Introduction to Tensor Networks: PEPS, Fermions, and · PDF fileIntroduction to Tensor...

Román Orús

Institut für Physik, Johannes Gutenberg-Universität, Mainz (Germany)!

School on computational methods in quantum materials Jouvence, June 5 2014

www.romanorus.com/JouvenceTutorial2014.pdf

Introduction to Tensor Networks: PEPS, Fermions, and More

Some reviews J. Eisert, Modeling and Simulation 3, 520 (2013), arXiv:1308.3318 N. Schuch, QIP, Lecture Notes of the 44th IFF Spring School 2013, arXiv:1306.5551 R. Orus, arXiv:1306.2164 J. I. Cirac, F. Verstraete, J. Phys. A: Math. Theor. 42, 504004 (2009) F. Verstratete, J. I. Cirac, V. Murg, Adv. Phys. 57,143 (2008) J. Jordan, PhD thesis, www.romanorus.com/JordanThesis.pdf G. Evenbly, PhD thesis, arXiv:1109.5424 U. Schollwöck, RMP 77, 259 (2005) U. Schollwöck, Annals of Physics 326, 96 (2011)

Outline

1) Generalities

2) 2d PEPS

3) 1d MERA: Basics

4) Some further topics

2.1) Basics

2.2) Computing

2.3) Fermions

Entanglement obeys area-law

Entanglement key resource in quantum information

€

(L > ξ)€

E

Reduced density matrix of subsystem A

Entanglement entropy (von Neumann entropy)

€

A

€

E

2d system

For many ground states

teleportation, quantum algorithms, quantum error correction, quantum cryptography…

€

S(A) ~ LdGeneric state (volume)

€

S(A) ~ Ld−1Ground states of (most) local Hamiltonians (area)

Srednicki, Plenio, Eisert, Dreißig, Cramer, Wolf…

In d dimensions

Locality of interactions area-law

Many-body Hilbert space is far too large

Number of basis states in the Hilbert space ~ O(101023

)

Everyday material, ~ Avogadro number ~ O(1023)NSystem of spins 1/2 N

Number atoms in the observable universe ~ O(1080 )Compare to…

Set of relevant states (area-law, ground states of local Hamiltonians)

Set of product states (mean field)

Hilbert space is a convenient illusion

O(101023

)“Exploration” time ~ sec.

O(1017 )Age of the universe ~ sec. Compare to…

Most states here are not even reachable by a time evolution with a

local Hamiltonian in polynomial time!!!

Poulin, Qarry, Somma, Verstraete,

PRL 106 170501 (2011)

We need a language to target the relevant corner of quantum states directly

Tensor Networks

DNA person

A new language

Tensors are local building blocks for the quantum state (like a DNA, or LEGO)

Ψtensor quantum state

A new language

Tensor network diagrams

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f (A,B,C,D)tensor contraction

vector

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v

€

Amatrix

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ABmatrix product

€

tr(ABC)trace of matrix product

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Ψi1i2i3i4 i5i6i7i8i9

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i1

€

i2

€

i9…

Tensor (multidimensional array of complex numbers)

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i4

€

i5

€

i7

€

i8€

i2

€

i1

€

i3

€

i9€

i6replacement

€

Aαβi9

€

α

€

β

€

Bβγδµi8

€

γ

€

δ

€

µ

physical index (local basis)

summed ancillary bond index 1…D (carries entanglement)

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Cαγδµi8i9 = Aαβ

i9 Bβγδµi8

β

∑β =1,2,…, D

€

O(pN ) O(poly(N, p, D))Efficient representation, satisfies area-law,

and targets low-energy eigenstates of local Hamiltonians

€

Ψ = Ψi1i2 ...iNi's∑ i1 ⊗ i2 ⊗⊗ iN

p-level systems

Break the wavefunction locally

Tensor Network

Important families of Tensor Network states

Multiscale Entanglement Renormalization Ansatz

(MERA) any-d scale

invariant systems Entanglement Renormalization

RG

Other: MPO, MPDO, TTN, PEPO…

Projected Entangled Pair States (PEPS),

Tensor Product States (TPS) 2d,3d… systems

Tensor Product Variational Approach, PEPS, Infinite-PEPS algorithm, Tensor-Entanglement Renormalization, TRG/SRG/HOTRG/HOSRG…

Matrix Product States (MPS) 1d systems DMRG, Power Wave Function Renormalization Group, Time Evolving Block Decimation…

c.f., Miles Stoudenmire & Ulrich Schollwöck‘s talk

Multiscale Entanglement Renormalization Ansatz

(MERA) Entanglement Renormalization

RG

Projected Entangled Pair States (PEPS),

Tensor Product States (TPS) 2d,3d… systems

Tensor Product Variational Approach, PEPS, Infinite-PEPS algorithm, Tensor-Entanglement Renormalization, TRG/SRG/HOTRG/HOSRG…

This lecture

any-d scale invariant systems

Outline

1) Generalities

2) 2d PEPS

3) 1d MERA: Basics


2.1) Basics

2.2) Computing

2.3) Fermions

From MPS to PEPS Matrix Product States (MPS) 1d systems

But we want to go beyond 1d systems!!!

Very painful for DMRG...

This lecture

From MPS to PEPS

2d systems

Matrix Product States (MPS) 1d systems

Projected Entangled Pair States (PEPS), Tensor Product States (TPS)

3d systems

and so on…

Two exact examples

star operator

plaquette operator €

i

Simplest known model with “topological order”

Kitaev, 1997

Ground state (and in fact all eigenstates) are PEPS with D=2

1

1 1

1 1 1

2 2

2 2 2

2 2

1 1 2

1 1

2 2

= = = = 1

And another tensor rotated 90 ̊

An exact example: Kitaev’s Toric Code

Resonating Valence Bond State

+ + …

Equal superposition of all possible nearest-neighbor singlet coverings of a lattice (spin liquid)

Proposed to understand high-TC superconductivity

SU(2) singlets Anderson, 1987

It is a PEPS with D=3

1

1 3

3 3

= = 1 And rotations 2

2 3

3 3

PEPS…

F. Verstraete, I. Cirac, cond-mat/0407066

and infinite PEPS (iPEPS)

assuming translation invariance

J. Jordan, RO, G. Vidal, F. Verstraete, I. Cirac, Phys. Rev. Lett. 101, 250602 (2008)

PEPS obey 2d area-law

in1..D

in

prefactor size of the boundary

1..D

Exact contraction is inneficient N. Schuch et al, PRL 98, 140506 (2007)

1..D


(double indices)

1..D2

Exact contraction is inneficient

times

computing time ~ Exponential amount of time!

Mathematical statement: exact contraction of a PEPS is a #P-Hard problem (harder than NP-Complete)

Applies also to expectation values of observables

N. Schuch et al, PRL 98, 140506 (2007)

Critical correlation functions

+ permutations

1 1

1 1

= 1 1

2 1

=

1 2

2 1

= 2 2

2 1

=

2 2

2 2

=

It is a PEPS with D=2 (left as exercise):

At the correlation length is infinite:

Expectation values are those of the classical 2d Ising model

F. Verstraete et al, PRL 96, 220601 (2006)

World of states

World of Hamiltonians

PEPS to/from Hamiltonians

Hamiltonian with local interactions

Generic PEPS with finite D

Ground state of a gapped Parent Hamiltonian

with local interactions

e.g. D. Perez-Garcia et al, QIC 8, 0650 (2008)

Thermal states can be approx. by a PEPS with finite D

M. Hastings, PRB 73, 085115 (2006)

PEPS target the relevant corner of the Hilbert space (area-law)

MPS vs PEPS

MPS in 1d

1d area law

Exact contraction is efficient

Finite correlation length

to/from 1d Hamiltonians

2d area law

Exact contraction is inefficient

Finite and infinite correlation lengths


PEPS in 2d

Outline

1) Generalities

2) 2d PEPS

3) 1d MERA: Basics


2.1) Basics

2.2) Computing

2.3) Fermions

PEPS as ansatz: variational optimization

Variational optimization (e.g. finite PEPS)

Optimize over each tensor individually and sweep over the entire system (as in DMRG)

Minimization of quadratic function

Generalized eigenvalue problem

Once and are known, we can solve this problem efficiently

Approximate calculation of and

... ...

... ...

e.g. F. Verstraete, I. Cirac, cond-mat/0407066

e.g. calculation of

e.g. calculation of

MPS

e.g. calculation of

MPS MPO

1d problem: use a 1d method for MPS (e.g., DMRG or TEBD)

1..D2

e.g. calculation of

MPSup

1..χ

e.g. calculation of

1..χ

MPSup

MPSdown

1..χ

e.g. calculation of

1..χ

MPSup

MPSdown

1..χ

0d problem: exact!

e.g. calculation of

1..χ

1..χ

1..χ 1..χ

is the environment of tensor is computed similarly, but sandwitching with the Hamiltonian

Valid also for any expectation value

1d problem: use DMRG!

0d problem: exact!

2d problem Dimensional reduction

Time evolution (real, imaginary)

... ...

Time evolution (e.g. imaginary)

Divide into small time-steps

... ...

Split the Hamiltonian (e.g. 2-body n.n.)

e.g. J. Jordan et al, PRL 101, 250602 (2008)

Divide into small time-steps

... ...

... ...


Split the Hamiltonian (e.g. 2-body n.n.)



2-body gates


2-body gates

Time evolution (e.g. imaginary) e.g. J. Jordan et al, PRL 101, 250602 (2008)


evolved state


Main idea 1..D


Different approaches to this problem: full update, simplified update, TPVA...

Full update:

Finite systems: optimize over all tensors in the PEPS (as before)

Infinite systems: optimize over tensors in the PEPS unit cell (iPEPS)

Require calculations of environments, like the one shown before.


Environments with infinite PEPS

Infinite systems

Finite PEPS

e.g. J. Jordan et al, PRL 101, 250602 (2008) R. Orús, G. Vidal, PRB 80 094403 (2009)

Unit cell of tensors is repeated periodically over the whole PEPS:

translational invariance

Infinite PEPS

€

A

€

A*

(double indices)

Let’s put it on the plane of the screen!

€

A

€

A*

(double indices)

Environment calculations Contraction of this infinite lattice

1-dim transfer matrix: dominant eigenvector?

Contracting the infinite 2d lattice

Can be approximated using infinite MPS

… … iTEBD, iDMRG, PWFRG, etc

1..χ


Coarse-graining approaches: TRG/SRG, HOSVD


corner transfer matrix (Baxter, 1968)

(nice spectral properties)

half-row transfer matrix

half-column transfer matrix

Directional version of the corner transfer matrix renormalization group (faster than 1d transfer matrix methods) T. Nishino, K. Okunishi, JPS Jpn. 65, 891 (1996)


Renormalized Corner Transfer Matrices

€

C1

€

C2

€

C3

€

C4€

T1

€

T2

€

T3€

T4

Renormalization (numerical)

(Baxter, 1968,1978)

1..χ

Corner transfer matrix and iPEPS

€

" C 1

€

C2

€

C3

€

" C 4€

T1

€

T2

€

T3€

" T 43) renormalization

Iterate in all directions until convergence

R. Orús, G. Vidal, PRB 80 094403 (2009), R. Orús, PRB 85, 205117 (2012)

€

C2

€

C3€

T1

€

T2

€

T3

€

˜ C 1

€

˜ C 4€

˜ T 4€

Z†

€

Z†

€

Z

€

Z

€

C1

€

C2

€

C3

€

C4€

T1

€

T2

€

T3€

T4

Example: left move

1) insertion

€

C1

€

C2

€

C3

€

C4€

T1

€

T2

€

T3€

T4

€

T1

€

T3

An example: Toric Code in arbitrary field

€

H= −J As − J Bpp∑

s∑ − hx σ i

x

i∑ − hy σ i

y

i∑ − hz σ i

z

i∑

€

i

€

s

€

p

S. Dusuel et al., PRL 106, 107203 (2011)

2nd order sheets, quantum Ising

universality class

1st order sheet

Self-dual point

Multicritical line

Multicritical point

Critical quantum Ising points

‘bare’ Toric Code

topological (deconfined)

phase

polarized phase

Phase diagram

Outline

1) Generalities

2) 2d PEPS

3) 1d MERA: Basics


2.1) Basics

2.2) Computing

2.3) Fermions

Fermionic 2d systems

Fermionic systems are extremely interesting physical systems, e.g. the 2d fermionic Hubbard model may be the key to understand the emergence of high-TC superconductivity

Fermions are a NUMERICAL MONSTER for Quantum MonteCarlo because of the sign problem!

but there is hope...

Unfortunately, fermionic systems are also amongst the most difficult to simulate, because of the sign problem in Quantum MonteCarlo (sampling of negative probabilities)

Tensor Networks + Fermions e.g., P. Corboz, R. Orús, B. Bauer, G. Vidal, PRB 81, 165104 (2010)

Bosons

€

ΨB (x1,x2) = ΨB (x2,x1)

€

bib j = b jbi

Symmetric wavefunction

Operators commute

Fermions

€

ΨF (x1,x2) = −ΨF (x2,x1)

€

cic j = −c jci

Antisymmetric wavefunction

Operators anticommute

Crossings in a TN +

Ignore crossings

_

Careful!!!!

The leading order of the computational cost is the same as in the bosonic case

€

T

€

P

€

P

€

P

€

P

€

P

€

T

€

=Use parity-preserving tensors

€

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

Symmetry of the Hamiltonian

€

i1

€

i2

€

j1

€

j2

€

i1

€

i2

€

i2

€

i1

€

XReplace crossings by

fermionic swap gates

€

Xi2i1 j1 j2= δi1 j1

δi2 j2S(P(i1),P(i2))

€

S(P(i1),P(i2)) =−1 if P(i1) = P(i2) = −1+1 otherwise# $ %

Fermionic operators anticommute

Tensor Network “fermionization” rules

physics is independent of the order

physics is independent of graphical projection

(different choices of Jordan-Wigner transformation, if mapping

to a spin system)

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1

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2

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4€

3

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7€

6

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9

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5

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8

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1

€

2

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3

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4

€

5

€

6

€

7

€

8

€

9

€

1

€

2

€

3€

4

€

5

€

6€

7

€

8

€

9

€

1

€

2

€

3

€

4

€

5

€

6

€

7

€

8

€

9

fermionic order ~ graphical projection of a PEPS

€

Ψ Ψ

€

Ψ

€

Ψ†

Contract as for bosons!

Example: scalar product of 3x3 PEPS

An example: 2d t-J model

€

H = −t ˜ c iσ† ˜ c jσ + hc( )

i, j σ∑ + J (SiS j −

14

nin j ) −µi, j∑ ni

i∑

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

P. Corboz, S. White, G. Vidal, M. Troyer, PRB 84 041108 (2011)

First simulations

€

(t /J = 3)Energy

Compatible results

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

Some improved simulations

Energy

Promising!

(t / J = 2.5)

P. Corboz, S. White, G. Vidal, M. Troyer, PRB 84 041108 (2011)

Outline

1) Generalities

2) 2d PEPS

3) 1d MERA: Basics


2.1) Basics

2.2) Computing

2.3) Fermions

From MPS to MERA Matrix Product States (MPS) 1d systems

But we want to do critical systems!!!

Also very painful for DMRG...

From MPS to MERA Matrix Product States (MPS) 1d systems

Multiscale Entanglement Renormalization Ansatz (MERA)

2d systems

and so on…

1d systems

This lecture

From MPS to MERA

1d systems

Matrix Product States (MPS) 1d systems

Multiscale Entanglement Renormalization Ansatz (MERA)

and so on…

2d systems

1d MERA

spatial dimension

holographic dimension

(RG) ... ...

1..χ

Tensors obey constraints

= u

u+

Unitaries (disentanglers)

= w

w+

Isommetries (coarse-grainings)

Reason:

entanglement is built locally at all length scales

entangle locally

coarse-grain entangle locally

coarse-grain

entangle locally

coarse-grain

L

L/2

L/4

L/8

...

leng

th/e

nerg

y sc

ale

top tensor (for finite system)

1

Extra dimension defines an RG flow: Entanglement Renormalization

Entropy of 1d MERA

„geodesic“ curve

Entanglement as boundary in holographic geometry:

Entanglement as boundary in holographic geometry:

Constant contribution at every layer

1d MERA can produce logarithmic violations to the area-law:

(like 1d critical systems!)

= u

u+

= w

w+

Norm of MERA

...

The norm is just the contraction of the top tensors

= u

u+

= w

w+

Norm of MERA

Expectation values

= u

u+

= w

w+

Expectation values

Only tensors inside of the causal cone contribute to the expectation value

„causal cone“ with bounded width

MPS vs 1d MERA

MPS in 1d

1d area law


Finite correlation length


Beyond 1d area law


Finite and infinite correlation lengths


MERA in 1d

Arbitrary tensors Constrained tensors

An example: 1d critical systems

G. Evenbly, G. Vidal, in "Strongly Correlated Systems. Numerical Methods", Springer, Vol. 176 (2013)

Critical Energies Critical XX Correlators

Outline

1) Generalities

2) 2d PEPS

3) 1d MERA: Basics


2.1) Basics

2.2) Computing

2.3) Fermions

PEPS & Entanglement Hamiltonians e.g. I. Cirac et al, PRB 83, 245134 (2011), N. Schuch et al, PRL 111, 090501 (2013)

(double indices)

1-dim transfer matrix: dominant eigenvector?

Can be approximated using infinite MPS

… … iTEBD, iDMRG, PWFRG, etc

It is also hermitian and positive by construction (up to finite-χ effects)

ρ = exp(−HE ) Who is ??? HE

Based on examples

Critical 2d systems 1d Hamiltonian, long-range

Gapped 2d systems, trivial phase 1d Hamiltonian, short-range

Gapped 2d systems, topological order Completely non-local (projector)

… …

Virtual indices of bra 1…D

Virtual indices of ket 1…D

Boundary virtual index 1...χ

Bulk Boundary Holographic principle

Contraction of 2d PEPS for gapped trivial phases could be approximated efficiently!

1d Entanglement Hamiltonian

MERA & AdS/CFT e.g. B. Swingle, PRD 86, 065007 (2012), G. Evenbly,G. Vidal, JSTAT 145:891-918 (2011)

For a scale-invariant MERA, the tensors of a critical model with a CFT limit correspond to a

„gravitational“ description in a discretized AdS space: „lattice“ realization of AdS/CFT correspondance

Bulk is a discretized AdS space

“branching” MERA G. Evenbly, G. Vidal, arXiv:1210.1895

1d MERA RG

1d branching MERA

Decoupling of degrees of freedom along RG (e.g. spin-charge separation), and allows

arbitrary scalings of the entanglement entropy

In 2d, ansatz for e.g., Fermi & Bose liquids,

RG

Lots of other things:

Chiral PEPS Entanglement measures

Time evolution Infinite systems

Classical systems Corner Transfer Matrices

Tensor Renormalization Group Symmetries 2d MERA 3d PEPS

MERA optimization Topological systems

Excited states Thermal states & dissipation Continuous tensor networks

Tensor Networks & Montecarlo Practical implementations

Blablabla...

Ask me if interested!

Thank you!

www.romanorus.com/JouvenceTutorial2014.pdf

Introduction to Tensor Networks: PEPS, Fermions, and · PDF fileIntroduction to Tensor...

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