Guifre Vidal- Entanglement Renormalization

Guifre Vidal

International Workshop on DMRG and related stuff

Entanglement Renormalization

Beijing, China, August 27th 2010

Outline

• Frustrated antiferromagnets (2+1 D) *

• Entanglement renormalization

• MPS and MERA

Introduction

• Quantum critical systems (1+1 D) *, **

Applications

• interacting fermions

• anyons

• topological order

• lattice gauge theory

other applications (not in this talk):

* Glen Evenbly

** Robert Pfeifer

http://www.physics.uq.edu.au/QuSim/index.php/Image:Gevenbly.jpg

http://www.physics.uq.edu.au/QuSim/index.php/Image:RPfeiferQuSimImage.jpg

Outline


• MPS and MERA

• Quantum critical systems (1+1 D)

• Frustrated antiferromagnets (2+1 D)

Introduction

Applications

1 2 ... N

d

N

1) What is a tensor network state/ansatz?

• quantum spin model: N spins

1 2

1 2

... 1 2

1 1 1

...N

N

d d d

i i i N

i i i

i i i

• state of the N spins (e.g. ground state of a local Hamiltonian)

difficulty:Nd coefficients

• variational ansatz: tensor network state

1 2 ... Ni i i

...1i

2i 3i

Ni

4iNd coefficients ( )O N coefficients

1i 2i 3i 4i ... Ni

• tensor network states:

1 2 3 4 ... Ni i i i i

Nd coefficients ( )O N “small” tensors

1i 2i 3i 4i ... Ni

N quantum spins

• Efficient representation of many-body states

2) What are tensor network states useful for?

• Characterize and classify phases of matter and phase transitions

e.g. symmetry-breaking and topological order, fixed points of RG, etc

(variational parameters = coefficients in the tensors)

• Variational ansatz for numerical approaches

H

3) Examples of tensor network states for 1D systems:

Matrix Product State MPS

Multi-scale Entanglement Renormalization Ansatz

MERA

Tree Tensor Network

TTN

• Wilson’s NRG(Numerical Renormalization Group)

• White’s DMRG(Density Matrix Renormalization Group)

• Entanglement Renormalization

• Morningstar-Weinstein’s COREno tensor network state comes out

• Dasgupta-Ma-Fischerrandom systems (strong disorder) perturbative (corrections to TTN)

4) Quick comparison

MPS

MERA

N sites

N sites

layers2log N

N

/ 2N

/ 4N

/ 8 1N

4) Quick comparison

MPS

MERA

• Number of tensors:

(1 1/ 2 1/ 4 1/ 8) 2N N

N tensors

2N tensors

N sites

N sites

layers2log N

4) Quick comparison

MERA

• Connectivity and correlations:

L

2log L

MPS

( ) LC L e

exponentialcorrelators

( ) qC L L

polynomialcorrelators

L

L

L

log logL L

log L

L

L

4) Quick comparison

MERA

• Entanglement entropy of a block:

MPS

( ) const.S L

constant

( ) logS L L

logarithmic

2log L

Outline


• MPS and MERA



Introduction

Applications


quantum spin system

W

wu

• Preservation of locality

• Coarse-graining transformation


• Preservation of locality

wu

u

u†

w

w†


• tensor network state

uw


• tensor network state with built-in coarse-graining

smallscale

largescale


• tensor network ansatz corresponding to a quantum circuit

small scaleentanglement/correlations

large scaleentanglement/correlations

00

00

0000

00000000

Outline


• MPS and MERA



Introduction

Applications

Applications

• Renormalization group flow andRG fixed points:

• topological order Aguado, Vidal, PRL 100, 070404 (2008)Koenig, Reichardt, Vidal, PRB 79, 195123 (2009) Chen, Gu, Wen, arXiv:1004.3835v2

• frustrated antiferromagnets Evenbly, Vidal, PRL 104, 187203 (2010)Lou, Harada, Kawashima (in preparation)

• Variational approach for strongly correlated systems in 2D

Vidal, PRL 101, 110501 (2008)Evenbly, Vidal, Phys. Rev. B 81, 235102 (2010) (free fermions)Evenbly, Vidal, New J. Phys. 12, 025007 (2010) (free bosons)Cincio, Dziarmaga, Rams, PRL 100, 240603 (2008) (Ising 8x8)Evenbly, Vidal, PRL 102, 180406 (2009) (Ising)Cincio, Dziarmaga, Oles, arXiv:1001.5457v1

• interacting fermions Corboz, Evenbly, Verstraete, Vidal, PRA 81, 010303(R) (2010) Pineda, Barthel, Eisert, PRA 81, 050303(R) (2010) Corboz, Vidal, PRB 80, 165129 (2009) Barthel, Pineda, Eisert, PRA 80, 042333 (2009)

• quantum critical systems

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, Phys. Rev. B 81, 235102 (2010)Evenbly, Vidal, New J. Phys. 12, 025007 (2010) Giovannetti, Montangero, Fazio, Phys. Rev. Lett. 101, 180503 (2008)Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)Montangero, Rizzi, Giovannetti, Fazio, Phys. Rev. B 80, 113103 (2009) Giovannetti, Montangero, Rizzi, Fazio, Phys. Rev. A 79, 052314(2009)Evenbly, Pfeifer, Pico, Iblisdir, Tagliacozzo, McCulloch,Vidal,arXiv:0912.1642

Evenbly, Corboz, Vidal, arXiv: 0912.2166Silvi, Giovannetti, Calabrese, Santoro, Fazio, arXiv: 0912.2893

• Anyonic systems Koenig, Bilgin, arXiv:1006.2478v1Pfeifer, Corboz, Buerschaper, Aguado, Troyer, Vidal, arXiv:1006.3532v2

Applications

• Renormalization group flow andRG fixed points:

• topological order Aguado, Vidal, PRL 100, 070404 (2008)Koenig, Reichardt, Vidal, PRB 79, 195123 (2009) Chen, Gu, Wen, arXiv:1004.3835v2

• frustrated antiferromagnets Evenbly, Vidal, PRL 104, 187203 (2010)Lou, Harada, Kawashima (in preparation)

• Variational approach for strongly correlated systems in 2D


• interacting fermions Corboz, Evenbly, Verstraete, Vidal, PRA 81, 010303(R) (2010) Pineda, Barthel, Eisert, PRA 81, 050303(R) (2010) Corboz, Vidal, PRB 80, 165129 (2009) Barthel, Pineda, Eisert, PRA 80, 042333 (2009)

• quantum critical systems

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, Phys. Rev. B 81, 235102 (2010)Evenbly, Vidal, New J. Phys. 12, 025007 (2010) Giovannetti, Montangero, Fazio, Phys. Rev. Lett. 101, 180503 (2008)Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)Montangero, Rizzi, Giovannetti, Fazio, Phys. Rev. B 80, 113103 (2009) Giovannetti, Montangero, Rizzi, Fazio, Phys. Rev. A 79, 052314(2009)Evenbly, Pfeifer, Pico, Iblisdir, Tagliacozzo, McCulloch,Vidal,arXiv:0912.1642

Evenbly, Corboz, Vidal, arXiv: 0912.2166Silvi, Giovannetti, Calabrese, Santoro, Fazio, arXiv: 0912.2893

• Anyonic systems Koenig, Bilgin, arXiv:1006.2478v1Pfeifer, Corboz, Buerschaper, Aguado, Troyer, Vidal, arXiv:1006.3532v2

Outline


• MPS and MERA



Introduction

Applications Glen Evenbly

Robert Pfeifer


http://www.physics.uq.edu.au/QuSim/index.php/Image:RPfeiferQuSimImage.jpg

Scale invariant MERA

(3)

(2)

(1)

(0)

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, Phys. Rev. B 81, 235102 (2010)Evenbly, Vidal, New J. Phys. 12, 025007 (2010) Giovannetti, Montangero, Fazio, Phys. Rev. Lett. 101, 180503 (2008)Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)Montangero, Rizzi, Giovannetti, Fazio, Phys. Rev. B 80, 113103 (2009) Giovannetti, Montangero, Rizzi, Fazio, Phys. Rev. A 79, 052314(2009)

Evenbly, Pfeifer, Pico, Iblisdir, Tagliacozzo, McCulloch,Vidal,arXiv:0912.1642Evenbly, Corboz, Vidal, arXiv: 0912.2166Silvi, Giovannetti, Calabrese, Santoro, Fazio, arXiv: 0912.2893

critical systems (1D)

critical exponentsOPE, CFT

boundary & defectsnon-local operators

computational cost: pNinhomogeneous system

logp Ntranslationinvariance

p

scaleinvariance

8 16 24 32 40 48 56 64 7210

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Refinement Parameter,

En

erg

y E

rro

r,

EIsing

3-state Potts

XX

Heisenberg

• Example I: accuracy in ground state energy (infinite chain)

Evenbly, Vidal, Phys. Rev. A 79, 144108 (2009)

Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)

• Example II: operator content of quantum Ising model

scaling operators/dimensions: scaling dimension

(exact )

scaling dimension

(MERA)error

0 0 ----

0.125 0.124997 0.003%

1 0.99993 0.007%

1/8 0.1250002 0.0002%

1/2 0.5 <10−8 %

1/2 0.5 <10−8 %

Iidentity

spin

energy

=36 =20

disorder

fermions

OPE for local & non-local primary fields

4( 6 10 )

1 / 2C

1/ 2C

/4 / 2iC e

/4 / 2iC e

C i

C i

fusion rules

I

I+

I

I

I

...

{I, , , , , }

{I, } {I, , }

{I, , } {I, , , }

local andsemi-localsubalgebras

Boundary critical phenomenasee also Silvi, Giovannetti, Calabrese, Santoro, Fazio, arXiv: 0912.2893

uw

• infinite chain (bulk)

bulk MERA

uw

w

MERA with boundary

• semi-infinite chain (boundary)

Evenbly, Pfeifer, Pico, Iblisdir, Tagliacozzo, McCulloch,Vidal, arXiv:0912.1642

2

0

3

4

1/ 2

1 1/ 2

2 1/ 2

Free BC

2

0

3

Fixed BC

1/ 8

1 1/ 8

2 1/ 8

Bulk

2

0

1

0

2

Boundary scaling operators/dimensions• Example: Ising model

2

0

3

4

1/ 2

1 1/ 2

2 1/ 2

Free BC

2

0

3

Fixed BC

1/ 8

1 1/ 8

2 1/ 8

Bulk

2

0

1

0

2

Boundary scaling operators/dimensions

scaling dimension

(exact )

scaling dimension

(MERA)error

0 0 ---

1/2 0.499 0.2%

1+1/2 1.503 0.18%

2 2.001 0.07 %

2+1/2 2.553 2.1%

scaling dimension

(exact )

scaling dimension

(MERA)error

0 0 ---

2 1.992 0.4%

3 2.998 0.07%

4 4.005 0.12 %

4 4.062 1.5%

Iidentity

spin

Iidentity

Free BC

Fixed BC

Outline


• MPS and MERA



Introduction

Applications

Glen Evenbly


• Frustrated antiferromagnets

Applications to 2D lattices:

topological order

interacting fermions


Aguado, Vidal, Phys. Rev. Lett. 100, 070404 (2008)Koenig, Reichardt, Vidal, Phys. Rev. B 79, 195123 (2009)

Evenbly, Vidal, Phys. Rev. Lett 104, 187203 (2010)Lou, Harada, Kawashima (in preparation)

Corboz, Evenbly, Verstraete, Vidal, PRA 81, 010303(R) (2010) Pineda, Barthel, Eisert, PRA 81, 050303(R) (2010) Corboz, Vidal, PRB 80, 165129 (2009) Barthel, Pineda, Eisert, PRA 80, 042333 (2009)

2D MERA

Heisenberg antiferromagneton Kagome latticewith 2D MERA

frustrated spins

Evenbly, Vidal, Phys. Rev. Lett. 104, 187203(2010)

Heisenberg antiferromagnet on Kagome lattice

,

i j

i j

H S S

• favours antiferromagnetic alignment i jS S

• Geometric frustration

• Ground state?

(Quantum Monte Carlo techniques suffer from sign problem)


Singh, Huse, Phys. Rev. B 76, 180407 (2007)

Series Expansion

Marston, Zeng (1991), Syromyatnikov, Maleyev (2002) Nikolic, Senthil (2003), Budnik, Auerbach (2004)

Jiang, Weng, Sheng, Phys. Rev. Lett. 101, 117203 (2008)

DMRG

Ran, Hermele, Lee, Wen, Phys. Rev. Lett. 98, 117205 (2007)

Gutzwiller ansatz

Spin Liquid ?Valence Bond Crystal ?

Ground state computation with 2D MERA

• minimization of

,

i j

i j

S S

• initial state:

• movie !!! (click here)


Kagome.avi

MERA solution (infinite lattice with 36-spin unit cell):

Order VBC

1 -0.375

2 -0.42187

3 -0.42578

4 -0.43155

5 -0.43208

Series Expansion (Singh, Huse)

VBC

8 -0.4298

14 -0.4307

20 -0.4314

26 -0.4319

32 -0.4322

MERA (Evenbly, Vidal)

N E0/N

48 -0.43591

108 -0.4316

N E0/N

192 -0.42866(2)

432 -0.42863(2)

DMRG(Jiang, Weng, Sheng)

Gutzwiller ansatz(Ran, Hermele, Lee, Wen)

-0.420

-0.426

-0.432

-0.423

-0.429

5th4th

3rd

2nd

81420

26energy

series expansion

MERA

32DMRG

Gutzwiller

108N432N

Conclusions

• Characterize and classify phases of matter

• Variational approach for many-body problems

• 2D frustrated antiferromagnets

• emerges from natural ansatz for

• emerges from natural ansatz for

MPS DMRG gapped systems

MERA entanglement renormalization

critical systems

• Tensor network states:

Efficient representation of many-body states

Applications:

also interacting fermions • anyons • topological order • lattice gauge theory

• 1D quantum critical point MERA conformal field theory

My talk next week (Thursday morning):

Beyond the entropic boundary law with entanglement renormalization

Evenbly, Vidal, in preparation

L

1( ) L logDS L L

0( ) log logS L L L L

MERA

1D

L

0( ) =const.S L D

MPS

2D

L

( )S L D

PEPS

• Boundary law for entanglement entropy 1( ) LDS L ( ) LDS L(and not )

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Guifre Vidal- Entanglement Renormalization