Time-dependent variational principle for quantum many-body systems

Transcription

1 Quantum Information in Quantum Many-Body Physics October 21, 2011 Centre de Recherches Mathematiques, Montréal Time-dependent variational principle for quantum many-body systems PRL 107, (2011) arxiv: arxiv: Jutho Haegeman, in collaboration with: Iztok Pižorn, Tobias J. Osborne, J. Ignacio Cirac, Frank Verstraete, Henri Verschelde

2 Variational Principle (time-independent: ground state) Ψ(z) Ĥ Ψ(z) E 0 min z Ψ(z) Ψ(z) Non-perturbative No sign problems Accuracy depends on adequacy of variational manifold = Ψ(z), z 2/25

3 Variational Principle (time-independent: ground state) Good variational ansätze for quantum many body systems?! Role of entanglement! Quantum lattice systems: (cf. PC s talk) " MPS, PEPS, MERA Quantum field theories: (cf. TJO s talk) " cmps, cmera 3/25

4 Variational Principle (time-independent: ground state) Good variational ansätze for quantum many body systems?! Role of entanglement! Quantum lattice systems: (cf. PC s talk) " MPS, PEPS, MERA Quantum field theories: (cf. TJO s talk) " cmps, cmera 3/25

5 Time-dependence? Quantum lattice systems: " Time-Evolving Block Decimation G. Vidal, PRL 91, (2003); PRL 93, (2004). Based on Lie-Trotter decomposition 1D: MPS, 2D: PEPS Lattices of finite and infinite size 4/25

6 Time-dependence? Quantum lattice systems: " Time-Evolving Block Decimation Breaks either translation invariance or internal symmetries State temporarily leaves manifold Truncation to original manifold is suboptimal for infinite-sized systems Manifestly discrete in time 5/25

7 Time-dependence? Quantum lattice systems: " Time-Evolving Block Decimation Breaks either translation invariance or internal symmetries State temporarily leaves original manifold How to implement for cmps? 6/25

8 Time-dependent variational principle iĥ Ψ(z) Dirac, 1930 Ψ(z) Exact time evolution in Hilbert space: (z) d dt Ψ(t) = iĥ Ψ(t) Projection onto the tangent plane : (z) i Ψ z(t) j Ψ z(t) ż j (t)= i i Ψ z(t) Ĥ Ψ z(t) 7/25

9 Time-dependent variational principle TDVP evolution never leaves : Linear differential equation in large space Non-linear differential equation in Inherently respects all symmetries: no Trotter decomposition! no Trotter error TDVP! = symplectic manifold Globally optimal, no truncation Time-dependent Hartree-Fock, Gross-Pitaevskii 8/25

10 TDVP for uniform MPS Can we apply TDVP to MPS efficiently? Uniform matrix product states: O(D 3 ) Ψ(A) =... A A A A A A A A A A... Tangent vectors: B i i Ψ(A) = Φ(B) (i = 1,..., dd 2 ) Φ(B) = A A A A A A A A A B... A A A A A A A A A B... A A A A A A A A A B /25

11 TDVP for uniform MPS Overlap of tangent vectors! metric of Φ(B) Φ( B) = B i i Ψ(A) j Ψ(A) B j Hopelessly complex? And we need to invert this d#d² d#d² matrix i Ψ z(t) j Ψ z(t) ż j (t)= i i Ψ z(t) Ĥ Ψ z(t)! Simplification through gauge fixing prescription 10/25

12 TDVP for uniform MPS Gauge invariance of MPS: Ψ(A) = Ψ(A G ) A G = G A G 1 (multiplicative) Gauge invariance of tangent vectors: Φ(B) = Φ(B + B X ) = Φ(B) + Φ(B X ) = 0 B = X X A A X (additive) 11/25

13 TDVP for uniform MPS! Gauge fixing prescription: fix D² components B l = 0 A or B A r = 0 with l A A A = l and r = r A 12/25

14 TDVP for uniform MPS! Gauge fixing prescription: fix D² components B B l = 0 or r = 0 A B A! Φ(B) Φ(B ) l r B! Now we can find a parameterisation of B such that the effective metric is the unit matrix. 13/25

15 TDVP for uniform MPS Imaginary-time evolution: Simple Euler update Translation invariant approximation of ground states with great accuracy Real-time evolution: Symmetric second-order implementation Accurate conservation of constants of motion 14/25

16 TDVP results Imaginary-time MPS example: S=1 Heisenberg AFM: Ĥ = n Ŝ x n Ŝ x n+1 + Ŝ y n Ŝ y n+1 + Ŝ z n Ŝ z n+1 First Schmidt coefficients for a ground state approximation with bond dimension D=128: S=1/2 S=3/2 S=5/2 15/25

17 TDVP results Real-time MPS example: S=1/2 Heisenberg AFM: Ĥ = n ˆσ x n ˆσ x n+1 + ˆσ y n ˆσ y n+1 + ˆσ z n ˆσ z n+1 Starting from initial MPS with Ψ(0) ˆσ z Ψ(0) = 0 Ψ(0) ˆσ x,y Ψ(0) = 0 h(a(t), A(t)) σ z (A(t), A(t))!1.66! σ x (A(t), A(t)) 2"10!11 0!2"10!11 TDVP TEBD t 16/25

18 TDVP results Imaginary-time cmps example: Ĥ = dx d ˆψ dx (Lieb-Liniger model) (x)d ˆψ dx (x) µ ˆψ (x) ˆψ(x)+g ˆψ (x) ˆψ (x) energy density e/ρ 3 error on energy density (a) (b) !2 10!4 10!6 10!8 D = 2 D = 4 D = 7 D = 14 D = 25 D = interaction strength γ= g/ρ ˆψ(x) ˆψ(x) 17 /25

19 Linearized TDVP Variational optimum satisfies Φ(B) Ĥ Ψ(A ) B i i E(A )=0! Steady state solution of TDVP equations Linearize TDVP around ground state: A(t)=A + ηb(t)! First order linear differential equation in B(t) and B(t) : B(t)=B + e iωt + B e +iωt 18/25

20 Linearized TDVP ω i Ψ(A ) j Ψ(A ) 0 j B+ 0 i Ψ(A ) j Ψ(A j ) B = i Ψ(A ) Ĥ j Ψ(A ) i j Ψ(A ) Ĥ Ψ(A ) j B+ i j Ψ(A ) Ĥ Ψ(A ) i Ψ(A ) Ĥ j j Ψ(A ) B 19/25

21 Linearized TDVP ω i Ψ(A ) j Ψ(A ) 0 j B+ 0 i Ψ(A ) j Ψ(A j ) B = i Ψ(A ) Ĥ j Ψ(A ) i j Ψ(A ) Ĥ Ψ(A ) j B+ i j Ψ(A ) Ĥ Ψ(A ) i Ψ(A ) Ĥ j j Ψ(A ) B! ω i Ψ(A ) j Ψ(A ) B j = i Ψ(A ) Ĥ j Ψ(A ) B j Application of variational principle in linear subspace (A ). (Rayleigh - Ritz equations) 19/25

22 Ansatz for excitations Φ(B) = +... B... B B /25

23 Ansatz for excitations Φ(B) = e ik(n 1)... B... + e ikn... B... + e ik(n+1)... B ! Ansatz for excitations with momentum k Combines: Östlund - Rommer ansatz Single-mode approximation (Feynman-Bijl ansatz) 20/25

24 Ansatz for excitations: MPS results S=1 Heisenberg AFM: Ĥ = n Ŝ x n Ŝ x n+1 + Ŝ y n Ŝ y n+1 + Ŝ z n Ŝ z n+1 D = 30 : Efficient implementation: D = 208 excitation energy ω / Δ Haldane π/4 π/2 3π/4 π momentum k S=0 S=1 S=2 S=3 Haldane = /25

25 Ansatz for excitations: cmps results Lieb-Liniger (cmps approximation with D = 33) (b) π 2π momentum p / ρ 3π energy ΔE / ρ2 energy ΔE / ρ2 3 (c) 8 energy ΔE / ρ2 (a) 0 0 π 2π 3π momentum p / ρ 0 π 2π 3π momentum p / ρ increasing! = g / " 22 /25

26 Ansatz for excitations: topologically non-trivial Ξ(B) = e ik(n 1)... B... + e ikn... B... + e ik(n+1)... B ! Ansatz for domain-walls with momentum k Contains: Mandelstam ansatz 23/25

27 Ansatz for excitations: topologically non-trivial XXZ model: H = excitation energy energy gap ΔXXZω (a) n x x σ n σ n+1 y y + σ n σ n+1 z z + σ n σ n+1 1 (b) y ~ x1/2 10! y~x !0.050 Δ= 1.6 Δ= 1.3 Δ= 1.2 Δ= 1.1 Δ= π 10!4 10!6 ( ) - Δ(D) error on gap ΔXXZ XXZ !8 1.2!2π/ π/ !10 π/3 2π/3 momentum anisotropy parameter Δ error onk energy density e(d) - e( ) XXZ XXZ π 24 /25

28 Ansatz for excitations: topologically non-trivial XXZ model: Ĥ = n ˆσ x n ˆσ x n+1 + ˆσ y n ˆσ y n+1 + ˆσ z n ˆσ z n+1 energy gap Δ XXZ !0.05 (a) anisotropy parameter Δ (b) 10!10 y ~ x 1/2 Δ= 1.6 Δ= 1.3 Δ= 1.2 Δ= 1.1 Δ= 1 y ~ x error on energy density e (D) XXZ - e( ) XXZ 1 10!2 10!4 10!6 10!8 error on gap Δ ( ) XXZ - Δ(D) XXZ 24/25

29 Conclusions: Systematic framework for time evolution and excitations by exploiting the (c)mps tangent plane Efficient implementation possible! Symmetry inherently respected!what about PEPS, (c)mera,...? 25/25

30 Thank you! Questions?