Fermionic tensor networks Philippe Corboz, ETH Zurich / EPF Lausanne, Switzerland Collaborators: University of Queensland: Guifre Vidal, Roman Orus, Glen Evenbly, Jacob Jordan University of Vienna: Frank Verstraete ETH/EPFL: Bela Bauer, Matthias Troyer, Frederic Mila Acknowledgments: Luca Tagliacozzo, Robert Pfeifer P. Corboz, G. Evenbly, F. Verstraete, G. Vidal, PRA 81, 010303(R) (2010) P. Corboz, G. Vidal, Phys. Rev. B 80, 165129 (2009) P. Corboz, R. Orus, B. Bauer, G. Vidal. PRB 81, 165104 (2010) P. Corboz, J. Jordan, G. Vidal, arxiv:1008:3937
Attack the sign problem sign problem cost exp( N k B T )
Overview: tensor networks in 1D and 2D 1D MPS Matrix-product state 1D MERA Multi-scale entanglement renormalization ansatz 1 2 3 4 5 6 7 8 Related to the famous densitymatrix renormalization group (DMRG) method i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 i 17 i 18 and more 1D tree tensor network... 2D (i)peps (infinite) projected pair-entangled state 2D MERA and more 2D tree tensor network String-bond states Entangledplaquette states...
Fermions in 2D & tensor networks Simulate fermions in 2D? Before April 2009: NO! Since April 2009: YES! P. Corboz, G. Evenbly, F. Verstraete, G. Vidal, arxiv:0904:4151 C. V. Kraus, N. Schuch, F. Verstraete, J. I. Cirac, arxiv:0904.4667 C. Pineda, T. Barthel, J. Eisert, arxiv:0905.0669 P. Corboz, G. Vidal, Phys. Rev. B 80, 165129 (2009) T. Barthel, C. Pineda, J. Eisert, PRA 80, 042333 (2009) Q.-Qian Shi, S.-Hao Li, J-Hui Zhao, H-Qiang Zhou, arxiv:0907.5520 P. Corboz, R. Orus, B.Bauer, G. Vidal, PRB 81, 165104 (2010) S.-Hao Li, Q.-Qian Shi, H-Qiang Zhou, arxiv:1001:3343 I. Pizorn, F. Verstraete. arxiv:1003.2743 Z.-C. Gu, F. Verstraete, X.-G. Wen. arxiv:1004.2563...
Outline Short introduction to tensor networks Idea: efficient representation of quantum many-body states Examples: Tree Tensor Network, MERA, MPS, PEPS Fermionic systems in 2D & tensor networks Simple rules Computational cost compared to bosonic systems Results (ipeps) & comparison with other methods Free and interacting spinless fermions t-j model Summary: What s the status?
Tensor networks: Efficient representation of QMB states 6 site system Ψ = Tensor/multidimensional array Big tensor i 1 i 2 i 3 i 4 i 5 i 6 Ψ i1 i 2 i 3 i 4 i 5 i 6 i 1 i 2 i 3 i 4 i 5 i 6 D j1 j 2 j 3 j 1 j 2 j 3 possible?? B j 2 i 3 i 4 C j 3 A j 1 Ψ i1 i 2 i 3 i 4 i 5 i 6 i 5 i 6 i 1 i 2 i 3 i 4 i 5 i 6 Ψ i1 i 2 i 3 i 4 i 5 i 6 Ψ ~exp(n) many numbers vs i1 i 2 Ṽ i 1 i 2 i 3 i 4 i 5 i 6 Entanglement entropy General state S(L) L D Why is this (volume) Contraction: A j1 i1 i 2 B j 2 i 3 i 4 C j 3 j 1 j 2 j 3 Ψ Ψ poly( χ,n) numbers {, } dimension χ χ different states tensor network i 5 i 6 D j1 j 2 j 3 = Ψ i1 i 2 i 3 i 4 i 5 i Ground state (local Hamiltonian) 6 S(L) L D 1 Efficient (area law) representation! Note: Some (critical) ground states exhibit a logarithmic correction to the area law
Tree Tensor Network (1D) isometry L 2 τ coarsegrained lattice χ L 1 coarsegrained lattice L 0 microscopic lattice
The MERA (The Multi-scale Entanglement Renormalization Ansatz) G. Vidal, PRL 99, 220405 (2007), PRL 101, 110501 (2008) # relevant local states χ disentangler isometry L 2 τ coarsegrained lattice χ L 1 coarsegrained lattice χ 0 = d L 0 microscopic lattice KEY: Disentanglers reduce the amount of short-range entanglement Efficient ansatz for critical and non-critical systems in 1D
2D MERA (top view) Evenbly, Vidal. PRL 102, 180406 (2009) Original lattice Apply disentanglers Accounts for arealaw in 2D systems S(L) L χ τ = const Coarse-grained lattice Apply isometries
2D MERA represented as a 1D MERA Typical network ρ (lower half) 7 8 9 4 5 6 1 2 3 1 2... 9 w 1 w 2 w 3 w 4 u x u y u 3 4 1 2 1 2 3 4 1 2 3 4 1 2 2 1 1 2 1 2 3 4 1 2 3 4 1 2 H 1 2 3 4 3 4 1 2 1 2 3 4 1 2 3 4 ρ (upper half) Crossing lines play an important role for fermions!
MPS & PEPS 1D MPS 2D (i)peps Matrix-product state (Related to DMRG) D (infinite) projected pair-entangled state D 1 2 3 4 5 6 7 8 Physical indices (lattices sites) S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995) F. Verstraete, J. I. Cirac, cond-mat/0407066 Reproduces area-law in 1D S(L) =const Reproduces area-law in 2D S(L) L
Summary: Tensor network algorithms Structure (ansatz) Find the best (ground) state Ψ Compute observables Ψ O Ψ iterative optimization of individual tensors (energy minimization) imaginary time evolution Contraction of the tensor network exact / approximate
Bosons vs Fermions Ψ B (x 1,x 2 )=Ψ B (x 2,x 1 ) symmetric! Ψ F (x 1,x 2 )= Ψ F (x 2,x 1 ) antisymmetric! ˆbiˆbj = ˆb jˆbi operators commute ĉ i ĉ j = ĉ j ĉ i operators anticommute Crossings in a tensor network + ----=+ ignore crossings take care!
The swap tensor # Fermions even even odd even even odd odd odd Parity RULE: + + + -- Parity P of a state: { P = +1 P = 1 (even parity), even number of particles (odd parity), odd number of particles Replace crossing by swap tensor B i 1 i 2 j 2 j 1 B i 1i 2 j 2 j 1 = δ i1,j 1 δ i2,j 2 S(P (i 1 ),P(i 2 )) S(P (i 1 ),P(i 2 )) = 1 +1 if P (i 1 )=P (i 2 )= 1 otherwise Use parity preserving tensors: T i1 i 2...i M =0 if P (i 1 )P (i 2 )...P(i M ) = 1
Example Bosonic tensor network Fermionic tensor network EASY!!!
Fermionic operator networks State of 4 site system Ψ = i 1 i 2 i 3 i 4 Ψ i1 i 2 i 3 i 4 i 1 i 2 i 3 i 4 { 0, 1} Bosons Tensor Ψ i1 i 2 i 3 i 4 j 1 j 2 A j 1 i1 i 2 D j1 j 2 B j 2 i 3 i 4 bosonic tensor network Contract i 1 i 2 i 3 i 4 i 1 i 2 i 3 i 4 Fermions Tensor + fermionic ops ˆΨ i 1 i 2 i 3 i 4 i 1 i 2 i 3 i 4 ˆD fermionic j 1 j 2 operator  ˆB network Contract + operator calculus  = A j 1 i1 i 2 i 1 i 2 j 1 = A j 1 i1 i 2 ĉ i 1 1 ĉ i 2 2 00 ĉj 1 1
Fermionic operator network Use anticommutation rules to evaluate fermionic operator network: Easy solution: Map it to a tensor network by replacing crossings by swap tensors
Cost of fermionic tensor networks?? ρ (lower half) w 1 w 2 w 3 w 4 u x u y u First thought: Many crossings many more tensors larger computational cost?? H NO! Same computational cost ρ (upper half)
The jump move = Jumps over tensors leave the tensor network invariant Follows form parity preserving tensors [ ˆT,ĉ k ]=0, if k/ sup[ ˆT ] Allows us to simplify the tensor network Final cost is the same as in a bosonic tensor network
Example of the jump move absorb now contract as usual! absorb
Message: Taking fermionic statistics into account is easy! Replace crossings by swap tensors & use parity preserving tensors Computational cost does not depend a priori on the particle statistics, but on the amount of entanglement in the system!
Computational cost Leading cost: O(D k ) MPS: k =3 PEPS: k 10...12 2D MERA: k = 16 polynomial scaling but large exponent! How large does have to be? D It depends on the amount of entanglement in the system! χ Bond dimension: D i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 i 17 i 18
Classification by entanglement Entanglement low high gapped systems gapless systems systems with 1D fermi surface S(L) L log L band insulators, valence-bond crystals, s-wave superconductors,... Heisenberg model, p-wave superconductors, Dirac Fermions,... free Fermions, Fermi-liquid type phases, bose-metals?
Overview: Results / benchmarks Free spinless fermions Finite systems (MERA, TTN) Finite systems (PEPS) Infinite systems (ipeps) Interacting spinless fermions Finite systems (MERA, TTN) Finite systems (PEPS) Phase diagram of t-v model (ipeps) t-j model Benchmark (ipeps) Phase diagram (ipeps) Corboz, Evenbly, Verstraete, Vidal, PRA 81, 010303(R) (2010), Corboz, Vidal, PRB 80, 165129 (2009) Pineda, Barthel, Eisert, arxiv:0905.0669 Kraus, Schuch, Verstraete, Cirac, arxiv:0904.4667 Pizorn, Verstraete. arxiv:1003.2743 Z.-C. Gu, F. Verstraete, X.-G. Wen. arxiv: 1004.2563 Corboz, Orus, Bauer, Vidal, PRB 81, 165104 (2010) Shi, Li, Zhao, Zhou, arxiv:0907.5520 Li, Shi, Zhou, arxiv:1001:3343
Non-interacting spinless fermions: infinite systems (ipeps) H free = rs [c rc s + c sc r γ(c rc s + c s c r )] 2λ r c rc r 4 3 10 2 γ 2 critical gapped Relative error of energy 10 3 10 4 10 5 10 6 10 7 γ=0 D=2 γ=0 D=4 γ=1 D=2 γ=1 D=4 γ=2 D=2 γ=2 D=4 D: bond dimension 1 1D Fermi surface 0 0 1 2 3 4 λ Li et al., PRB 74, 073103 (2006) 1 1.5 2 2.5 3 λ fast convergence with D in gapped phases slow convergence in phase with 1D Fermi surface
Correlators C(r) 0 0.05 0.1 γ=1, λ=1 (critical) exact D=2 D=4 D=6 5 0 5 x 10 3 γ=1, λ=3 (gapped) C(r) C(r,D) C ex (r) 0 2 4 6 8 10 10 0 10 1 10 2 10 3 10 4 10 5 10 2 10 3 10 4 0 2 4 6 8 10 0 2 4 6 8 10 r 10 0 10 2 10 4 10 6 10 8 10 10 10 3 10 4 10 5 10 6 10 7 10 8 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 r
Phase diagram of interacting spinless fermions (ipeps) Ĥ = t i,j ĉ i ĉj + H.c. µ i ĉ i ĉi + V i,j ĉ i ĉiĉ jĉj Restricted Hartree-Fock (HF) results: Woul&Langmann. J. Stat Phys. 139, 1033 (2010) metal (gapless) charge-density-wave (CDW, gapped) metal (gapless) 1st order PT 1st order PT metal (gapless) phase separation CDW phase separation metal (gapless) 0 0.5 1 n (filling)
ipeps vs Hartree-Fock Woul, Langmann. J. Stat Phys. 139, 1033 (2010) Corboz, Orus, Bauer, Vidal. PRB 81, 165104 (2010) 3 unstable phase separation region unstable phase separation region HF D=4 D=6 V 2 CDW line 1 metal metal 0 0 0.2 0.4 0.6 0.8 1 n qualitative agreement Phase boundary moves away from HF result with increasing D
ipeps vs Hartree-Fock Woul, Langmann. J. Stat Phys. 139, 1033 (2010) Corboz, Orus, Bauer, Vidal. PRB 81, 165104 (2010) 0.4 0.6 V =2 crossing of the energy of the two phases E s 0.8 1 1.2 HF metal HF CDW D=4 metal D=4 CDW D=6 metal D=6 CDW 1.4 0 0.5 1 1.5 2 µ D=4 result ~ HF result D=6: lower (better) energies in the metal phase
Adding a next-nearest neighbor hopping Ĥ = t i,j ĉ i ĉj + H.c. µ i ĉ i ĉi + V i,j ĉ i ĉiĉ jĉj Corboz, Jordan, Vidal, arxiv:1008:3937 t i,j ĉ iσĉjσ + H.c. Hartree-Fock phase diagram: Woul, Langmann. J. Stat Phys. 139, 1033 (2010) V =2 a) t = 0.4 metal PS CDW 0 0.274(1) 0.368(1) 0.5 n metal PS CDW Is b) there a stable doped CDW phase beyond Hartree-Fock? n 0 0.308 0.343 0.350 0.5 D=4 D=6 D=8 NO... (at least not for these parameters)
0.5 Adding a next-nearest neighbor hopping E s <n> S 2 2 0.4 0.2 0.3 0.4 3 0.6 2 0.8 1 HF metal HF CDW D=4 metal D=4 CDW D=6 metal D=6 CDW D=8 metal D=8 CDW Corboz, Jordan, Vidal, arxiv:1008:3937 n * =0.350(3) n * =0.343(3) doped CDW region n * =0.307(3) 0.5 0 0.5 1 µ
t-t -J model H t-j = t ijσ c iσ c jσ t ijσ c iσ c jσ + J ij(s i S j 1 4 n in j ) µ i n i Comparison with: L. Spanu, M. Lugas, F. Becca, S. Sorella. PRB 77, 024510 (2008). variational Monte Carlo (VMC) (Gutzwiller projected ansatz wf) state-of-the-art fixed node Monte Carlo (FNMC) 1 1.2 J/t =0.4, t /t = 0.2 1.4 E s 1.6 1.8 2 2.2 How about striped phases?? need larger unit cell!! D=2 D=4 D=6 D=8 VMC N=98 VMC N=162 FNMC N=98 FNMC N=162 D=8 results in between VMC and FNMC 0.9 0.92 0.94 0.96 0.98 1 n Corboz, Jordan, Vidal, arxiv:1008:3937
Summary: status This is useless! D 12 scaling is as bad as exponential! YES, we have the holy grail! We can now solve everything!
Summary: status Variational ansatz with no (little) bias & controllable accuracy. Accuracy depends on the amount of entanglement in the system Accurate results for gapped systems Competitive compared to other variational wave functions Systematic improvement upon mean-field solution - For which bond dimension is it converged? - Limited accuracy for gapless systems - no L log L scaling - High computational cost Combine with Monte Carlo sampling (Schuch et al, Sandvik&Vidal, Wang et al.) Exploit symmetries of a model (Singh et al, Bauer et al.) Improve optimization/contraction schemes