Fermionic tensor networks

Bosons Fermionsvs

Philippe Corboz, Institute for Theoretical Physics, ETH Zurich

P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009) : fermionic 2D MERAP. Corboz, R. Orus, B.Bauer, G. Vidal, PRB 81, 165104 (2010) : fermionic PEPS

‣ 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

Outline

Introduction to Fermions

Lattice with N sites

V : local Hilbert space....

1 2 3 4 5 N-1 N{|0⇤, |1⇤}

order!

Basis states |n1, . . . , nN � = (c†1)n1(c†2)

n2 . . . (c†N )nN |0�

ni � {0, 1}

c†i

ci

c†i |0⇥ = |0 · · · 1 · · · 0⇥

ci|0 · · · 1 · · · 0⇥ = |0⇥Fermionic creation

annihilation ops

site i

Anticommutation relations

c†i c†i |0⇥ = �c†i c

†i |0⇥ = 0cicj = �cj ci

{ci, cj} = cicj + cj ci = 0,n

ci, c†j

o

= �ij

Fermionic basis

Basis states |n1, . . . , nN � = (c†1)n1(c†2)

n2 . . . (c†N )nN |0�

Examples:

c†1c†2|0� = |110� c†1|010� = |110�

c†2|010� = 0 c2|011� = |001�

c†3|010� = |011�

c†3|010⇥ = c†3c†2|0⇥ = �c†2c

†3|0⇥ = �|011⇥

Fermionic operators acting on basis states

c†i |n1...ni...nN ⇥ = �ni,0P (1)P (2)...P (i� 1)|n1...ni+1...nN ⇥

ci|n1...ni...nN ⇥ = �ni,1P (1)P (2)...P (i� 1)|n1...ni�1...nN ⇥

General rule:

c†i ci+k|...ni...ni+k...⇥ = �ni,0�ni+k,1P (i + 1)P (i + 2)...P (i + k � 1)|...ni+1...ni+k�1...⇥

P |1⇥ = (�1) |1⇥P |0� = (+1) |0�

Jordan-Wigner transformation!

c†i ci+k b†i �zi+1�

zi+2...�

zi+k�1 bi+k

Fermionic operator acting on 2 sites

(Hardcore) bosonic or spin operator acting on k+1 sites

Parity symmetry

Fermionic observables and Hamiltonians have parity symmetry

[H, P ] = 0 [O, P ] = 0

➡ they can be written as an even polynomial of ‘s c†, c

A parity preserving fermionic operator commutes with any other fermionic operator if their supports are disjoint

[O, ck] = 0, if k /� sup[O]

(�1)2 = +1

(�1)2N = +1

c†3c4 c†k = c†k c†3c4

k /� {3, 4}

{

O

Local fermionic operators

c†i ci+k b†i �zi+1�

zi+2...�

zi+k�1 bi+k

non-local paritypreserving op

No negative sign arises when acting with a local, normal ordered, parity preserving operator on a state

Example:

= c†3c4 c†1c†2c

†4|0� = c†1c

†2 c†3c4 c†4|0� = |1110�c†3c4|1101�

JWT

c†i ci+1 b†i bi+1local parity preserving op

(no ‘s are involved)JWT

�z

acts in the same way as the (hardcore) bosonic operator

Jordan-Wigner transformation in 1D

1D1 2 3 4 easy!

c†2c3 � b†2b3

no string of ‘sfor nearest-neighbor operators

�z

★ 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

Jordan-Wigner transformation in 2D

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16 2-site local operator

k-site non-local operator

c†1c5 � b†1�z2�z

3�z4 b5

2D

★ In two dimensions some local fermionic operators get mapped into non-local bosonic operators

★ This looks complicated for large (or even infinite) 2D systems!

Can we use a “fermionic” tensor network ansatz to represent fermionic wave functions directly?

Yes, it’s possible!Corboz et al, PRA 81 (2010)

Breakthrough in 2009: Fermions with 2D tensor networks

How to take fermionic statistics into account?

P. Corboz, G. Evenbly, F. Verstraete, G. Vidal, Phys. Rev. A 81, 010303(R) (2010)

C. V. Kraus, N. Schuch, F. Verstraete, J. I. Cirac, Phys. Rev. A 81, 052338 (2010)

C. Pineda, T. Barthel, and J. Eisert, Phys. Rev. A 81, 050303(R) (2010)

P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009)

T. Barthel, C. Pineda, and J. Eisert, Phys. Rev. A 80, 042333 (2009)

Q.-Q. Shi, S.-H. Li, J.-H. Zhao, and H.-Q. Zhou, arXiv:0907.5520

P. Corboz, R. Orus, B.Bauer, G. Vidal, PRB 81, 165104 (2010)

I. Pizorn, F. Verstraete, Phys. Rev. B 81, 245110 (2010)

Z.-C. Gu, F. Verstraete, X.-G. Wen. arXiv:1004.2563

Different formulations:

fermionic operators anticommute

cicj = �cj ci

3x3 PEPS

Crossings in 2D tensor networks

€

Ψ

€

؆��|�

⇥

Crossings appear when projecting the 3D network onto 2D!

w1 w2 w3 w4

uux uy

� (lower half)

� (upper half)

H

Example network of the 2D MERA

Bosons vs Fermions

�B(x1, x2) = �B(x2, x1)

symmetric!

�F (x1, x2) = ��F (x2, x1)

antisymmetric!

+ --Crossings in a tensor network

ignore crossings take care!

--=+

operators anticommutebibj = bj bi

operators commute

cicj = �cj ci

The swap tensor

RULE:

# Fermions

+

even even

+

evenodd

--oddodd

+

oddeven Parity

(even parity), even number of particlesParity of a state: P

(odd parity), odd number of particles

P = +1

P = �1{Replace crossing by swap tensor

i1 i2

j2 j1

B

S(P (i1), P (i2)) =��1+1

if P (i1) = P (i2) = �1otherwise

Bi1i2j2j1

= �i1,j1�i2,j2S(P (i1), P (i2))

Parity symmetry

even odd

• Decomposing local Hilbert spaces into even and odd parity sectors

• Label state by a composite index i = (p, �p)parity sector 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!

• Choose all tensors to be parity preserving!

Ti1i2...iM = 0 if P (i1)P (i2) . . . P (iM ) �= 1=

P P P

P

• Fermionic systems exhibit parity symmetry! [H, P ] = 0

➡ Parity preserving operator T commutes with any operator K if their support is disjoint

Example!

ExampleBosonic tensor network Fermionic tensor network

EASY!!!

Fermionic “operator networks”

Tensor

State of 4 site system

bosonictensor network

{|0⇤, |1⇤}

i1 i2 i3 i4

�i1i2i3i4

i1 i2 i3 i4

j1 j2

Bj2i3i4Aj1

i1i2

Dj1j2

Contract

Boso

ns

|�� =�

i1i2i3i4

�i1i2i3i4 |i1i2i3i4�

Tensor + fermionic ops

i1 i2 i3 i4 i1 i2 i3 i4

j1 j2fermionic operator networkA B

D Contract+

operator calculus

�

Ferm

ions

A = Aj1i1i2

|i1i2ihj1| = Aj1i1i2

c†i11 c†i22 |0ih0|cj110

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

O = t c†1c3

|�� = |011�

���| = �110|

0

0

= �tc†1I3c†2|0⇥�tc†1c3c

†3c

†2|0⇥ ={

�tc†1c†2I3|0⇥ = (�t)|110⇥

⇥110|(±t)|110⇤ = ± t

➡ 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

Example

c†2c†3|0�

�c†3c†2|0⇥

A simple example: , ���|O|�⇥ O = t c†1c3

Cost of fermionic tensor networks??

w1 w2 w3 w4

uux uy

� (lower half)

� (upper half)

H

First thought:Many crossings many more tensors larger computational cost??

NO!Same computational cost

The “jump” move

• Jumps over tensors leave the tensor network invariant• Follows form parity preserving tensors

=

[T , ck] = 0, if k /� sup[T ]

• Allows us to simplify the tensor network• Final cost of contraction is the same as in a bosonic tensor network

Example of the “jump” move

Example of the “jump” move

Example of the “jump” move

Example of the “jump” move

absorb

absorb

Example of the “jump” move

now contract as usual!

Fermionic (i)PEPS

€

Ψ

€

Ψ†

‣ No crossings anymore

‣ Fermionic character taken into account locally!

‣ Only small modification to bosonic PEPS

€

≡

��|�

⇥

Example of the “jump” move

absorb

absorb

Fermionic (i)PEPS: expectation values

environment

compute environment approximately (e.g. using MPS)

Connect two-body operator

Contract this network!

insert swap tensors

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)!

Computational cost

It depends on the amount of entanglement in the system!

• How large does have to be?D

• Leading cost: O(Dk)polynomial scaling but large exponent!

MPS: k = 3PEPS: k ⇡ 10

2D MERA: k = 16

The less entangled the system, the better the accuracy!

0

1

2

3

1 1.5 2 2.5 310−6

10−5

10−4

10−3

λ

Enta

ngle

men

t en

trop

yEr

ror

of e

nerg

y

H(�)Spinless fermion model

fixed bond dimension D

Classification by entanglement (in 2D)

high

Entanglement

low gapped systemsband insulators, valence-bond crystals, s-wave superconductors, ...

gapless systemsHeisenberg model, p-wave superconductors, Dirac Fermions,...

systems with “1D fermi surface”

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)

Non-interacting spinless fermions: infinite systems (iPEPS)

fast convergence with D in gapped phases

slow convergence in phase with 1D Fermi surface

Hfree =�

�rs⇥

[c†rcs + c†scr � �(c†rc†s + cscr)]� 2⇥

�

r

c†rcr

0 1 2 3 40

1

2

3

4

λ

γ critical gapped

1D Fermi surface

1 1.5 2 2.5 310−7

10−6

10−5

10−4

10−3

10−2

λ

Rel

ativ

e er

ror o

f ene

rgy

γ=0 D=2γ=0 D=4γ=1 D=2γ=1 D=4γ=2 D=2γ=2 D=4 D: bond dimension

Li et al., PRB 74, 073103 (2006)

Non-interacting fermions (2D MERA)

2D MERA is scalable!

101 102

10−3

10−2

L

rel.

erro

r of e

nerg

y

λ=1

λ=1.5

λ=2

λ=2.5

λ=3

� = 1, ⇥ = 4

critical phase

gapped phase

Layers1234

Size6x6

18x1854x54

162x162

Error constant with L�

Nearest-neighbor hopping Heisenberg interaction

t-J model

Ht-J = �tX

hiji�

c†i� cj� + H.c. + JX

hiji

(SiSj �14ninj) constraint: only one

electron per site!

Does it reproduce “striped” states observed

in some cuprates? ??

DMRG (wide ladders): YES!Variational Monte CarloFixed-node Monte Carlo NO!

t-J model: iPEPS resultsM. Lugas, L. Spanu, F. Becca, S. Sorella. PRB 74, 165122 (2006)• Comparison with:

‣ variational Monte Carlo (VMC) (Gutzwiller projected wave functions)

‣ state-of-the-art fixed node Monte Carlo (FNMC)

0 0.05 0.1 0.15 0.2 0.25−1.55

−1.5

−1.45

−1.4

−1.35

δ

E hole

VMCFNMCiPEPS D=8 2x2 unit cell

J/t = 0.4iPEPS with 2x2 unit cell

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

Corboz, White, Vidal, Troyer, PRB84, 2011

0 0.05 0.1 0.15 0.2 0.25−1.55

−1.5

−1.45

−1.4

−1.35

δ

E hole

VMCFNMCiPEPS D=8 2x2 unit celliPEPS D=8 8x2 unit celliPEPS D=8 12x2 unit celliPEPS D=10 12x2 unit cell

• Comparison with:

‣ variational Monte Carlo (VMC) (Gutzwiller projected wave functions)

‣ state-of-the-art fixed node Monte Carlo (FNMC)

M. Lugas, L. Spanu, F. Becca, S. Sorella. PRB 74, 165122 (2006)

J/t = 0.4

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

AF order

Large hole

density

iPEPS: lower (better) variational energy than VMC/FNMC

t-J model: iPEPS results

DMRG, 18x10, cylindric BC, m=5000

Corboz, White, Vidal, Troyer, PRB84, 2011

� = 0.1

0 0.05 0.1 0.15 0.2−1.65

−1.6

−1.55

−1.5

−1.45

−1.4

1/D

E h

Doping δ=0.1

uniform statestriped state

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)

Solution within reach! Work in progress...

VMC+Lanczos for N=162

Hu, Becca, Sorella, PRB 85 (2012)

?who wins

States are still competing at very low energies!

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