entanglement renormalization and conformal field theory
TRANSCRIPT
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Guifre Vidal
Les Houches 4th Mar 2010
Workshop
Physics in the plane
Entanglement Renormalizationand
Conformal Field Theory
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Brisbane
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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
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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
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• 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
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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
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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
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disentangler
wisometry
udisentangler
1,2, ,
w
†wII
u†u
k=1,2,…,r
isometry
MERA (multi-scale entanglement renormalization ansatz)
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isometry0 0
unitary
(2)
(1)
(0)
0000 00 00 00 00
0000
00
0N
MERA as a quantum circuit
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• 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‟
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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
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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)
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Lτ+1
Lτo
1o
RG transformation
L3
L2
L1
L0
The MERA defines a coarse-graining transformation
• transformation of local operators
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The MERA defines a coarse-graining transformation
RG transformationL L’
o o’
o
o
disentangler
Iu†u
w
†wI
isometry
oo’
o
Ascending superoperator
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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
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L3
L2
L1
L0 0
1
2
3
• descending superoperator
0 1 2
1( ) 1 1 1 Lτ+1
Lτ
''
( ')L ( ')C ( ')R
'
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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)
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(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
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(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
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(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
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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
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• 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
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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
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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
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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
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• non-local scaling operators (bulk)
Recent developments:
• boundary critical phenomena
• defects (in bulk)
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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
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gV
gV
• “the symmetry commutes with the coarse-graining”
• non-local operators of the form
o
gV ogV o
can be “locally” coarse-grained
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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
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• Example: Ising model
Non-local scaling operators Scale invariant MERA (bulk)
local non-local
I identity
spin
energy
disorder
fermions
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• 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
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• 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
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• Example: quantum XX model
=54 =32
(exploiting U(1) symmetry)
XX 1 1( )i i i iH X X YY
G = U(1) symmetry
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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
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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
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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)
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Bulk expectation values in the presence of a boundary
( )rr
• bulk
( )r ( )rr r2
1( ) ( ')
| ' |r r
r r
• boundary
1( )
| |r
r
( ) 0r
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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
( )
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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
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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
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Lw RwTw
Finite system with two boundaries
• finite MERA with two boundaries
• Energy spectrum
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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
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Lattice Defects:
-XX -XX -XX -XX -XX -XX -XX
Z Z Z Z Z Z Z Z Z
-αXX
Hamiltonian:
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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
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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
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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
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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
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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)
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• 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