Fermionic tensor networks

Transcription

1 Fermionic tensor networks Philippe Corboz, Institute for Theoretical Physics, ETH Zurich Bosons vs Fermions P. Corboz and G. Vidal, Phys. Rev. B 80, (2009) : fermionic 2D MERA P. Corboz, R. Orus, B.Bauer, G. Vidal, PRB 81, (2010) : fermionic PEPS

2 Outline Reminder: Calculus with fermionic operators Jordan Wigner transformation: 1D vs 2D Fermionic tensor networks Mapping a fermionic operator network onto a familiar tensor network Simple, intuitive rules Same computational cost Benchmarks Free Fermions are the hardest case! Application: the t-j model Infinite PEPS can compete with other variational approaches Summary

3 Introduction to Fermions Lattice with N sites Fermionic creation annihilation ops N-1 N ĉ i ĉ i ĉ i 0 = ĉ i = 0 V : local Hilbert space site i { 0, 1 } Anticommutation relations {ĉ i, ĉ j } =ĉ i ĉ j +ĉ j ĉ i =0, n o ĉ i, ĉ j = ij ĉ iĉ j = ĉ j ĉ i ĉ i ĉ i 0 = ĉ i ĉ i 0 =0 n i {0, 1} Basis states n 1,...,n N =(ĉ 1 )n 1 (ĉ 2 )n 2...(ĉ N )n N 0 order!

4 Fermionic basis Basis states n 1,...,n N =(ĉ 1 )n 1 (ĉ 2 )n 2...(ĉ N )n N 0 Examples: ĉ 1ĉ 2 0 = 110 ĉ = 110 ĉ =0 ĉ = 001 ĉ = 011 ĉ =ĉ 3ĉ 2 0 = ĉ 2ĉ 3 0 = 011

5 Fermionic operators acting on basis states General rule: ˆP 0 = (+1) 0 ˆP 1 =( 1) 1 ĉ i n 1...n i...n N = ni,0 ˆP (1) ˆP (2)... ˆP (i 1) n 1...n i +1...n N ĉ i n 1...n i...n N = ni,1 ˆP (1) ˆP (2)... ˆP (i 1) n 1...n i 1...n N ĉ i ĉi+k...n i...n i+k... = ni,0 n i+k,1 ˆP (i + 1) ˆP (i + 2)... ˆP (i + k 1)...n i +1...n i+k 1... Jordan-Wigner transformation! ĉ i ĉi+k ˆb i ˆzi+1ˆzi+2...ˆzi+k 1 ˆb i+k Fermionic operator acting on 2 sites (Hardcore) bosonic or spin operator acting on k+1 sites

6 Parity symmetry Fermionic observables and Hamiltonians have parity symmetry [Ĥ, ˆP ]=0 [Ô, ˆP ]=0 they can be written as an even polynomial of ĉ, ĉ s A parity preserving fermionic operator commutes with any other fermionic operator if their supports are disjoint [Ô, ĉ k]=0, if k/sup[ô] ( 1) 2N = +1 ĉ 3ĉ4 ĉ k = ĉ k ĉ 3ĉ4 { Ô k/{3, 4} ( 1) 2 = +1

7 Local fermionic operators non-local parity preserving op ĉ i ĉi+k JWT ˆb i ˆzi+1ˆzi+2...ˆzi+k 1 ˆb i+k local parity preserving op ĉ i ĉi+1 JWT ˆb iˆb i+1 (no ˆz s are involved) acts in the same way as the (hardcore) bosonic operator Example: ĉ 3ĉ =ĉ 3ĉ4 ĉ 1ĉ 2ĉ 4 0 =ĉ 1ĉ 2 ĉ 3ĉ4 ĉ 4 0 = 1110 No negative sign arises when acting with a local, normal ordered, parity preserving operator on a state

8 Jordan-Wigner transformation in 1D no string of z s for nearest-neighbor operators 1D easy! ĉ 2ĉ3 ˆb 2ˆb 3 Local fermionic operators get mapped into local bosonic operators We can easily map the fermionic Hamiltonian onto a (hardcore) bosonic Hamiltonian and then solve the bosonic (spin) model

9 Jordan-Wigner transformation in 2D 2D site local operator k-site nonlocal operator ĉ 1ĉ5 ˆb 1 z 2 z 3 z 4ˆb 5 In two dimensions some local fermionic operators get mapped into non-local bosonic operators This looks complicated for large (or even infinite) 2D systems! Yes, it s possible! Corboz et al, PRA 81 (2010) Can we use a fermionic tensor network ansatz to represent fermionic wave functions directly?

10 Breakthrough in 2009: Fermions with 2D tensor networks How to take fermionic statistics into account? ĉ i ĉ j = ĉ j ĉ i fermionic operators anticommute Different formulations: P. Corboz, G. Evenbly, F. Verstraete, G. Vidal, Phys. Rev. A 81, (R) (2010) C. V. Kraus, N. Schuch, F. Verstraete, J. I. Cirac, Phys. Rev. A 81, (2010) C. Pineda, T. Barthel, and J. Eisert, Phys. Rev. A 81, (R) (2010) P. Corboz and G. Vidal, Phys. Rev. B 80, (2009) T. Barthel, C. Pineda, and J. Eisert, Phys. Rev. A 80, (2009) Q.-Q. Shi, S.-H. Li, J.-H. Zhao, and H.-Q. Zhou, arxiv: P. Corboz, R. Orus, B.Bauer, G. Vidal, PRB 81, (2010) I. Pizorn, F. Verstraete, Phys. Rev. B 81, (2010) Z.-C. Gu, F. Verstraete, X.-G. Wen. arxiv:

11 Crossings in 2D tensor networks 3x3 PEPS Ψ Ψ Example network of the 2D MERA (lower half) w 1 w 2 w 3 w 4 u x u y u Crossings appear when projecting the 3D network onto 2D! H (upper half)

12 Bosons vs Fermions B(x 1,x 2 )= B (x 2,x 1 ) symmetric! ˆbiˆbj = ˆb jˆbi operators commute F (x 1,x 2 )= F (x 2,x 1 ) antisymmetric! ĉ i ĉ j = ĉ j ĉ i operators anticommute Crossings in a tensor network =+ ignore crossings take care!

13 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 = i1,j 1 i 2,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

14 Parity symmetry Fermionic systems exhibit parity symmetry! [Ĥ, ˆP ]=0 Choose all tensors to be parity preserving! T i1 i 2...i M =0 if P (i 1 )P (i 2 )...P(i M ) =1 P P P P = Decomposing local Hilbert spaces into even and odd parity sectors even odd parity sector Label state by a composite index i =(p, p) enumerate states in parity sector p tensor with a block structure (similar to a block diagonal matrix) Easy identification of the parity of a state! Parity preserving operator T commutes with any operator K if their support is disjoint

15 Example Bosonic tensor network Fermionic tensor network EASY!!!

16 Fermionic operator networks { 0, 1 } State of 4 site system = i 1 i 2 i 3 i 4 i 1 i 2 i 3 i 4 i 1 i 2 i 3 i 4 Bosons Tensor i 1 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 ˆ 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 ihj 1 = A j 1 i1 i 2 ĉ i 1 1 ĉ i 2 2 0ih0 ĉj 1 1 0

17 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

18 Example A simple example: Ô, Ô = t ĉ 1ĉ3 = ĉ 2ĉ 3 0 Ô = t ĉ 1ĉ3 ĉ 3ĉ 2 0 { tĉ 1ĉ3ĉ = tĉ 3ĉ 2 0 = 1 I 3ĉ 2 0 = tĉ 1ĉ 2 I 3 0 =( 110 (±t) 110 = ± t t) 110 All involved anticommutations to evaluate a fermionic operator network are represented by a crossing The crossings play the key role to evaluate a fermionic tensor network

19 Cost of fermionic tensor networks?? (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 (upper half)

20 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 of contraction is the same as in a bosonic tensor network

21 Example of the jump move

22 Example of the jump move

23 Example of the jump move

24 Example of the jump move absorb absorb

25 Example of the jump move now contract as usual!

26 Fermionic (i)peps Ψ Ψ No crossings anymore Fermionic character taken into account locally! Only small modification to bosonic PEPS

27 Example of the jump move absorb absorb

28 Fermionic (i)peps: expectation values environment compute environment approximately (e.g. using MPS) Connect two-body operator insert swap tensors Contract this network!

29 Summary: Fermionic TN Simulate fermionic systems with 2D tensor networks 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)!

30 Error of energy Entanglement entropy Computational cost Leading cost: O(D k ) MPS: k =3 PEPS: k 10 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! Spinless fermion model H( ) fixed bond dimension D λ The less entangled the system, the better the accuracy!

31 Classification by entanglement (in 2D) Entanglement low high gapped systems gapless systems systems with 1D fermi surface band insulators, valence-bond crystals, s-wave superconductors,... Heisenberg model, p-wave superconductors, Dirac Fermions,... free Fermions, Fermi-liquid type phases, bose-metals S(L) L log L Free fermions (or Fermi-liquid phases) are among the hardest problem for a 2D tensor network (in real space)

32 Non-interacting spinless fermions: infinite systems (ipeps) H free = [c rc s + c sc r (c rc s + c s c r )] 2 c rc r 4 rs r γ 2 critical gapped Relative error of energy 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 λ Li et al., PRB 74, (2006) λ fast convergence with D in gapped phases slow convergence in phase with 1D Fermi surface

33 Non-interacting fermions (2D MERA) rel. error of energy =1, =4 λ=1 λ=1.5 λ=2 λ=2.5 λ=3 Layers critical phase gapped phase Size 6x6 18x18 54x54 162x L Error constant with L 2D MERA is scalable!

34 t-j model H t-j = t X hiji c i c j + H.c. + J X hiji (S i S j 1 4 n in j ) constraint: only one electron per site! Nearest-neighbor hopping Heisenberg interaction? Does it reproduce? striped states observed in some cuprates? DMRG (wide ladders): YES! Variational Monte Carlo Fixed-node Monte Carlo NO!

35 t-j model: ipeps results Corboz, White, Vidal, Troyer, PRB84, 2011 Comparison with: M. Lugas, L. Spanu, F. Becca, S. Sorella. PRB 74, (2006) variational Monte Carlo (VMC) (Gutzwiller projected wave functions) state-of-the-art fixed node Monte Carlo (FNMC) 1.35 VMC FNMC ipeps D=8 2x2 unit cell J/t =0.4 ipeps with 2x2 unit cell 1.4 A B A B A B E hole 1.45 A C B D A C B D A C B D C D C D C D 1.5 A B A B A B C D C D C D δ

36 t-j model: ipeps results Corboz, White, Vidal, Troyer, PRB84, 2011 E hole Comparison with: VMC FNMC ipeps D=8 2x2 unit cell ipeps D=8 8x2 unit cell ipeps D=8 12x2 unit cell ipeps D=10 12x2 unit cell M. Lugas, L. Spanu, F. Becca, S. Sorella. PRB 74, (2006) variational Monte Carlo (VMC) (Gutzwiller projected wave functions) state-of-the-art fixed node Monte Carlo (FNMC) J/t =0.4 Large hole density AF order 1.5 DMRG, 18x10, cylindric BC, m= δ ipeps: lower (better) variational energy than VMC/FNMC

37 t-j model: Homogeneous vs striped state Study energies as a function of the bond dimension D Larger D thanks implementation of symmetries! Bauer, Corboz, Orus, Troyer, PRB 83 (2011) uniform state striped state =0.1 E h who? 1.6 wins /D VMC+Lanczos for N=162 Hu, Becca, Sorella, PRB 85 (2012) States are still competing at very low energies! Solution within reach! Work in progress...

38 Summary: Fermionic tensor networks Fermionic Hamiltonians can be mapped onto a (hardcore) bosonic model (or spin model) via a Jordan-Wigner transformation In 2D local fermionic Hamiltonians get mapped into non-local bosonic Hamiltonians We can use a fermionic operator network to represent fermionic wave-functions in 2D We can account for the fermionic anticommutation rules by replacing crossings by swap tensors + use parity preserving tensors Same computational cost Free fermions (with a 1D Fermi-surface) violate the area law! Infinite PEPS yields competitive variational energies for the t-j model Promising way to simulate the 2D Hubbard model