Tensor Networks, MERA & 2D MERA ✦ Classify Tensor Network Ansatz According to Its Entanglement Scaling

Lecture 1: tensor network states (MPS, PEPS & iPEPS, Tree TN, MERA, 2D MERA)

Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18 Outline

‣ Lecture 1: tensor network states ✦ Main idea of a tensor network ansatz & area law of the entanglement entropy ✦ MPS, PEPS & iPEPS, Tree tensor networks, MERA & 2D MERA ✦ Classify tensor network ansatz according to its entanglement scaling

‣ Lecture II: tensor network algorithms (iPEPS) ✦ Contraction & Optimization

‣ Lecture III: Fermionic tensor networks ✦ Formalism & applications to the 2D Hubbard model ✦ Other recent progress Motivation: Strongly correlated quantum many-body systems

High-Tc Quantum magnetism / Novel phases with superconductivity spin liquids ultra-cold atoms

Challenging!tech-faq.com

Typically:

• No exact analytical solution Accurate and efﬁcient • Mean-ﬁeld / perturbation theory fails numerical simulations • Exact diagonalization: O(exp(N)) are essential! Quantum Monte Carlo

• Main idea: Statistical sampling of the exponentially large conﬁguration space • Computational cost is polynomial in N and not exponential

Very powerful for many spin and bosonic systems Quantum Monte Carlo

• Main idea: Statistical sampling of the exponentially large conﬁguration space • Computational cost is polynomial in N and not exponential

Very powerful for many spin and bosonic systems

Example: The Heisenberg model

. . . Sandvik & Evertz, PRB 82 (2010): . . . H = SiSj system sizes up to 256x256 i,j 65536 ⇥ Hilbert space: 2 Ground state sublattice magn. m =0.30743(1) has Néel order

...... Quantum Monte Carlo

• Main idea: Statistical sampling of the exponentially large conﬁguration space • Computational cost is polynomial in N and not exponential

Very powerful for many spin and bosonic systems BUT Quantum Monte Carlo & the negative sign problem

Bosons Fermions (e.g. 4He) (e.g electrons)

(x ,x )= (x ,x ) (x ,x )= (x ,x ) B 1 2 B 2 1 F 1 2 F 2 1 symmetric! antisymmetric!

this leads to the infamous ✓ negative sign problem t (poly(N/T )) t (exp(N/T )) sim O sim O cannot solve large systems at low temperature! Strongly correlated fermionic systems

ˆ 2D Hubbard model H = t cî†cˆj + U nî nî ⇥ i,j , i • High-temperature superconductors ⇤⌅ Hopping between On-site repulsion between nearest-neighbor sites electrons with opposite spin

Is it the relevant model of high-temperature superconductors? Quantum Monte Carlo & the negative sign problem

Non-frustrated Frustrated spin systems spin systems

frustrated ? all bonds satisfied not all bonds satisfied

this leads to the infamous ✓ negative sign problem t (poly(N/T )) t (exp(N/T )) sim O sim O cannot solve large systems at low temperature! To make prog ress in strongly correlated systems it is essential to negative develop new accurate numerical methods! sign problem • DMFT / DCA • Diagrammatic Monte Carlo • Tensor network algorithms • Fixed-node Monte Carlo • Series expansion • Density Matrix Embedding Theory • Variational Monte Carlo • Functional renormalization group • Coupled-cluster methods • ... Overview: tensor networks in 1D and 2D 1D MPS 1D MERA Matrix-product state Multi-scale entanglement renormalization ansatz

and more 1 2 3 4 5 6 7 8 ‣ 1D tree tensor Underlying ansatz of the network density-matrix renormalization ‣ correlator product states group (DMRG) method i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18 ‣ ... 2D PEPS 2D MERA and more projected entangled-pair state ‣ Entangled- plaquette states ‣ 2D tree tensor network ‣ String-bond states ‣ ... Diagrammatic notation l

c i v i M j i A j i B j ... k k number vector vi matrix Mij Aijk Bijkl rank-0 tensor rank-1 tensor rank-2 tensor rank-3 tensor rank-4 tensor

★ We don’t need to write down formulas with tensors with many indices!

i j Example 1: M v = i u

Mijvj = ui j X ★ Connected lines: sum over corresponding indices! Diagrammatic notation

i j Example 2: u M v = c ★ sum over all connected indices: contraction of a tensor network uiMijvj = c ij X

B j Example 3: i = T A C ★ The rank of the k resulting tensor u v u v corresponds to the number of open legs in the network AuikBijCvjk = Tuv Xijk Diagrammatic notation

★ Hard to write down with all indices...

★ We know the result is going to be a number Introduction to tensor networks

➡ Aim: Efﬁcient representation of quantum many-body states

V : local Hilbert space Lattice with .... e.g. , N sites 1 2 3 4 5 N-1 N {| ⇧ | ⇥⇧}

Full Hilbert dimension 2N .... space VVVVV VV grows exponentially with N

Hamiltonian Hˆ = hîj sum of local terms ij ⇥ Represent the = i i i ground state | ⇤ i1i2...iN | 1 ⇥ 2 ⇥ ···⇥ N ⇤ i i ...i 1 2 N i , k ⇤ { ⇥} 2N coefficients Complexity ~exp(N) many numbers inefficient! Tensor network ansatz for a wave function

2 basis states per site: , {| "i | #i} Lattice: 26 basis states 1 2 3 4 5 6 26 coefﬁcients State: = i i i i i i | ⇥ i1i2i3i4i5i6 | 1 2 3 4 5 6⇥ i i i i i i 1 23 4 5 6 Tensor/multidimensional array Tensor network: matrix product state (MPS) Why is athisb c d e Big tensor i1i2i3i4i5i6 A B C D E F D bond dimension i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 possible??i6

Aa BabCbcDcdEdeF e = ˜ i1i2i3i4i5i6 i1 i2 i3 i4 i5 i6 i1i2i3i4i5i6 ⇡ abcdeX Efﬁcient exp(N) many numbers vs poly(D ,N) numbers representation! “Corner” of the Hilbert space

Ground states (local H)

★ GS of local H’s are less entangled than a Hilbert random state in the Hilbert space space ★ Area law of the entanglement entropy Splitting in the middle Left side Right side

Big tensor i1i2i3i4i5i6

i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 i6 Singular value decomposition U s V† = UsV †

skk 0 l r l, r 1,...,M l r 2 { } M =2N/2 diagonal matrix!

lr = UlkskkVrk⇤ Xk = l r = UlkskkV ⇤ l r lr rk| i| i | i | i| i lr k Xlr X X Schmidt = skk uk vk | i| i decomposition Xk How many relevant singular values?

M how many non-zero = s u v U s V† | i kk| ki| ki singular values? Xk l r ★ Special cases:

s11 =1,skk = 0 for k>1

=1u v Product state | i | 1i| 1i

1 1 s11 = ,s22 = ,skk = 0 for k>2 p2 p2 1 1 = u1 v1 + u2 v2 Entangled state | i p2| i| i p2| i| i

1 Maximally skk = , for all k pM entangled state How many relevant singular values? bond dimension D = M how many relevant = s u v U s V† | i kk| ki| ki singular values? Xk skk l r

˜ = s u v | i⇡| i kk| ki| ki Ground state (local H) Xk random state: keeping the largest all singular values important singular values minimizes the error ˜ k ||| i| i|| M KEY IDEA OF relevant singular values truncate this part DMRG! Reduced density matrix

M A B = s u v | i kk| ki| ki Xk ★ Reduced density matrix of left side: describes system on the left side

2 ⇢ =tr [⇢]=tr [ ]= u u k = s probability A B B | ih | k| kih k| kk Xk

★ Entanglement entropy: S(A)= tr[⇢ log ⇢ ]= log A A k k Xk ‣ Product state: S(A)= 1 log 1 = 0

... 1 1 ‣ Maximally entangled state: S(A)= log = logM M M Xk How large is S in a ground state? How does it scale with system size? . . . Area law of the entanglement entropy E . . .

1D 2D A E A E ...... L

L ...... # relevant states Entanglement entropy S(A)= tr[⇥ log ⇥ ]= log A A i i exp(S) i ⇠

General (random) state Ground state (local Hamiltonian)

d d 1 S(L) L (volume) S(L) L (area law) ⇠ ⇠

Critical ground states: 1D S(L)=const = const (all in 1D but not all in 2D) 2D S(L) L ⇥ exp(L) 1D S(L) log(L) ⇠ 2D S(L) L log(L) ⇠ MPS & PEPS 1D MPS Matrix-product state Bond dimension 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)

✓ Reproduces area-law in 1D S(L)=const MPS & PEPS 1D MPS Matrix-product state Bond dimension D

➡ One bond can contribute at most log(D) to the A E entanglement entropy L

rank(⇢ ) D S(A) log(D)=const A  

✓ Reproduces area-law in 1D S(L)=const MPS & PEPS 1D 2D MPS can we use Matrix-product state an MPS? Bond dimension D

1 2 3 4 5 6 7 8 Physical indices (lattices sites) L

S. R. White, PRL 69, 2863 (1992) Fannes et al., CMP 144, 443 (1992) Östlund, Rommer, PRL 75, 3537 (1995)

!!! Area-law in 2D !!! ✓ Reproduces area-law in 1D S(L) L S(L)=const D exp(L) ⇠ MPS & PEPS

1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) Bond dimension D Bond dimension 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 Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115

✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D S(L)=const S(L) L PEPS: Area law

one “thick” bond of dimension DL

DL A B

...... L

S(A) L log D L  ⇠ each cut auxiliary bond can contribute (at most) log D to the entanglement entropy The number of cuts scales ✓ Reproduces area-law in 2D with the cut length S(L) L MPS & PEPS

1D 2D PEPS (TPS) MPS projected entangled-pair state Matrix-product state (tensor product state) Bond dimension D Bond dimension D

1 2 3 4 5 6 7 8 Physical indices (lattices sites)

✓ Reproduces area-law in 1D ✓ Reproduces area-law in 2D S(L)=const S(L) L Infinite PEPS (iPEPS) 1D 2D iMPS iPEPS infinite matrix-product state infinite projected entangled-pair state

A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A

Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008)

★ Work directly in the thermodynamic limit: No finite size and boundary effects! Infinite PEPS (iPEPS) 1D 2D iMPS iPEPS infinite matrix-product state infinite projected entangled-pair state

A B A B A B B A B A B A A B A B A B A B A B A B B A B A B A A B A B A B B A B A B A

Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008)

★ Work directly in the thermodynamic limit: No ﬁnite size and boundary effects! iPEPS with arbitrary unit cells 1D 2D iMPS iPEPS inﬁnite matrix-product state with arbitrary unit cell of tensors

D A B C D A H E F G H E A B C D A B D A B C D A H E F G H E D A B C D A H E F G H E

here: 4x2 unit cell

PC, White, Vidal, Troyer, PRB 84 (2011)

★ Run simulations with different unit cell sizes and compare variational energies Hierarchical tensor networks (TTN/MERA)

MERA

MPS

“ﬂat” i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11i12 i13 i14 i15i16 i17 i18

tensors at different length scales

★ Powerful ansatz for critical systems! (reproduces S(L) ~ logL scaling) Real-space renormalization group transformation

coarse- effective model 2 2 grained H L lattice

coarse- effective model 1 1 grained H L lattice

microscopic microscopic model H0 L0 lattice Tree Tensor Network (1D)

1D systems (non-critical) S(L)=const

= const

coarse- isometry 2 grained L lattice

coarse- 1 grained L lattice

= d microscopic 0 L0 lattice # relevant local states Tree Tensor Network (1D)

1D critical systems S(L) log(L) poly(L) coarse- isometry 2 2 grained L lattice

coarse- 1 1 grained L lattice

= d microscopic 0 L0 lattice # relevant local states The MERA (The multi-scale entanglement renormalization ansatz) G. Vidal, PRL 99, 220405 (2007) G. Vidal, PRL 101, 110501 (2008) 1D systems (critical) KEY: disentanglers reduce the S(L) log(L) amount of short-range entanglement = const

coarse- isometry disentangler 2 grained L lattice

coarse- 1 grained L lattice

= d microscopic 0 L0 lattice # relevant local states MERA: Properties

Let’s compute O O: two-site operator | | ⇥

O two-site operator

| MERA: Properties

Causal cone Isometries are isometric

w = I

w† ⇥ | O | Disentanglers

are unitary

u = I u† MERA: Properties

Causal cone Isometries are isometric

w = I

w† ⇥ | O | Disentanglers

are unitary

u = I u†

Efﬁcient computation of expectation values of observables! Different types of MERA’s Figures by G. Evenbly

8 3-to-1 blocking O (χ )

9 O χ 2-to-1 blocking ( )

7 O (χ ) 2-to-1 blocking

TRADEOFF: computational cost vs efﬁciency of coarse-graining MERA: Entanglement entropy

S(A) n(A) log() 

n(A)=2 S(A) const ! ⇠ T

n(A) 2T 2 log L ⇡ ⇡ 2 S(A) log(L) ⇠

Reproduces log(L) scaling of 1D critical systems n(A)=4L S(A) L ! ⇠

ﬁgures from Evenbly & Vidal, J Stat Phys 145 (2011) Power-law decaying correlations

-how accurately do MPS and MERA approximate quantum XX model: H XX YY ground states in terms of (critical, c=1) XX=+∑(σσrr++ 1 σσ rr 1 ) r correlators?

poly decay

exp decay However, critical systems can still be studied with MPS!

slide from Glen Evenbly Scale invariant MERA z

Translational invariance: same tensors along x Scale invariance (at criticality): same tensors along z 2D MERA (top view) Evenbly, Vidal. PRL 102, 180406 (2009)

Original lattice Apply disentanglers

✓Accounts for area- law in 2D systems S(L) L = const

Coarse-grained lattice Apply isometries Different structures of the 2D MERA...

Evenbly & Vidal, PRL 102, 180406 (2009) 2D MERA on the Kagome lattice

Evenbly & Vidal, PRL 104, 187203 (2010) 1D vs 2D MERA

same number of decreasing number connections in each layer of connections Evenbly and G. Vidal, J Stat Phys 145, 891(2011). Branching MERA: beyond area law scaling in 2D

G. Evenbly and G. Vidal, Physical Review Letters 112, (2014). Summary: Tensor network ansätze MPS PEPS 2D MERA

TN ansatz (variational)

1D MERA

➡ A tensor network ansatz is an efﬁcient variational ansatz for ground states of local H where the accuracy can be systematically controlled with the bond dimension

➡ Different tensor networks can reproduce different entanglement entropy scaling: ★ MPS: area law in 1D ★ MERA: log L scaling in 1D (critical systems) ★ PEPS/iPEPS: area law in 2D ★ 2D MERA: area law in 2D ★ branching MERA: beyond area law in 2D (e.g. L log L scaling) (Evenbly & Vidal, 2014) Comparison: MPS in 2D vs iPEPS

Snake MPS vs (i)PEPS Bond dimension D

★ Scaling of algorithm: D3 ★ Large / inﬁnite systems (scalable)! ★ Simpler algorithms & implementation ★ Much fewer variational parameters because much more natural 2D ansatz ★ Very accurate results for “small” L - Algorithms more complicated - inaccurate beyond certain L because D~exp(L) - Large cost of roughly D10 Comparison MPS & iPEPS: 2D Heisenberg model

iPEPS D=6 (variational optimization)

Stoudenmire & White, Ann. Rev. CMP 3 (2012)

iPEPS D=6 in the similar MPS D=3000 on thermodynamic limit accuracy ﬁnite Ly=10 cylinder

~ 2’600 variational pars. ~ 18’000’000 4 orders of magnitude fewer⇠ parameters (per tensor) iMPS vs iPEPS on inﬁnite cylinders: Heisenberg model J. Osorio Iregui, M. Troyer & PC, PRB 96 (2017)

W=11: D=5 iPEPS comparable to m=4096 MPS iMPS vs iPEPS on inﬁnite cylinders: Hubbard model (n=1) J. Osorio Iregui, M. Troyer & PC, PRB 96 (2017)

MPS in 2D and iPEPS provide complementary results!!!

W=7: D=11 iPEPS comparable to m=8192 MPS Classiﬁcation by entanglement (2D)

It depends on the amount of • How large does have to be? D entanglement in the system! Entanglement

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

gapless systems Heisenberg model, d-wave / p-wave SC, with area law Dirac Fermions, ... high systems with free Fermions, Fermi-liquid “1D fermi surface” type phases, bose-metals?

S(L) L log L Non-interacting spinless fermions (old iPEPS results) Corboz, Orús, Bauer, and Vidal, PRB 81 (2010) 4 H = [c†c + c†c (c†c† + c c )] 2⇥ c†c free r s s r r s s r r r rs r 3 ⇥

−2 γ 2 critical gapped 10

−3 10 1 1D Fermi −4 surface 10 γ=0 D=2 0 0 1 2 3 4 γ=0 D=4 −5 λ 10 γ=1 D=2 Li et al., γ=1 D=4 −6 PRB 74, 073103 (2006) 10 γ=2 D=2 Relative error of energy γ=2 D=4 D: bond dimension −7 10

1 1.5 2 2.5 3 λ

fast convergence with D slow convergence in phase in gapped phases with 1D Fermi surface Non-interacting fermions (2D MERA) Layers Size 1 6x6 2 18x18 3 54x54 4 162x162 =1, ⇥ =4 λ=1

−2 critical 10 λ=1.5 phase

λ=2

−3 λ=2.5 gapped 10 rel. error of energy λ=3 phase

1 2 10 10 L Error constant 2D MERA is with L scalable!