Optimal multi-configuration approximation of an N-fermion wave function
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The document discusses methods for approximating fermionic wave functions using Slater determinants, addressing a significant problem in quantum many-body physics. It presents an iterative algorithm for optimizing these approximations, highlights the mathematical challenges involved, and explores the implications for multi-configuration time-dependent Hartree-Fock (MCTDHF) calculations. Additionally, it touches on ground state structures and entanglement measures associated with fermions in one-dimensional lattices.
Optimal multi-configuration approximation of an N-fermion wave function
- 1. Optimal multi-configuration approximation of an N-fermion wave function Jiang-min Zhang In collaboration with Marcus Kollar December 12, 2013
- 2. Outline A basic (but surprisingly overlooked) problem How to approximate a given fermionic wave function with Slater determinants A simple iterative algorithm converges monotonically and thus definitely easily parallelized Some analytic results Mathematically interesting and challenging Multi-configuration time-dependent Hartree Fock Spinless fermions in 1D
- 3. A Basic Problem The simplest type of wave function of a fermionic system is the Slater wave function: f(x1, x2, . . . , xN ) = 1 √ N! φ1(x1) φ1(x2) · · · φ1(xN ) φ2(x1) φ2(x2) · · · φ2(xN ) · · · · · · · · · · · · φN (x1) φN (x2) · · · φN (xN ) . But not every fermionic wave function is in the Slater form: f(x1, x2) = 1 3 φ1(x1) φ1(x2) φ2(x1) φ2(x2) + 1 6 φ3(x1) φ3(x2) φ4(x1) φ4(x2) = 1 2 ψ1(x1) ψ1(x2) ψ2(x1) ψ2(x2) . A natural question in the spirit of approximation: What is the best Slater approximation of a given fermionic wave function?
- 4. Significance of the Question Mathematically a very interesting and very challenging problem Like the celebrated “N-representability” problem Geometric measure of entanglement in many-body systems of identical particles The most widely used entanglement measure is based on the Schmidt decomposition f(x1, x2, . . . , xN ) = j λjψj(x1)Ψj(x2, . . . , xN ). The N indistinguishable particles are split into two parts artificially; Indistinguishable particles treated as distinguishable! A slater wave function is an entangled state! How strong is the correlation between the electrons? Distance from a free-particle system Basis of multi-configuration time-dependent Hartree Fock (MCTDHF). K. Byczuk, et al., Phys. Rev. Lett. 108, 087004 (2012). P. Thunstr¨om, et al., Phys. Rev. Lett. 109, 186401 (2012).
- 5. Mathematical Formulation (the single-configuration case) N fermions are distributed in L ≥ N orbitals. Given a wave function f, f(. . . , xp, . . . , xq, . . .) = −f(. . . , xq, . . . , xp, . . .), 1 ≤ xi ≤ L, find N orthonormal single-particle orbitals φi (1 ≤ i ≤ N) to construct a Slater determinant wave function S(x1, . . . , xN ) = 1 √ N! detAN×N , Aij = φi(xj), so that the overlap between f and S I ≡ | f|S |2 = N! dx1 · · · dxN f∗ (x1, · · · , xN )φ1(x1)φ2(x2) · · · φN(xN ) 2 is maximized. A crucial feature: Each orbital appears only once!
- 6. Mathematical Formulation (the multi-configuration case) N orbitals might be insufficient! Take M > N orbitals. Out of {φ1, φ2, . . . , φM }, CN M Slater determinants can be constructed, SJ ∝ φj1 ∧ φj2 ∧ . . . ∧ φjN , with J being an N-tuple J ≡ (j1, j2, . . . , jN ), 1 ≤ j1 < j2 . . . < jN ≤ M. Maximize the projection of f on the subspace spanned by the SJ ’s, I = J |ηJ |2 , with ηJ ≡ f|SJ = √ N! dx1 · · · dxN f∗ (x1, · · · , xN )φj1 (x1)φj2 (x2) · · · φjN (xN ).
- 7. An “educated” idea Suppose one needs to maximize function f(α, β, γ), α, β, γ ∈ R. An idea based on calculus: h1(α, β, γ) ≡ ∂f ∂α = 0, h2(α, β, γ) ≡ ∂f ∂β = 0, h3(α, β, γ) ≡ ∂f ∂γ = 0. not object-oriented: only stationary, not maximal complicated nonlinear equations to solve even more complicated in case of constraints K. J. H. Giesbertz, Chemical Physics Letters 591, 220 (2014).
- 8. A “less-educated” idea (walking upstairs) A middle-school student’s idea: fix β and γ to get a function fβ,γ(α) ≡ f(α, β, γ). Maximize it with respect to α. ⇒ f ↑. fix α and γ, maximize f with respect to β. ⇒ f ↑. fix α and β, maximize f with respect to γ. ⇒ f ↑. Repeat the procedure above. The value of f ↑ all the way. Two important factors to take into account: fβ,γ(α) should be easy to maximize pitfalls of local maxima (solution: multiple runs with random initial values)
- 9. Illustration in the two-fermion case (N = 2) For a given wave function f(x1, x2) = −f(x2, x1), try to find two orthonormal single-particle orbitals {φ1, φ2}, so that the Slater determinant S(x1, x2) = 1√ 2 (φ1(x1)φ2(x2) − φ2(x1)φ1(x2)) approximates f best. Equivalently, maximize the absolute value of I ≡ dx1dx2f∗ (x1, x2)S(x1, x2) = √ 2 dx1dx2φ1(x1)f∗ (x1, x2)φ2(x2) = dx1φ1(x1)g∗ 1(x1) g1(x1) ≡ √ 2 dx2f(x1, x2)φ∗ 2(x2) = dx2φ2(x2)g∗ 2(x2) g2(x2) ≡ √ 2 dx1f(x1, x2)φ∗ 1(x1) The procedure: Carry out the two steps alternatively fix φ2 (and calculate g1) and update φ1 as φ1 ∝ g1 fix φ1 (and calculate g1) and update φ2 as φ2 ∝ g2 Luckly, φ1 ⊥ φ2 is satisfied automatically!
- 10. Trial I: a ring state Consider such a state with N = 3 fermions in L = 6 orbitals: f = 1 √ 3 (|123 + |345 + |561 ), |ijk ≡ a† i a† ja† k|vac For the single-configuration approximation (N = M = 3), analytically Imax = 4/9 = 0.44444 . . . a transitory plateau at I = 1/3
- 11. Trial II: another ring state Consider such a state with N = 4 fermions in L = 9 orbitals: f = 1 √ 2 |1234 + 1 √ 3 |4567 + 1 √ 6 |7891 , For the single-configuration approximation (N = M = 4), analytically Imax = 1/2 = 0.5 local maxima at I = 1/3 and I = 1/6.
- 12. Some Analytic Results I N-fermions in L-orbitals, approximated using M orbitals: If L = N + 1, the wave function must be a Slater determinant If M = L − 1, just drop the least occupied natural orbital If N = 2, for fermions, the wave function has the canonical form f(x1, x2) = α Cα 2 (ψ2α−1(x1)ψ2α(x2) − ψ2α(x1)ψ2α−1(x2)), with α Cα = 1, and {ψi} being the natural orbitals. Take the M most occupied natural orbitals Let λi be the occupation of the ith natural orbital, λi ≥ λi+1, Imax = 1 N M i=1 λi, N = 2, Imax ≤ 1 N M i=1 λi, N ≥ 3.
- 13. Some Analytic Results II (single-configuration) f = a|12 . . . N + b|N + 1, N + 2, . . . , 2N , N ≥ 2, Imax = max(|a|2 , |b|2 ). f = a|12 . . . N + b|N, N + 2, . . . , 2N , N ≥ 3, Imax = max(|a|2 , |b|2 ). A always occupied orbital can be factorized away Two together-going orbitals allow breaking down the wave function into two parts f = 1 2 (|123 + |145 + |256 + |346 ), Imax = 1 2 . f = 1√ 6 (|123 + |234 + |345 + |456 + |561 + |612 ), Imax = 3 4 . f = a|123 + b|345 + c|561 , Imax ≥ 4 9 . The equality is achieved when and only when |a|2 = |b|2 = |c|2 = 1 3 . A conjecture: min Imax = 4/9 for (N, M, L) = (3, 3, 6).
- 14. 1D Spinless Fermions (e.g., spin-polarized electrons) N spinless fermions on an L-site 1D lattice, with nearly-neighbor-interaction, and open boundary condition, ˆH = L−1 i=1 −(ˆc† i ˆci+1 + ˆc† i+1ˆci) + U ˆniˆni+1. Ground state structure of the ground state Time-evolving state after a quantum quench: Initially the fermions are confined to the Li sites on the left end and then suddenly released into the whole lattice. check the algorithm of Multi-configuration time-dependent Hartree Fock (MCTDHF)
- 15. Ground state (repulsive interaction U > 0) 5 10 15 20 25 0.8 0.85 0.9 0.95 1 L Imax (a) N = M = 5U=1 U=3 U=5 10 15 20 25 0.8 0.85 0.9 0.95 1 L Imax (b) N = M = 6U=1 U=2 U=4 Important features: L = N and L = N + 1, Imax = 1 irrespective of U. L → +∞, Imax → 1. In the large-U limit, a local maximum develops at L = 2N − 1. charge-density-wave: The N fermions reside every other lattice site.
- 16. Ground state (attractive interaction U < 0) 30 60 90 120 150 0.6 0.7 0.8 0.9 1 L Imax (a) N = M = 2 U=−1.95 U=−2 U=−2.05 20 40 60 80 100 0.4 0.5 0.6 0.7 0.8 0.9 1 L Imax (b) N = M = 3 U=−1.95 U=−2 U=−2.05 Bifurcation at Uc = −2: |U| < |Uc|, no bound fermionic pair formation |U| > |Uc|, bound fermionic pair formation Profile of |f(x1, x2)| at U = −3
- 17. Multi-configuration time-dependent Hartree Fock Time evolution of a many-body system is difficult! The conventional approach: Time-independent basis, chosen a priori |ψ(t) = J CJ (t)eJ , with eJ being a many-body basis vector constructed out of time-independent single-particle orbitals. Hilbert space exponentially large! Now a very smart idea: Adaptively chosen basis, |ψ(t) J CJ (t)eJ (t), with eJ (t) constructed out of time-dependent single-particle orbitals. significantly diminished Hilbert space!
- 18. Multi-configuration time-dependent Hartree Fock For N spinless fermions on an L-site lattice, M L time-dependent single-particle orbitals are taken, {φ1(t), φ2(t), . . . , φM (t)}, out of which ˜D = M! N!(M−N)! Slater determinants SJ (t) can be constructed. The variational wave function is |ψ(t) = J CJ (t)SJ (t). Evolution of the coefficients CJ (t) and the orbitals φi(t) is determined by the Dirac-Frenkel variational principle δ dt i ψ| ∂ ∂t ψ − ψ|H|ψ = 0. A natural question: can the wave function really be well approximated by using only M L orbitals?
- 19. Evolution of Imax Initially N = 3 fermions are confined to the Li left-most sites (|ψ(t) first evolved by ED, and then approximated using the algorithm) Two different cases (U = 1): 0 0.2 0.4 0.6 0.8 1 Imax (a) N = 3, Li = 3, L = 25 0 5 10 15 20 25 30 0 1 2 t Eint/U (c) 0.7 0.8 0.9 1 Imax (b) N = 3, Li = 5, L = 25 0 50 100 150 0 0.5 t Eint/U (d) From bottom to top, M increases from 3 to 8. Line M = 3 coincides with line M = 4. Reduction of Hilbert space: C3 25 = 2300 to C3 8 = 56.
- 20. Evolution of density distribution—comparison of ED and MCTDHF Initially N = 3 fermions are confined to the Li left-most sites (U = 1) 0 0.2 0.4 ˆni (a1) t = 5 0 0.1 0.2 (a2) t = 10 0 10 20 0 0.1 0.2 0.3 0.4 i ˆni (a3) t = 15 0 10 20 0 0.1 0.2 0.3 i (a4) t = 20 0 0.1 0.2 0.3 0.4 ˆni (b1) t = 5 0 0.1 0.2 (b2) t = 10 0 10 20 0 0.2 0.4 0.6 iˆni (b3) t = 15 0 10 20 0 0.1 0.2 0.3 i (b4) t = 20 Blue line with circles: ED results Red line with squares: MCTDHF with M = 3 orbitals Green line with aestrisks: MCTDHF with M = 8 orbitals
- 21. Conclusions and outlooks A problem relevant in the MCTDHF context Numerically, a simple iterative algorithm is proposed a quantitative approach to geometric measure of entanglement but the idea is inapplicable to bosons! Analytically, several scattered nontrivial results have been obtained A lot of open questions and conjectures MCTDHF gauged (checked). An immediate problem: For the Laughlin wave function (zi = xi + iyi) f(z1, z2, . . . , zN ) = 1≤zi<zj ≤N (zi − zj)3 N i=1 exp(−|zi|2 ), how does Imax scale with N (assuming M = N)? J. M. Zhang and Marcus Kollar, arXiv:1309.1848 (2013).