated magnetism context. Here, we are interested mainly in how those nuclei couple to the electronic spins—the hyperfine interaction—and how the electronic spins’ behavior is thereby reflected in that of the nucleus. NMR is a rich and fairly complex subject, and we will only review the simplest and most common applications here. Generally, experiments study the response of the nucleus to externally applied AC fields which are resonant or nearly resonant with a nuclear transition. One can thereby measure the energy of this transition, as well as the rates at which nuclei relax amongst and within these levels. One typically identifies two relaxation times: T₁, which is the time needed to relax from one Zeeman level to another in a uniform DC field, and T₂, which is the time needed to dephase oscillations of the magnetization normal to the DC field.

From the point of view of electronic magnetism, one is mostly interested in the transition frequency and the longitudinal relaxation time T₁. The transition frequency is basically set by the effective Zeeman field seen by a nucleus. This is the sum of contributions from an external applied field and internal hyperfine fields. The latter is proportional to the magnetization of the electronic states which are coupled to that particular nucleus. This ‘shift’ of the nuclear resonance frequency is known as the Knight shift. In the linear response regime, the magnetization of the electronic states is just proportional to their (local) susceptibility, so the Knight shift just gives a measure of the local electronic susceptibility. This makes it a useful probe to compare with bulk uniform susceptibility. In a magnetically ordered state, different nuclear sites may become inequivalent, resulting in splitting of a nuclear transition into multiple peaks, or even a broad shape. Through a careful analysis, NMR can give a great deal of information on the presence and arrangement of static electronic moments.

The relaxation rate 1/T₁ is determined by the ability of the nucleus to emit energy and decay from an excited Zeeman state. Such decay requires the energy to be deposited elsewhere into some excitation outside the nucleus. This can be in the lattice or, more interestingly, in the electronic spins. The latter process arises through the hyperfine coupling, and according to Fermi’s golden rule analysis, probes the density of electronic spin excitations at the transition frequency of the nucleus. This NMR frequency is very low on magnetic scales, and can usually be considered to be just infinitesimally above zero. Hence, formally, the relaxation obeys $1/{{T}_{1}}\sim T\chi _{\text{local}}^{\prime \prime}\left({{0}^{+}},T\right)$, where ${{\chi}_{\text{local}}}(\omega,T)$ is the AC local susceptibility of the electronic spins. Since 1/T₁ probes the imaginary part of the susceptibility, which is dissipative and hence only non-zero if there are excitations, it is a true probe of the low-energy density of states of the spin system, even if the uniform susceptibility and Knight shift fail due to broken spin-rotation invariance.

We note that there can be subtleties in the NMR relaxation, since what is actually measured is the time evolution of the polarization after some initialization or pulse sequence. The relaxation times are deduced by exponential fits of this time dependence. This may not be a simple exponential if there are multiple environments for the nucleus, in which case more interpretation is required. A positive view on this is that NMR can be a good way to detect inhomogeneous or multiple environments for nuclei, which may point to a disordered or complex (e.g. incommensurate) magnetic state.

In muon spin resonance, or μSR, one implants positively charged muons into a sample, and uses the muon spin, which is like that of an electron, as a two-level system to perform NMR-like experiments. The disadvantage is the need for a facility to produce muons, but there are some advantages. It can be used on materials for which there are no magnetic nuclear isotopes. It can be used straightforwardly for zero-field measurements: muons are inserted with a definite spin state, and then precess in whatever local field they experience after stopping in the crystal. From measurements of the muon precession, one can determine the local field it experiences, or rather the distribution of local fields experienced by the ensemble of stopped muons. One may also measure relaxation times, perform spin-echo experiments, etc, similarly to the way one does in NMR.

An issue which sometimes arises in μSR is that, because the muon has a positive charge, it may actually perturb the sample significantly. However, there are certainly often clear outcomes. The observation of spontaneous oscillations of polarization in zero-field μSR is incontrovertible evidence of magnetic order. The interested reader is referred to reviews for further information [195, 196].

Light interacts with matter through both its electric and magnetic field, and can probe magnetic systems in several ways. The simplest effect is a one-photon absorption, which is due to the linear coupling of electromagnetic radiation to the current or charge density. Since we are concerned with Mott insulators, the usual electron-hole excitations are not present at low energies. Below the gap, one-photon effects are due to the linear interaction of the electric field with the electric polarization, and the magnetic field with the magnetic dipole operator. Naïvely, the polarization operator does not involve the magnetic degrees of freedom, but due to spin-lattice interactions and spin–orbit coupling, it may in fact couple to the spins. Various examples of this have been discussed in the literature [197, 198]. This means that magnetic excitations quite generically lead to non-zero absorption within the Mott gap [199, 200]. This fact seems to not be widely appreciated, but is true not only for QSLs, but also for ordinary ordered antiferromagnets. In most cases, due to the time-reversal odd nature of the spin operator, this polarization coupling involves terms even in spin operators, which means that even in such ordered states, the sub-gap weight is multi-magnon in origin, and hence continuum-like. However, the effect has been predicted to be significantly larger in some gapless QSLs.

The magnetic dipole operator can be proportional to a single spin and so, in principle, this can couple to the same sorts of excitations observed, e.g. in neutron scattering. Due to the very long wavelength of light, however, one-photon processes of this type couple only to zero-momentum spin excitations. The inability to scan momentum is, to an extent, made up for by excellent energy resolution. This has been quite beautifully demonstrated in CoNb₂O₆ [201] and Yb₂Ti₂O₇ [202].

Two-photon transitions, known as Raman scattering, in which light is scattered from one energy to another, are commonly studied as probes of both lattice and spin excitations. The latter is due to the quadratic coupling of the electromagnetic field to the spins, usually to ‘bond’ operators involving pairs of spins [203]. In general, the signals measured by Raman are continua and, involving as they do correlations of bond operators, somewhat complex objects. The contributions from spins must also be separated from those of phonons, etc. Due to these complexities, applications to QSLs are limited, and open to multiple interpretations [204, 205].

The thermal conductivity of a material can, like the electrical conductivity, be expressed by an Einstein relation as the product of the thermal diffusion constant and specific heat, which plays the role of the density of states. Due to the second factor, it is closely related to the heat capacity, which we discussed in section 7.2.3. However, because of the first factor, only excitations which are mobile contribute significantly to the thermal conductivity. Thus, the thermal conductivity is useful to determine if excitations which contribute to specific heat are extended or localized. This is important because an insulator may quite commonly have localized in-gap electronic states, which might masquerade in a purely thermodynamic measurement as spin excitations.

In general, all propagating excitations can contribute to thermal conductivity, and nominally add in the conductivity as parallel conduction channels. A phonon contribution is always present. Ideally, in a QSL with gapless fermionic spinons, these excitations are expected to add a power-law dependence (as a function of temperature), indicative of the type of low-energy excitations present in the system. If the contribution is sufficiently large, as in the case of a spinon Fermi surface (due to its high density of states) it would be expected to dominate over the lattice term at low enough temperatures. A large thermal conductivity of magnetic origin in an insulator may thus be indirect evidence of fermionic quasiparticles, a surprising feature of a spin system. However, one should be clear that thermal conductivity itself is completely agnostic as to the quantum numbers of the excitations, so that, without some corroborating evidence, one has no reason to suppose a measured thermal conductivity signal is related to spins. Comparison of the data with that in a high magnetic field, where magnetic excitations are quenched at low energies, can help disentangle phonon contributions from magnetic ones.

The thermal Hall effect, analogous to the electrical Hall effect, is the flow of a thermal current normal to a temperature gradient. This requires time-reversal symmetry breaking, usually induced by a field, although it could occur spontaneously. The difficult thermal Hall conductivity measurement has been reported in some compounds. A priori the thermal Hall effect requires some degree of freedom besides the lattice, since it is hard to see how phonons couple significantly to an applied magnetic field. In an insulator, there can be both orbital and Zeeman contributions that affect the spins and, through various means, induce a thermal Hall effect. In general, we may expect that thermal Hall current can still include a lattice contribution, if the lattice degrees of freedom are scattered anisotropically by magnetic or electronic degrees of freedom which sense the field. It may also include a direct contribution due to transverse heat current carried by spins.

A quantized thermal Hall effect (the direct contribution) is a sharp characteristic of chiral topological phases, such as the chiral spin liquid, which have chiral edge states [53]. It has been suggested that an unquantized direct contribution might be characteristic of some gapless QSLs [206]. To our knowledge, no clear signature of a QSL has been observed by thermal Hall measurements to date.

Recognizing the lack of smoking-gun signatures in standard measurements of magnetic systems, theorists have proposed unusual ways one might look for QSLs. These include:

• A state with a spinon Fermi surface probably differs most dramatically from an ordinary insulator, and various specific experiments may be envisioned to look for this Fermi surface. For example, one might observe Friedel oscillations from an impurity [207], or observe RKKY oscillations of exchange bias in a magnetic heterostructure [208]. Angle-resolved photoemission might reveal the Fermi surface above the Mott gap [209].
• More generally, there are ideas for experiments which exploit the relations between the quasiparticles of a spin liquid and excitations of ‘nearby’ phases. For example, one may obtain a superconductor by condensing a bosonic ‘holon’ (which is topologically an e particle, but endowed with electric charge) in a Z₂ spin liquid. In this picture, the m particle is converted into a superconducting vortex. This led to a proposal to measure persistent vorticity on quenching between a Z₂ QSL and a superconductor, or the possibility of observing vortices with double the minimal flux in a nearby superconductor [12, 210]. Such a measurement was even attempted experimentally [211], with a null result. Similarly, a fermionic ε particle in the QSL can be converted into a Bogoliubov–de Gennes quasiparticle in a superconductor, and one may contemplate an oscillation experiment to detect injection of this quasiparticle from a superconductor in contact with a Z₂ QSL [212].
• In ultra-cold atoms, non-local measurements might actually be possible that detect QSLs more intrinsically. For example, using a quantum gas microscope, one can make simultaneous measurement of the state of every lattice site. This allows one to reconstruct the amplitudes of every term in the wavefunction, and thereby reconstruct the full wavefunction if some knowledge of the phases is given. This has been used to measure the string order in 1D boson Mott insulators [213]. There are also protocols to measure the second Renyi-entanglement entropy by producing a ‘twinned’ state composed of two copies of a physical system, and then carrying out a specific quantum evolution by controlling interactions between these copies—see [214]. Interference between such twins has been measured [215], and very recently the proposed experiment has actually been carried out for a four-site chain of bosons [216].

The honeycomb iridate materials form a family whose simplest members are Na₂IrO₃ and Li₂IrO₃, in which Ir⁴⁺ , which is in a d⁵ configuration, occupies the sites of layered honeycomb lattices (separated by a plane of sodium or lithium atoms). These compounds rose in prominence with the suggestion [217, 218] that they might be governed to a good approximation by the remarkable soluble Kitaev honeycomb Hamiltonian discussed in section 6.1.1. The basic physics descends from that of the Ir⁴⁺ ion which, in the octahedrally coordinated oxygen environment, can be considered to have a single hole in a threefold degenerate t_2g shell. Due to strong spin–orbit coupling, the orbital degeneracy is lifted in favor of a non-degenerate Kramer’s doublet ground state: an effective S  =  1/2 spin (this is, in the simplest limit, called a ’ j  =  1/2 state’ in the iridate literature). Owing to the edge-sharing nature of the octahedra in these materials, there are two approximately 90 degree Ir–O–Ir bonds connecting each pair of spins, and a superexchange analysis in the strong spin–orbit limit predicts a Kitaev-type interaction between neighboring spins [217], in addition to a Heisenberg term arising from direct Ir–Ir overlap. Thus, the ‘minimal’ model for these materials is largely believed to be a S  =  1/2 Hamiltonian consisting of the sum of the Kitaev one plus Heisenberg terms. A QSL ground state is possible in some (unfortunately narrow) range of parameters for this model.

These materials were eventually shown to order magnetically [219, 220]. Nevertheless, one may hope to observe some remnant of QSL physics either above the ordering transition, or in the higher energy excitations, or both. There are indications of strong fluctuations at least. Na₂IrO₃ is the best characterized, and has a Curie–Weiss temperature of ${{ \Theta }_{\text{CW}}}\approx -120$K, and inelastic neutron scattering confirms that the excitation spectrum disperses on the scale of 6–10 meV, which is comparable. However, the Néel temperature of ${{T}_{\text{N}}}\approx 18$ K is significantly lower, giving it a moderate frustration parameter of f  =  6.6. Moreover, the static moment is estimated at $0.22\mu$B from elastic neutron scattering, much reduced from the full 1μB spin-1/2 moment. Quite recently rather direct evidence that some direction-dependent spin interactions are substantial has been uncovered by diffuse magnetic x-ray scattering, which measures equal-time spin correlations above T_N. Experiments show that different spin components are correlated at different momenta, consistent with substantial Kitaev coupling [221].

Complications arise when one looks in more detail at the ordered state, which is of ‘zigzag’ type, a collinear arrangement of spins that doubles the unit cell. This was not expected from the simplest theoretical model, leading to two main speculations on an appropriate modified Hamiltonian. The zigzag state can be captured by the simplest nearest-neighbor Kitaev–Heisenberg model, but only if the Kitaev term has a anti-ferromagnetic sign and the Heisenberg one is ferromagnetic, which does not seem so natural. Alternatively, it may occur with ferromagnetic Kitaev interactions and an antiferromagnetic Heisenberg coupling, if second- and third-neighbor exchange are significant. These are not the only possibilities, and more complicated Hamiltonians have been explored [222]; yet another potential pitfall is that (Na,Li)₂IrO₃ do not actually have exact hexagonal symmetry: the true space group is C2/m.

Yet a different theoretical view is provided by ab initio calculations, which propose a ‘quasi-molecular orbital’ picture of these materials as band rather than Mott insulators, and argue that spin–orbit coupling has only moderately significant effects [223, 224]. This may not be incompatible with the spin model including longer-range exchange. However, a band picture usually fails when the gap is much larger than the magnetic exchange scale, which seems to be the case given the gap of 340 meV in Na₂IrO₃ measured by angle-resolved photoemission spectroscopy (ARPES).

The lithium analog, Li₂IrO₃ actually occurs in several structures: α-Li₂IrO₃, which is a true honeycomb like the sodium compound, and two 3D polytopes, β-Li₂IrO₃ and γ-Li₂IrO₃, which locally have the coordination of a honeycomb with each IrO₆ octahedra sharing three edges with other octahedra, but arranged into more complex non-planar structures. These 3D structures also admit soluble Kitaev-like Hamiltonians with Z₂ QSL phases, that have been studied theoretically. Experimentally, all three materials order antiferromagnetically. The order in the 3D ($\beta,\gamma$) materials is completely determined from single crystal resonant elastic x-ray scattering, while the magnetic structure of the α type is not yet known. Both 3d materials exhibit incommensurate non-coplanar spiral structures with a very similar wavevector, and similar ordering temperatures $\sim 40$ K, while the α polytope orders at ${{T}_{\text{N}}}\sim 15$ K, close to that of the sodium material. The Curie–Weiss temperature for the α compound is estimated as ${{ \Theta }_{\text{CW}}}\approx -33$ K and for the β material ${{ \Theta }_{\text{CW}}}\approx +40$ K, neither of which would suggest significant frustration. Single-crystal measurements of the γ compound show significant strong temperature-dependent anisotropy which cannot be fitted well by a Curie–Weiss form, but does clearly indicate directional-dependent interactions. A priori, the higher ordering temperatures and lower Curie–Weiss temperatures of the lithium materials may suggest they are further away from a QSL state than the sodium compound. The larger ordered moment, estimated at $0.47\mu$B for the β compound [225], seems to support this picture.

The 4d transition metal material α-RuCl₃ has emerged quite recently as another possible Kitaev material, outside the iridate family. In general, stronger correlations (i.e. larger Hubbard U) but weaker spin–orbit coupling is expected for 4d ions relative to 5d ones. However, it can very well be sufficient to generate strong anisotropy (which even occurs for 3d ions in some situations). Optical and x-ray studies, backed up with density functional theory, support the interpretation of this material as a spin–orbit-assisted Mott insulator with Ru³⁺ ions in a $\text{t}_{2g}^{(5)}$ state, just as in the iridates [226]. The optical gap is $\sim 0.3$ eV, also comparable to that of Na₂IrO₃. However, if we adopt the usual superexchange approach, the octahedra in α-RuCl₃ are actually less distorted than in Na₂IrO₃, and also the Ru are more separated than the Ir, both of which probably favor a more ideal realization of nearest-neighbor Heisenberg–Kitaev physics.

Unfortunately, however, like the iridates, α-RuCl₃ clearly does not have a QSL ground state. Early studies showed antiferromagnetic ordering at 15 K [227], and more recently multiple transitions have been observed [228–230]. It is not clear if both are intrinsic or may reflect different structural domains in a sample [231]. It is plausible that this is related to stacking faults due to the weak van der Waals bonding of this compound. The susceptibility gives a ferromagnetic Curie–Weiss temperature of ${{ \Theta }_{\text{CW}}}\approx 23$–40 K [227, 232], which is not much larger than T_N, but given that it is ferromagnetic and the ordering is antiferromagnetic, there may be some frustration. Elastic neutron scattering [230] seems to support zigzag order as in Na₂IrO₃. Very recent inelastic powder neutron scattering observes a high-energy ($\sim 6.5$ meV) feature which persists above T_N and which seems similar to what is expected from the Kitaev model [231].

In general, while none of these honeycomb-like materials seem to be an ideal QSL, they may be close enough to be interesting. It would be interesting to see if remnants of QSL physics might be observable at high energy or high temperature (it is an interesting theoretical problem whether we would even expect this), or if any one of these materials is really close to a QSL state, it could be somehow driven into a real QSL in some way, e.g. by pressure, field, or chemical modification.

The rare-earth pyrochlore oxides form a venerable class of compounds studied for their frustrated magnetism [233]. Such materials have chemical formula R₂M₂O₇, with R a rare-earth ion, and M a transition-metal ion such as Ti or Sn. In many of these compounds, much like in the honeycomb iridates from the previous section, very strong spin–orbit coupling and crystal fields split the single-ion spectrum into low- or zero-degeneracy manifolds. When the single-ion ground manifold is a doublet, well separated in energy from the first-excited manifold, the effective theory should be that of a spin-1/2. Sometimes the interactions amongst these effective spins are extremely classical: they involve (to very high accuracy) only a single component of each spin, so that all terms in the Hamiltonian commute. This is the situation in the so-called classical spin ices, such as Ho₂Ti₂O₇ and Dy₂Ti₂O₇. However, this is not always the case. Strong quantum effects are present in quite a few materials, e.g. Yb₂Ti₂O₇, Er₂Ti₂O₇, Tb₂Ti₂O₇, Pr₂Zr₂O₇. Recognition of these materials as fully quantum-effective S  =  1/2 systems became widespread only a few years ago, stimulated by careful investigation of Yb₂Ti₂O₇ by inelastic neutron scattering.

In general, the high degree of localization of the f-orbitals means that exchange is very short-range, and apart from a weaker dipolar interaction, can be taken to an excellent approximation to be restricted to nearest neighbors. This restriction, and the high cubic symmetry of these compounds, allows for a rather complete understanding of the underlying spin Hamiltonian. For most ions (exceptions occur only if the low-lying doublet has unusual symmetry, e.g. Nd³⁺ [80]), the most general nearest-neighbor Hamiltonian takes the form [234, 235]

$$\begin{eqnarray}\begin{array}{*{35}{l}} H & = & \underset{\langle ij\rangle}{\sum}\,\left[{{J}_{zz}}{\mathsf {S}}_{i}^{z}{\mathsf {S}}_{j}^{z}-{{J}_{\pm }}\left({\mathsf {S}}_{i}^{+}{\mathsf {S}}_{j}^{-}+{\mathsf {S}}_{i}^{-}{\mathsf {S}}_{j}^{+}\right) \right. \\ {} & {} & +\,{{J}_{\pm \pm }}\left[{{\gamma}_{ij}}{\mathsf {S}}_{i}^{+}{\mathsf {S}}_{j}^{+}+\gamma _{ij}^{\ast}{\mathsf {S}}_{i}^{-}{\mathsf {S}}_{j}^{-}\right] \\ {} & {} & \left. +\,{{J}_{z\pm }}\left[{\mathsf {S}}_{i}^{z}\left({{\zeta}_{ij}}{\mathsf {S}}_{j}^{+}+\zeta _{ij}^{\ast}{\mathsf {S}}_{j}^{-}\right)+i\leftrightarrow j\right]\,\right], \end{array}\end{eqnarray} \tag{ 92 }$$

Here, the spin components are taken along local axes, and the coefficients γ and ζ are constant phases related to a particular choice of local ‘transverse’ axes, which ensure the model has all the proper symmetries. All such details can be found in, e.g. [193]. Equation (92) should provide an excellent description of a wide set of these rare-earth pyrochlore materials. It has three dimensionless parameters (ratios of exchanges), so the problem of finding QSLs in these materials reduces to the problem of finding them in this 3D space.

What do we know about this problem? It is most studied (and most interesting) for the case J_zz  >  0, where this Ising interaction is frustrated, and this defines what one may call ‘quantum spin ice’. The first two terms of H are exactly the XXZ model of equation (81) discussed in section 6.2.2. The existence of a stable spin-liquid phase in the latter model was established in the ${{J}_{\pm }}/{{J}_{zz}}\ll 1$ limit, where the model was mapped to a pure U(1) compact gauge theory, as discussed in section 6.2.2. This can be seen as the restriction of a full gauge theory with matter to the sector with no gauge charge, physical when the gauge charges are gapped. In fact, away from the perturbative limit the gauge charges are important, and at least virtual fluctuations must be included. This is accomplished in [193], where a novel parton construction, different from the standard ones discussed in section 4, was introduced in a way designed to match the perturbative limit of [43]:

$$\begin{eqnarray}&&{\mathsf {S}}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}^{z}={{E}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}},\quad \quad {\mathsf {S}}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}^{+}= \Phi _{\mathbf{r}}^{\dagger}{{\text{e}}^{\text{i}{{A}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}}}}{{ \Phi }_{{{\mathbf{r}}^{\prime}}}},\end{eqnarray} \tag{ 93 }$$

with the constraint $\text{div}E={{Q}_{\mathbf{r}}}$, where ${{ \Phi }_{\mathbf{r}}}={{\text{e}}^{-\text{i}{{\phi}_{\mathbf{r}}}}}$ (and so $\Phi _{\mathbf{r}}^{\dagger}{{ \Phi }_{\mathbf{r}}}=1$, a constraint in a sense ‘analogous’ to the constraints ${{b}^{\dagger}}b=2S$ and ${{f}^{\dagger}}f=1$ from the Schwinger boson and Abrikosov fermion rewritings of the spin, equations (44) and (45), respectively) is a rotor variable, and $\left[{{\phi}_{\mathbf{r}}},{{Q}_{\mathbf{r}}}\right]=i$, which is invariant under the combination of ${{A}_{\mathbf{r}}}\to {{A}_{\mathbf{r}{{\mathbf{r}}^{\prime}}}}+{{\chi}_{\mathbf{r}}}-{{\chi}_{{{\mathbf{r}}^{\prime}}}}$ and ${{ \Phi }_{\mathbf{r}}}\to {{ \Phi }_{\mathbf{r}}}{{\text{e}}^{\text{i}{{\chi}_{\mathbf{r}}}}}$. Gauss’ law commutes with the Hamiltonian, a sign of the U(1) gauge symmetry generated by $U\left(\mathbf{r}\right)={{\text{e}}^{\text{i}\left(\text{div}E-{{Q}_{\mathbf{r}}}\right)}}$. Here, the partons live on the centers of the tetrahedra, and have a very physical interpretation, in view of the physics of classical spin ice.

In the original formulation of the theory [193], the spinons were taken to be bosons. This has the advantage of allowing a simple description of the ‘all-in–all-out’ tetrahedral configurations (i.e. with two spinons on a tetrahedron) as well as one of the transitions in terms of single-spinon condensation. The Hamiltonian we obtain through this procedure is that of spinons hopping (${{J}_{\pm }}$ and ${{J}_{z\pm }}$ terms) as well as interacting (${{J}_{\pm \pm }}$ term) in a fluctuating background. This reformulation is exact, when the constraints discussed above (rotor and Gauss’ law) are enforced. This reformulation allows for the implementation of a mean field theory—dubbed gauge mean field theory (gMFT)—which can describe quantum liquid phases. In particular, $\langle \Phi \rangle =0$ should be characteristic of a quantum liquid phase, as $\langle \Phi \rangle \ne 0$ is the criterion for Higgs condensation, which induces a conventional time-reversal-breaking magnetically ordered phase. Such a mean-field approach was carried out for some large subregions of the phase space of equation (92) in [193, 236]. But, we should emphasize, this is only a mean field theory, and while it agrees with solid results in some limits, is not expected to be accurate everywhere. And indeed, the Hamiltonian includes some particularly difficult limits, such as the isotropic Heisenberg antiferromagnet (see section 6.4.5). Other parton constructions are possible, as are other mean field theories [237]. In our opinion, more computational efforts, e.g. variational Monte Carlo calculations, are sorely needed here.

What about the experimental situation? Among the known ‘quantum’ pyrochlore materials, Yb₂Ti₂O₇, Tb₂Ti₂O₇, and Pr₂Zr₂O₇ are most discussed as spin-liquid candidates. Yb₂Ti₂O₇ is the most studied. It is clearly a highly quantum, effective S  =  1/2 system, for which direct evidence is provided by the large inelastic weight and clearly defined sharp spin-wave excitations in high field, with a bandwidth of the order of an meV [235, 238]. It seems also to be in the quantum-spin-ice category. This is based, in part, on the exchange couplings, which were determined by the very direct technique of measuring the single magnon spectrum in a large (5T) field, for which the spins are nearly fully polarized in the ground state. Because the ground state is rendered trivial by the strong field, one can straightforwardly and accurately calculate the magnon spectrum for arbitrary exchange couplings, and then fit the result to experiment [235, 239]. The technique is well suited to this class of materials because exchange interactions are relatively weak, hence the necessary fields are accessible and compatible with experiment, and large single crystals are available. This experiment determined all the exchanges and in particular concluded that J_zz  >  0 was the strongest coupling [235], which supports the quantum-spin-ice picture (though this has recently been contested in [240] and by Coldea). Other direct evidence for spin-ice physics is [241], which reported the presence of ‘pinch point’ like features in equal-time neutron scattering at higher temperature, consistent with the picture of a transition to a classical spin-ice regime [242].

However, it is also clear that Yb₂Ti₂O₇ does not realize the ideal Coulomb QSL state possible for equation (92). High-quality samples show a pronounced specific heat anomaly indicating a sharp phase transition at about ${{T}_{c}}\approx 240$ mK. This seems largely consistent with a ‘splayed’ (non-collinear) ferromagnetic ground state, which is expected according to both the classical approximation and gMFT, given the exchanges fitted in [235]. Many experiments do, indeed, support ferromagnetic order [241, 243–245]. However, the transition temperature itself is suppressed, ${{T}_{\text{MF}}}/{{T}_{c}}\approx 14$, using Curie–Weiss mean field theory to determine T_MF given the fit parameters. Moreover, spin-wave excitations are strangely absent in the zero-field ordered phase: inelastic neutron scattering shows instead only diffuse continuum excitations [235, 238]. Muon spin resonance also shows dynamics different from that of usual ferromagnets [246].

Interpretation of the zero-field behavior is not settled. We believe quantum fluctuations play an essential role in at least the unconventional excitations, though the connection to Coulombic QSLs is far from clear. A priori, even gMFT does not place the zero-field state close to a QSL phase: rather it is near a phase boundary between two ordered states. So perhaps proximity to this boundary is important [247]. We also note that a large sample dependence due to the disorder which is known to exist in this family of compounds. In particular, it was explicitly shown that a single crystal of Yb₂Ti₂O₇ had approximately $2 \%$ ‘stuffed’ (off-stoichiometric) Yb⁺³ ions, i.e. extra Yb ions lacunar Ti ones [248]. So disorder may play some role in the zero field spin dynamics, though we are inclined to think it is a secondary one. Given continued experimental activity [202, 249, 250], we expect Yb₂Ti₂O₇ to be an interesting and evolving topic in the near future.

Tb₂Ti₂O₇ has the status of a long-standing puzzle and was the first pyrochlore to be suggested to belong to the ‘quantum spin ice’ class. Theoretically, however, much less is known about this compound. This is largely due to the fact that it is unclear whether the full low-energy properties may be accounted for by a simple two-level (effective S  =  1/2) approach. Indeed, in contrast to Yb³⁺ in Yb₂Ti₂O₇, whose lowest crystal-field doublet lies 700 K below the first excited one, the first two doublets in Tb₂Ti₂O₇ are only separated by an energy of approximately 20 K. While this may seem relatively large compared to an exchange energy of a few Kelvins, inelastic neutron-scattering data seems to show what can be interpreted as substantial ‘mixing’ between the crystal field levels. Experimentally, Tb₂Ti₂O₇ has a Curie–Weiss temperature reported to be between  −19K and  −14K. Remarkably, for a long time, no magnetic order or signature of a magnetic phase transition had been reported down to 50 mK, with neutron scattering only displaying a diffuse signal [251–256]. Recently, however, thorough investigations have pointed to both the existence of more structure than initially thought in neutron scattering, like features reminiscent of ‘pinch points’ (though quite different from those seen in classical spin ices) point towards power-law correlations [257] and relatively sharp magnetic Bragg peaks in quasi-elastic scattering in field-cooled samples [257–259], as well as an anomaly in the μSR frequency shift at T  =  0.15 K [260]. In fact, despite quite a large sample dependence, most crystals display a sharp upturn in the specific heat below about 400 mK [261]. While an upturn in the specific heat is expected because of the nuclear contributions, the minimum, which occurs in [260] at a temperature close to where the kink in μSR is observed, could also be indicative of a transition. A magnetic transition at 150 mK would yield a frustration parameter of $f\sim 125$. Thermal conductivity—both diagonal and Hall—provide a new vantage point, which also seems to indicate the presence of exotic magnetic excitations [262, 263]. Finally, Tb³⁺ contains an even number of electrons and, therefore, the crystal-field doublets are not Kramers doublets: their degeneracy is enforced not by time-reversal symmetry, but rather by (possibly-approximate) crystal symmetries. As a consequence, the levels may be split by, e.g. Jahn–Teller distortions, although this last fact is still at the center of a controversy.

Pr₂Zr₂O₇ was also studied as a potential quantum-spin-ice candidate [264]. While no signs of long-range magnetic order were seen in inelastic neutron scattering, pinch-point-like features were observed. The first excited crystal-field doublet lies at about 90 K [264], so that a spin-1/2 model is expected to be appropriate, as pointed out by Onoda [265]. This compound is, however, due to the similar sizes of the Pr³⁺ and Zr⁴⁺ ions, more prone to site inversion and, therefore, to disorder than the other, aforementioned, compounds. Moreover, like in Tb₂Ti₂O₇, the non-Kramers nature of the low-lying Pr³⁺ doublet allows for its splitting, for example, by structural disorder. Random doublet splitting has recently been advocated experimentally. Theoretically, local symmetry breaking, even when random, rather than leading to a glassy phase, can be shown to instead give rise to the quantum-spin-ice phase, even in the absence of quantum spin exchanges, as well as to a ‘Mott glass’ phase, depending on the strength of the disorder [266]. Whether this is indeed the physics at play in the samples of Pr₂Zr₂O₇ grown so far will require more careful investigation, but this compound definitely stands as a promising candidate for the realization of a disorder-induced quantum-spin liquid, a scenario possible thanks to the the fact that entanglement does not rely on the existence of any symmetries (such as translational).

The materials discussed here are only a few examples (the best studied for their possible quantum-spin-ice behavior) in the large family of rare-earth pyrochlore compounds. Intense experimental (new materials, and more detailed experiments) and theoretical investigations in this family of compounds, as well as in that of spinels–where the same physics could occur–are under way. In this context, new interesting developments are more than likely, and there is hope for a quantum-spin liquid to be discovered.

There are a great variety of organic molecular crystals which have been studied as nominally simple examples of correlated electron systems [267]. Because of the large size of the organic molecules, there are many atoms in the unit cell, and the electronic structure is, in principle, quite complex. However, it is generally thought that each organic molecule, or pair of molecules, hosts a single partially occupied molecular orbital which is near the Fermi energy. In this way, such materials are conceptually reduced to simple Hubbard-like models. Many structures can be produced, by varying both the molecules containing the partially filled orbitals, other molecules in the structure, end-groups on these molecules, and the pattern of the molecular arrangements. Diverse examples of both conductors and insulators have been studied.

Out of this plethora of materials, two (a third κ-H₃(Cat-EDT-TTF)₂, mentioned below, is much less investigated) have emerged as excellent empirical QSLs: κ-(ET)₂Cu₂(CN)₃, and EtMe₃Sb[Pd(dmit)₂]₂—we will abbreviate these to κ-ET and dmit, respectively. These are quasi-2D nominally half-filled Mott insulators, in which molecular units host single orbitals which appear magnetic and are arranged into an approximate triangular lattice, each such layer being separated by non-magnetic ions. Each of these two compounds are specific members of large families with the same conducting molecules, where all the differences within each family occur within the non-magnetic ions. Based on quantum chemistry calculations, and more recently density functional theory, an extended Hubbard model is typically considered an adequate theoretical description. Unlike inorganic frustrated magnets based on 4f and 3d ions, which are ‘strong’ Mott insulators well described by entirely localized spins, here, the electrons are close to a metal–insulator transition, and a spin-only description may be insufficient. Evidence for this ‘weak’ Mott insulating behavior comes from optics and transport, which observe a negligible or very small gap, and pressure studies, which are able to drive a Mott transition to a metallic state with relatively small pressures. This suggests that a pure spin Hamiltonian with very short-range interactions is a bad approximation for these materials, and that either an extended Hubbard model or perhaps a long-range exchange model is better.

What is the evidence for QSLs in these materials? First, they are insulating, as shown in transport and optics, so it is reasonable to regard them as Mott insulators. Second, both magnetic susceptibility and NMR indicate the presence of local moments, with spin S  =  1/2 per molecular unit, and effective antiferromagnetic exchange interactions of the order of $J\sim 200$–250 K, in the sense the susceptibility is progressively suppressed below a Curie law as temperature is lowered. However, despite the relatively large exchange coupling, there is no indication of magnetic ordering. Thermodynamic quantities—the susceptibility and heat capacity—show no singular features which would clearly signal phase transitions. More directly, NMR measurements, which are the most sensitive probes in these systems, show the absence of any spontaneous static magnetic fields developing down to very low temperature of 20 mK in both materials. The combination of these measurements seems to conclusively rule out long-range antiferromagnetic order at a temperature $\approx {{10}^{4}}$ times lower than J!

However, this is negative evidence. What do we actually know about the ground state and low-energy excitations? The main evidence for interesting physics comes from thermal measurements—heat capacity and thermal conductivity—at temperatures below $\sim$1 K. In these particular salts, in contrast to others in their families, a large linear term in the specific heat versus temperature, ${{c}_{v}}\sim \gamma T$, is observed [268, 269]. This is similar to the Sommerfeld contribution in a metal and is, in fact, comparable to that in a metal (if normalized per spin)—something unexpected in an insulator. The former indicates some low-energy excitations with a large density of states, and has been interpreted as evidence for fermionic quasiparticles, which can sustain a large constant density of states by the Pauli principle. In a Mott insulator, these cannot be electrons, so it is tempting to think this may be an indication of a spinon Fermi sea. It was also noted that the uniform susceptibility χ (measured both by magnetization and NMR Knight shift) saturates to a substantial constant at low temperature, which is also typical for a metal. Furthermore, the Wilson ratio,

$$\begin{eqnarray}&&{{R}_{\text{W}}}=\frac{4{{\pi}^{2}}k_{\text{B}}^{2}\chi}{3{{\left(g{{\mu}_{\text{B}}}\right)}^{2}}\gamma},\end{eqnarray} \tag{ 94 }$$

is order one in both materials, which is expected for free fermions, and may indicate that the same degrees of freedom control both the susceptibility and the specific heat. This seems like a non-trivial consistency check. On the other hand, the specific heat data, at least for dmit, has a strange feature: the Sommerfeld coefficient is modified by a factor of two by partial isotopic substitution of deuterium for hydrogen in the layers between the dmit molecules [269], which one would think would has nothing to do with the spins.

A different concern is whether the signals might arise from localized single-particle states, since specific heat and (to a first approximation) susceptibility are related just to density of states. Therefore, a measurement which proves the delocalized nature of the excitations is important. This is the thermal conductivity κ. Measurements on dmit over the range of $0.1~\text{K}<T<0.3~\text{K}$ fit well to a T ³ phonon contribution plus a Sommerfeld-like term, linear in temperature, attributed to the spin subsystem [270]. This gives some weight to the idea that the low-energy excitations are delocalized fermionic spinons, which is quite exciting. In κ-ET, the non-phonon contribution appears to be strongly suppressed at the bottom of this temperature range, leading to the suggestion of a gap opening [271].

Theoretically, modeling of these results has been based on the triangular-lattice Hubbard model. A non-magnetic insulating phase was first proposed based on numerical simulations [180, 272]. Later, following the specific heat measurements, theorists specifically proposed the spinon Fermi sea coupled to an emergent U(1) gauge field [181, 186]—see section 6.6. As discussed, this is broadly in agreement with experiment. Notably, this QSL state is described by a strongly interacting field theory, so there are theoretical uncertainties. Nevertheless, theory predicts some even more dramatic phenomena than free fermionic spinons. In fact, it finds not a linear specific heat, but one scaling as T^2/3 [186]. Given the limited range of data, and the presence of a nuclear Schottky term, it is unfortunately not possible to distinguish this from the linear Sommerfeld behavior.

However, these measurements are quite indirect: they really indicate only that there are some excitations carrying heat, and that these are mobile. There is no direct indication that these are actually associated with the spins. Due to the limited range of temperature of the measurement, one cannot directly see that these excitations emerge when the spins become correlated, which is what one would expect. The restriction to very low temperature is, unfortunately, necessary, because the spin-1/2 moments are very dilute in the large organic crystal, and so their contribution to the energy is fully swamped by the lattice one above a few K. Doubts have been raised about the specific heat measurements due to possible contamination by nuclear contributions. Moreover, it is known that molecular rotational tunneling excitations can contribute entropy and thermal conductivity at low temperatures [273, 274]. The most direct probe which definitely measures the spins is NMR, but beyond the demonstration of the absence of ordered moments, it has not clarified the situation (there is plenty of data, but these do not seem to fit into the expectations from the U(1) spinon theory).

There are many unexplained phenomena. The NMR relaxation in both materials becomes non-exponential below a few K, which is one of several indications of inhomogeneity (possibly field-induced)—see below. The low-frequency conductivity is quite large and roughly power-law in form [275, 276], which is, in fact, expected from the spin Fermi-surface model [199, 200], but in detail the behavior does not agree with this theory. A number of experiments point to some sort of phase transition or crossover at 6 K in κ-ET [277, 278]. Some transition is also indicated at 1 K in dmit based on NMR [279, 280], and based on the field dependence of the thermal conductivity below this temperature, it was suggested that there are some spin excitations with a gap of this order, which opens at this temperature. Motivated by these apparent transitions, theorists have proposed pairing instabilities of the spinon Fermi-surface state [281, 282].

Most of these features are associated with quite low temperature. So it is possible some of them are subsidiary phenomena sensitive to small details of the Hamiltonian. It would be more ideal to measure the spin state between 1 K and room temperature, as the larger scales are the natural ones for spins with $J\sim 200$ K. For example, it would be very desirable to probe the correlations of spins directly through neutron scattering. However, since organics contain a large concentration of hydrogen atoms, which strongly scatter neutrons, such measurements are extremely difficult and have not been successful in κ-ET or dmit so far. In general, apart from NMR, we have no direct probes of the spins in this range.

A basic question is that of the mechanism for the spin-liquid behavior. If these materials are truly QSLs, why only these two organics, out of the vast families? Originally, based on chemical modeling of the hopping parameters, it was proposed that the key variable is spatial anisotropy: organics which are closest to isotropic fall into the spin-liquid phase. However, recent density functional-based theory finds rather different hopping parameters, putting this mechanism into question [283, 284]. These calculations also estimate that on-site Hubbard interactions are significantly stronger than the earlier models do, and are accompanied by more significant off-site terms, which may further complicate the picture. Some evidence against the idea that spatial anisotropy is the key ingredient comes from newer experiments on κ-H₃(Cat-EDT-TTF)₂, a third organic material which shows QSL behavior in the susceptibility and torque magnetometry, but which is estimated to be much more 1D [285].

One open issue is the extent and influence of disorder and inhomogeneity. For many years these materials have been viewed as extremely clean, with a mean free path estimated from thermal conductivity in dmit of 1μm [270]. Non-exponential relaxation in NMR below 6 K in κ-ET and between 1–10 K in dmit suggest some inhomogeneity, but it was suggested this is field-induced. However, there are several indications of disorder or inhomogeneity in more recent experiments on κ-ET, based on e.g. transport and optics [286], electron spin resonance [287], and muon spin resonance (muSR) [288]. Whether any of these phenomena are also present in dmit, and whether they are important to the spin physics [289], remains unclear to the authors.

The kagomé lattice has long been thought theoretically to be a sweet spot for spin-liquid physics. In the domain of the most quantum S  =  1/2 spins, however, material realizations are sparse, and all suffer from various complications departing from the theoretical ideal. They come from various copper minerals. For example, there is volborthite, Cu₃V₂O₇(OH)₂⋅2H₂O, which is very clean, but lacks threefold symmetry and is quite spatially anisotropic, and there is strong evidence for competing exchanges very far from the simple nearest-neighbor antiferromagnet. Vesignieite, BaCu₃V₂O₈(OH)₂, also lacks threefold symmetry, but is much less characterized. Kapellasite, Cu₃Zn(OH)₆Cl₂, has threefold symmetry, but is believed to have competing exchanges up to third neighbor, including ferromagnetic interactions between nearest neighbors [290].

The simplest and most heavily studied of the quantum kagomé materials is herbertsmithite, with chemical formula ZnCu₃(OH)₆Cl₂. Synthesized in 2005 [291], a review as of 2010 can be found in [292]. It has the ideal kagomé threefold symmetry, and all indications are that the Cu²⁺ ions interact predominantly by a nearest-neighbor antiferromagnetic Heisenberg exchange of order $J\sim 180$ K (as reinforced by first-principles calculations in [293]), yet avoids ordering and spin-freezing down to temperatures as low as 50 mK, as observed by NMR [294], μSR, neutron scattering [295, 296], and susceptibility measurements [297]. These early experiments all observed a large density of low-energy states, e.g. by specific heat [298], indicating a gapless state.

Structurally, herbertsmithite can be understood as a depleted pyrochlore lattice, with non-magnetic Zn atoms substituting for 1/4 of the pyrochlore sites, and forming triangular layers which then divide the remaining Cu sites into kagomé planes. However, the substitution is not perfect, and it is believed that between 5–10 percent of these triangular sites are actually occupied by Cu rather than Zn. At a first approximation, these inter-layer Cu ions form quasi-free spins, which are then responsible for the low-energy states. More controversial is the extent of Zn atoms in the kagomé planes and, in general, the feedback of both types of defects on the physics of the spins within the kagomé layers. Optimistically, the kagomé layers may be almost unaffected by the defects, so that any measurement yields simply the sum of some contribution from the inter-layer Cu spins and a signal from ideal kagomé planes. Less optimistically, the fragile nature of the ideal kagomé state may be altered by the disorder.

The first single crystals were grown in 2011, allowing a next stage of studies. For example, single-crystal measurements of susceptibility anisotropy suggest a 10% easy-axis Ising exchange anisotropy [299]. Single-crystal inelastic neutron scattering [300] observes only diffuse weight which persists to zero energy, and an absence of any sharp modes. This is consistent with a QSL state, and was interpreted as evidence for gapless spinon excitations. The extremely smooth spectral weight, however, which lacks any sharp features such as excitation edges, and has weight down to zero energy at all momenta, is not expected for any of the model QSL states, including both the gapped Z₂ QSL found by DMRG, and the U(1) Dirac favored variationally which, though gapless, has low-energy states only at isolated momentum points. High field measurements have been carried out up to 55T, which show a rather linear magnetization versus field, and no indication of any field-induced transitions. Moreover, the large low-temperature, quasi-linear specific heat seems to persist throughout the field range, which is interesting because presumably the inter-layer spins must polarize and develop a Zeeman gap in this regime, implying that these low-energy excitations must come from the kagomé layers. Terahertz optics also observe some in-plane AC conductivity $\sigma (\omega )\sim {{\omega}^{\beta}}$ with $\beta \approx 1.5$ for $30~\text{K}<\hslash \omega /{{k}_{\text{B}}}<67$ K, which may be related to gapless electric dipole excitations of the spins [199].

Various avenues have been explored in order to reconcile the ‘gapless’ scattering with theory. It was suggested that herbertsmithite might be perturbed sufficiently by interactions beyond nearest-neighbor, e.g. Dzyaloshinskii–Moriya coupling, to drive it close to a quantum critical point to a nearby ordered phase [301]. Further neighbor exchange might also serve the same purpose. Another suggestion is that the ground state is a Z₂ QSL, but the bosonic spin-less m excitations have anomalously low energy and a very flat dispersion, and the ‘shake-off’ effects of these m particles completely smooths out the intrinsic excitation spectrum of the spinons [302]. A less appealing, but highly plausible explanation is that the observed excitations are strongly influenced by disorder. The very weak momentum dependence, indeed, suggests the excitations are rather local, and with this in mind the high degree of site disorder is hard to discount. Numerical calculations including disorder have been carried out [303].

While all the above measurements point to a gapless state of some kind, a very recent NMR experiment, which utilizes the single crystal to select a specific advantageous field orientation, claims to observe a spin gap at low field in the kagomé plane [304]. This might bring experiments on herbertsmithite more in line with DMRG calculations for the Heisenberg model, but reconciliation of this preliminary result with prior experiments may then become the issue. This requires careful modeling [305].