Guifre Vidal

Les Houches 4th Mar 2010

Workshop

Physics in the plane

Entanglement Renormalizationand

Conformal Field Theory

Brisbane

Brisbane

Quantum Information

QuantumMany-Body Physics

Computational Physics

• many-body entanglement• computational complexity

•renormalization group•quantum criticality•frustrated antiferromagnets•interacting fermions•topological order

• simulation algorithmsstrongly correlated systems

1D,2D lattices

Research group at the University of Queensland

faculty postdocs students

Guifre VidalIan McCulloch

Philippe CorbozKaren DancerAndy FerrisRoman OrusLuca Tagliacozzo

Andrew BirrellGlen EvenblyXiaoli HuangJacob JordanShahla NikbakhtRobert PfeiferSukhi Singh

announcement: postdoctoral position available

• critical fixed point continuous quantum phase transition

CFT

Entanglement Renormalization/MERA

Outline

• Quantum Computation

MERA = Multi-scale Entanglement Renormalization Ansatz

• Renormalization Group (RG)

transformation

Scale invariant MERA RG fixed point

boundary defect bulk(local/non-local)

collaboration with

Glen Evenbly, R. Pfeifer, P. Corboz, L. Tagliacozzo, I.P. McCulloch

V. Pico (U. Barcelona)S. Iblisdir (U. Barcelona)

Scale invariant MERA Holographic principle

i1 i2

iN...

... ...

1 2

1 2

... 1 2

1 1 1

...N

N

d d d

GS i i i N

i i i

c i i i

• Lattice with N sites

1 2 ... Nd

N

PEPS(2D)

MPS (1D)

( )O N coefficients

tensor network ansatz/state

i1 i2i17

i18...

... ...

1 2

1 2

... 1 2

1 1 1

...N

N

d d d

GS i i i N

i i i

c i i i

• Lattice with N sites

1 2 ... Nd

N

MERA (multi-scale entanglement renormalization ansatz)

Vidal, Phys. Rev. Lett. 99, 220405 (2007); ibid 101, 110501 (2008)

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16 i17 i18

disentangler

wisometry

udisentangler

1,2, ,

w

†wII

u†u

k=1,2,…,r

isometry

MERA (multi-scale entanglement renormalization ansatz)

isometry0 0

unitary

(2)

(1)

(0)

0000 00 00 00 00

0000

00

0N

MERA as a quantum circuit

• Many possibilities

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16 i17 i18

u

w

k=1,2,…,R

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16

k=1,2,…,R

uw

examples: binary 1D MERA ternary 1D MERA

g

†gI

• Defining properties:

1) isometric tensors

2) past causal conewith bounded „width‟

What is the MERA useful for?

• ground states/low energy subspaces in 1D, 2D lattices

• at criticality: critical exponents/CFTN

p

(no translationinvariance !!! )

+ (scale invariant) boundary/defect/

interface

logp Ntranslationinvariance

(log( ))O N

Simulation costs:

pN

inhomogeneous

( )O N

p

translation& scale

invariance

(1)O

RG transformation

The MERA defines a coarse-graining transformation

L0 0

L1 1

L2 2

L3 3

Lτ+1

Lτ

isometry

disentangler

1

'

Entanglement renormalization

Vidal, Phys. Rev. Lett. 99, 220405 (2007); ibid 101, 110501 (2008)

Lτ+1

Lτo

1o

RG transformation

L3

L2

L1

L0

The MERA defines a coarse-graining transformation

• transformation of local operators

The MERA defines a coarse-graining transformation

RG transformationL L’

o o’

o

o

disentangler

Iu†u

w

†wI

isometry

oo’

o

Ascending superoperator

L3

L2

L1

L0 0o

1o

2o

3o

• ascending superoperator

0o 1o 2o

1 ( )o o

1 ( )Lo o 1 ( )Co o 1 ( )Ro o

o

1o

o o

1o1o

Lτ+1

Lτ

past causal conewith bounded width

Local operators remain local

L3

L2

L1

L0 0

1

2

3

• descending superoperator

0 1 2

1( ) 1 1 1 Lτ+1

Lτ

''

( ')L ( ')C ( ')R

'

L3

L2

L1

L0

• Description of the system at different length scales

• Ascending and descending super-operators: change of length scale (or time in a quantum computation)

(3)

(2)

(1)

(0)

Scale invariant MERA

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, arXiv:0710.0692 Evenbly, Vidal, arXiv:0801.2449Giovannetti, 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

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

critical systems (1D)

topologically ordered systems

(2D)

critical exponentsOPE, CFT

boundary & defectsnon-local operators

(3)

(2)

(1)

(0)

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)

Evenbly, Vidal, arXiv:0710.0692 Evenbly, Vidal, arXiv:0801.2449

MERA is a good ansatz for ground state at quantum critical point

• it recovers scale invariance! same state and Hamiltonian at each level[shown for Ising model, free fermions, free bosons]

• proper scaling of entanglement entropy

Scale invariant MERA

/2 log( )NS N log( )N

N

(3)

(2)

(1)

(0)

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)

Evenbly, Vidal, arXiv:0710.0692 Evenbly, Vidal, arXiv:0801.2449

MERA is a good ansatz for ground state at quantum critical point

• it recovers scale invariance! same state and Hamiltonian at each level[shown for Ising model, free fermions, free bosons]

• proper scaling of entanglement entropy

Scale invariant MERA

• polynomial decay of correlations

o 'o ''oconstant ascending superoperator

' ( )o oscaling

superoperator

( )o

Vidal, Phys. Rev. Lett. 101, 110501 (2008)• polynomial decay of correlations

Scale invariant MERA

log( )r

1 2| |r s s

log( )q zlog( ) log( )

2 1 2( , ) ,r z qC s s z r r

1o 2o

1o 2oz

1o 2o2z

• Eigenvalue decomposition Giovannetti, Montangero, Fazio, Phys. Rev. Lett. 101, 180503 (2008)

( )3scaling operator

scaling dimension 3log

1 1( )o zo 2 2( )o zo

• Eigenvalue decomposition Giovannetti, Montangero, Fazio, Phys. Rev. Lett. 101, 180503 (2008)

( )3scaling operator

scaling dimension 3log

Critical exponents can be extracted from the scaling superoperator(= quMERA channel, MERA transfer map)

• Connection to Conformal Field Theory Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)

spin lattice at quantum critical point

CFT

• central charge c

• primary fieldsp

conformal dimensions ( , )p ph hp p ph h

• operator product expansion OPE

p p pC

p p pC

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

• operator product expansion OPEfrom three point correlators

L3

L2

L1

L0( )x ( )y ( )z

( ) ( ) ( )| | | | | |

Cx y z

x y y z z x

Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)• Example: Ising model

Scale invariant MERA (bulk)

scaling operators/dimensions: scaling dimension

(exact )

scaling dimension

(MERA)error

0 0 ----

0.125 0.124997 0.003%

1 0.99993 0.007%

1.125 1.12495 0.005%

1.125 1.12499 0.001%

2 1.99956 0.022%

2 1.99985 0.007%

2 1.99994 0.003%

2 2.00057 0.03%

Iidentity

spin

energy

=36 =20

• Operator product expansion (OPE):

12C

0C C C

C4( 5 10 )

I

I+fusionrules

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,

E

Ising

3-state Potts

XX

Heisenberg

• Example: other models

• non-local scaling operators (bulk)

Recent developments:

• boundary critical phenomena

• defects (in bulk)

Non-local scaling operators Evenbly, Corboz, Vidal, arXiv: 0912.2166

uw

Ising Ising

N NZ H Z H

Ising 1i i iH X X Zexample[ ]

†NN

g gV H V H• global symmetry G g G

u†u

w

†wu u

†

gV†

gV

gV

w w

gV

• G-symmetric MERA

gV

gV

• “the symmetry commutes with the coarse-graining”

• non-local operators of the form

o

gV ogV o

can be “locally” coarse-grained

1

3

ou wo

scaling superoperator

' ( )o o

( )scaling operator

scaling dimension

3

3log

• local scaling operators

1

3

ou wgV

o

, , ,( )g g g g

,g

,g

,

, ,3 g

g gnon-local scaling operator

scaling dimension , 3 ,logg g

modified scaling superoperator

' ( )go o

• non-local scaling operators

• Example: Ising model

Non-local scaling operators Scale invariant MERA (bulk)

local non-local

I identity

spin

energy

disorder

fermions

• Example: Ising model

Non-local scaling operators Scale invariant MERA (bulk)

local non-localscaling dimension

(exact )

scaling dimension

(MERA)error

1/8 0.1250002 0.0002%

1/2 0.5 <10−8%

1/2 0.5 <10−8%

1+1/8 1.124937 0.006 %

1+1/2 1.49999 < 10−5%

1+1/2 1.49999 < 10−5%

2+1/8 2.123237 0.083 %

2+1/8 2.124866 0.006 %

2+1/8 2.125487 0.023 %

disorder

fermions

• Example: Ising model

Non-local scaling operators Scale invariant MERA (bulk)

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

• Example: quantum XX model

=54 =32

(exploiting U(1) symmetry)

XX 1 1( )i i i iH X X YY

G = U(1) symmetry

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

uw

• infinite chain (bulk)

bulk MERA

uw

w

w

u

w

MERA with boundary

• semi-infinite chain (boundary)

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

0 10 20 30 40 50 60

0.65

0.7

0.75

0.8

0.85

Distance from boundary

<z >

exact

MERA

bulk

Free boundary conditions: local magnetization• Example: Ising model

100

102

104

106

108

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

distance from boundary, d

<z>

MERA - 2/

<z>

exact - 2/

<Bulk

z>

MERA -2/

| <z>

exact - <

z>

MERA |

• Boundary effects still noticeable far away from the boundary

• MERA gets correct magnetization everywhere (approx. same accuracy as with bulk MERA without boundary)

Bulk expectation values in the presence of a boundary

( )rr

• bulk

( )r ( )rr r2

1( ) ( ')

| ' |r r

r r

• boundary

1( )

| |r

r

( ) 0r

o

o

o

L

L

L

L

uww

Boundary scaling operators/dimensions

o

ow

†w( )o o

boundary scaling superoperator

boundary scaling operator

boundary scalingdimension

3

3log

( )

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

Lw RwTw

Finite system with two boundaries

• finite MERA with two boundaries

• Energy spectrum

wu

Du

,D Rwu

w,D Lw

defect

LwLu

RuRw

Iu

,I Lw ,I Rw

left region

right regioninterface

defect / interface Evenbly, Pfeifer, Pico, Iblisdir, Tagliacozzo, McCulloch,Vidal,arXiv:0912.1642

Lattice Defects:

-XX -XX -XX -XX -XX -XX -XX

Z Z Z Z Z Z Z Z Z

-αXX

Hamiltonian:

10 20 30 40 500.5

0.55

0.6

0.65

0.7

0.75

Lattice Site

<

z >

= 1.5

= 1

= 0.8

= 0

= 2 decoupled semi-infinite chains with Free BC

2 coupled semi-infinite chainswith “Fixed” BC

Summary: scale invariant MERA,critical phenomena, and CFT

• Bulk

• Boundary

• Defect

• (local & non-local) scaling operators/dimensions• CFT: primary fields and OPE

• boundary scaling operators• BCFT: primary fields and OPE

• defect scaling operators/dimensions

• interface

• finite system with two boundaries

MERA and the AdS/CFT correspondence Swingle, arXiv:0905.1317

scale invariant MERA

vacuum of CFT

1+1 CFT 2+1 AdS

t

x

t

x

z

MERA and the AdS/CFT correspondence

1+1 CFT 2+1 AdS

t

x

(fixed t)

x

z

z

x

t

scale invariant MERA = lattice version of the holographic principle

MERA and the AdS/CFT correspondence

1+1 CFT 2+1 AdS

t

x x

z

t

Ryu ,Takayanagi, PRL96, 181602 (2006)

A

Holographic derivationof entanglement entropy

| | log | |A AS A

A (boundary of )A

A (geodesic line)

• critical fixed point continuous quantum phase transition

CFT

Entanglement Renormalization/MERA

Outline

• Quantum Computation

MERA = Multi-scale Entanglement Renormalization Ansatz

• Renormalization Group (RG)

transformation

Scale invariant MERA RG fixed point

boundary defect bulk(local/non-local)

collaboration with

Glen Evenbly, R. Pfeifer, P. Corboz, L. Tagliacozzo, I.P. McCulloch

V. Pico (U. Barcelona)S. Iblisdir (U. Barcelona)

Scale invariant MERA Holographic principle