guifre vidal- entanglement renormalization in two spatial dimensions: frustrated antiferromagnets...
TRANSCRIPT
Erwin Schrodinger Institute (ESI) Thu 13th Aug 2009
Workshop: Quantum Computation
and Quantum Spin Systems
Entanglement Renormalizationin two spatial dimensions:
frustrated antiferromagnets
Guifre Vidal
frustrated antiferromagnetsand interacting fermions
Entanglement Renormalization/MERA
Outline
• Quantum Computation
MERA = multi-scale entanglement renormalization ansatz
• Renormalization Group
transformation
• 1D systems • 2D systems
• spin systems(commuting variables)
• fermion systems(anticommuting variables)
Tensor Network representation of a many-body state
1 2
1 2
... 1 21 1 1
...N
N
d d d
GS i i i Ni i i
c i i i= = =
Ψ =∑∑ ∑⋯
1 2 ... N⊗ ⊗ ⊗H H H⋯d
N
1D TTNMPS (1D)
...1i
2i 3iNi
4i
Nd coefficients
1D TTN
PEPS
(2D)1D MERA
MPS (1D)
( )O N coefficients
Disclaimer
• No proofs (actually, no theorems) Merlin and Arthur
• Variational ansatz that is physically motivated (locality, entanglement)
• Numerical/analytical evidence that the MERA approximates ground states well in several particular cases.
• Counter-examples? Sure (but this is not the point)
Entanglement Renormalization/MERA
Outline
• Quantum Computation
MERA = multi-scale entanglement renormalization ansatz
• Renormalization Group
transformation
• 1D systems • 2D systems
• spin systems(commuting variables)
• fermion systems(anticommuting variables)
Tree Tensor Network (TTN)
• coarse-graining transformation:
=w
†w
I*( )ijk ijk
ijk
µ νµνδ=∑w w
LΨ
'L'Ψ
w
isometry:ijkµw
w
'L
LΨ
'Ψ
( 2 )L
(1)L
( 0 )L
( )0Ψ
( )1Ψ
( )2Ψ
Tree Tensor Network (TTN)
• quantum circuit:
LΨ
isometry0 0
unitary=
'L23' 0N⊗Ψ ⊗ 'L
LΨ
3' 0⊗Ψ ⊗
00 00 00 00 00
( 2 )L
(1)L
( 0 )L
00
0000 00 00 00 00 00 00 00
00 00
000
Ψ
0N⊗
Multi-scale Entanglement Renormalization Ansatz (MERA)
• Entanglement renormalization -- coarse-graining transformation:
LΨ
'Ψ 'L
=w
†w
Iw
isometry
=u
†u
unitary(disentangler)
u
Vidal, Phys. Rev. Lett. 99, 220405 (2007); ibid 101, 110501 (2008)
Ψ
'Ψ w
u
L
'L
( 2 )L
(1)L
( 0 )L
( )0Ψ
( )1Ψ
( )2Ψ
′L
L
L
w
u
w
′L
• why do we want disentanglers ?
without disentanglers: with disentanglers:
entangled
sites
unentangled
site
L
L u
Multi-scale Entanglement Renormalization Ansatz (MERA)
• quantum circuit
LΨ
isometry0 0
unitary=
( 2 )L
(1)L
( 0 )L
0000 00 00 00 00
0000
00
Ψ
0N⊗
Quantum Circuit as a Many-Body Ansatz
Ψ
0N⊗
“generic” quantum circuit ansatz
Ψ
0N⊗
tree quantum circuit ansatz
Quantum Circuit as a Many-Body Ansatz
• Entanglement ?
Ψ
0N⊗
NS ∼entanglement
(1)S O∼entanglement
Ψ
0N⊗
/2 2NNS ∼
entanglemententropy /2 (1)NS O∼
entanglemententropy
Quantum Circuit as a Many-Body Ansatz
Entanglement
Ψ
0N⊗
Ψ
0N⊗
NS ∼ (1)S O∼entanglement entanglement
Ψ
0N⊗
/2 2NNS ∼ /2 (1)NS O∼
entanglemententropy
entanglemententropy
/2 log( )NS N∼entanglemententropy
Quantum Circuit as a Many-Body Ansatz
• Causal cone/cost ?
( )O Nwidth ~ (1)Owidth ~
Ψ
0N⊗
Ψ
0N⊗
exp( )Ncost ~
( )O Nwidth ~
cost ~
(1)Owidth ~
(log )O N
Quantum Circuit as a Many-Body Ansatz
Causal cone
Ψ
0N⊗
Ψ
0N⊗
( )O Nwidth ~ (1)Owidth ~
Ψ
0N⊗
( )O Nexp( )Ncost ~
width ~ (1)Ocost ~
width ~
(1)O
(1)O
cost ~
width ~
(log )O N
Summary: 1D MERA = quantum circuit + small causal cones = variational ansatz
Ψ
0N⊗
• efficient representation parameters4( log )O NχN
log N
• efficient representation parameters4( log )O Nχ(translation invariance)
• efficient simulationcomputational
costs
8( log )O Nχ(translation invariance)
• non-trivially entangled /2 logNS N∼entanglement entropy
• good approximation to ground states, including critical systems
( 3 )L
( 2 )L
(1)L
( 0 )L
⋯ ⋯
⋮
Example: scale invariant MERA
critical exponents
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)
Aguado, Vidal, Phys. Rev. Lett. 100, 070404 (2008)Koenig, Reichardt, Vidal, Phys. Rev. B 79, 195123 (2009)
critical systems(1D)
topologically ordered systems
(2D)
OPE, CFT
1D critical systems
Computation of ground state energy (1D scale invariant MERA)
Evenbly, Vidal, Phys. Rev. B 79, 144108 (2009)
10101010-5-5-5-5
10101010-4-4-4-4
10101010-3-3-3-3
1D MERA, Ground-Energies1D MERA, Ground-Energies1D MERA, Ground-Energies1D MERA, Ground-Energies
Ene
rgy
Err
or,
Ene
rgy
Err
or,
Ene
rgy
Err
or,
Ene
rgy
Err
or,
∆∆ ∆∆EE EE
IsingIsingIsingIsingXXXXXXXXHeisenbergHeisenbergHeisenbergHeisenbergPottsPottsPottsPotts
8888 16161616 24242424 32323232 40404040 48484848 56565656
10101010-9-9-9-9
10101010-8-8-8-8
10101010-7-7-7-7
10101010-6-6-6-6
χχχχ
Ene
rgy
Err
or,
Ene
rgy
Err
or,
Ene
rgy
Err
or,
Ene
rgy
Err
or,
Bulk
1D critical systems
Scaling dimensions/critical exponents
scaling
dimension
scaling
dimension error
0 0 ----
CFT∆MERA∆
(I)
Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)
2
1(0) ( )r
rφ φ ∆∼ 1
x x zr r r
r r
H σ σ σ+= − −∑ ∑Quantum Ising chain
Iσ
ε
1/ 8
1 1 / 8+
2 1/ 8+2
0
1
0
2
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%
(I)( )σ( )ε
Entanglement Renormalization/MERA
Outline
• Quantum Computation
MERA = multi-scale entanglement renormalization ansatz
• Renormalization Group
transformation
• 1D systems • 2D systems
• spin systems(commuting variables)
• fermion systems(anticommuting variables)
1D MERA
2D MERA
w
u
L
'L
2-site causal cone
Evenbly, Vidal, Phys. Rev. B 79, 144108 (2009)
wv uL 'L
2D MERA
2x2-site causal cone
• efficient representation parameters( log )qO Nχ(translation invariance)
• efficient simulationcomputational
costs
'( log )qO Nχ(translation invariance)
• Entropic area law LxLS L∼entanglement entropy
• good approximation to ground states
2D MERA
Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, arXiv:0710.0692 Evenbly, Vidal, arXiv:0801.2449Cincio, Dziarmaga, Rams, Phys. Rev. Lett. 100, 240603 (2008)Evenbly, Vidal, Phys. Rev. Lett. 102, 180406 (2009)
Aguado, Vidal, Phys. Rev. Lett. 100, 070404 (2008)Koenig, Reichardt, Vidal, Phys. Rev. B 79, 195123 (2009)
Evenbly, Vidal, arXiv:0904.3383
Corboz, Evenbly, Verstraete, Vidal, arXiv:0904.4151Pineda, Barthel, Eisert, arXiv:0905.0669 Corboz, Vidal, arXiv:0907.3184Barthel, Pineda, Eisert, arXiv:0907.3689
topological order(analytical)
scalable simulations
frustrated spins
interacting fermions
• good approximation to ground states
Example: Heisenberg antiferromagnet on Kagome latticeEvenbly, Vidal, arXiv:0904.3383
,i j
i j
H S S= ⋅∑Hamiltonian:
• Favours antiferromagnetic alignment
• Geometrically frustrated
• Ground state?• Ground state?
(Quantum Monte Carlo techniques suffer from sign problem)
Example: Heisenberg antiferromagnet on Kagome lattice
- Valence Bond Crystal ? - Spin Liquid ?
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
2D MERA (!!?)
Example: Heisenberg antiferromagnet on Kagome latticeEvenbly, Vidal, arXiv:0904.3383
Example: Heisenberg antiferromagnet on Kagome lattice
MERA solution (infinite lattice with 36-spin unit cell):
Order VBC
0 -0.375
1 -0.375
2 -0.42187
3 -0.42578
Singh, Huse, Phys. Rev. B 76, 180407 (2007)
series expansion
Evenbly, Vidal, arXiv:0904.3383
3 -0.42578
4 -0.43155
5 -0.43208
VBC
8 -0.4298
14 -0.4307
20 -0.4314
26 -0.4319
1-layer MERA
χ-0.420
-0.426
-0.432
-0.423
-0.429
5th4th
3rd
2nd
8χ =14χ =20χ =
26χ =energy
series exp.
MERA
Entanglement Renormalization/MERA
Outline
• Quantum Computation
MERA = multi-scale entanglement renormalization ansatz
• Renormalization Group
transformation
• 1D systems • 2D systems
• spin systems(commuting variables)
• fermion systems(anticommuting variables)
Fermionic MERA
( 2 )L
(1)L
( 0 )L
0000 00 00 00 00
0000
00
Ψ
0N⊗
Corboz, Evenbly, Verstraete, Vidal, arXiv:0904.4151Corboz, Vidal, arXiv:0907.3184
Pineda, Barthel, Eisert, arXiv:0905.0669 Barthel, Pineda, Eisert, arXiv:0907.3689
( 0 )L
Ψ
quantum wire
1 2 2 1a a a a=Bosons: b =
Fermions:1 2 2 1a a a a= − f = swap
Fermionic MERA
quantum wire
1 2 2 1a a a a=Bosons:
Fermions: 1 2 2 1a a a a= −
b
f
=
= swap
parity sectors + + + + + − + −( ) ( )V V V+ −≅ ⊕
parity sectors
swap =+ + + +
swap =+ − + −
swap =+ − +−
swap = −−− − −
Fermionic MERA
Benchmark results1) free spinless fermions on 6x6 lattice (periodic BC)
† † † †
,
[ ( )] 2free r s r s r s r rr s r
H c c c c c c c cγ λ= − + −∑ ∑pairing potential
chemical potential
0 0
†( ) r r rC r c c+=0 0r r r+
Fermionic MERA
Benchmark results
† † † †free
,
[ ( )] 2r s r s r s r rr s r
H c c c c c c c cγ λ= − + −∑ ∑
2) interacting spinless fermions on a 6x6 lattice (PBC)
pairing potential
chemical potential
† †int free
,r r s s
r s
H H V c c c c= + ∑
interaction
Fermionic MERA
Benchmark results
† † † †free
,
[ ( )] 2r s r s r s r rr s r
H c c c c c c c cγ λ= − + −∑ ∑pairing potential
chemical potential
† †int free
,r r s s
r s
H H V c c c c= + ∑
2) interacting spinless fermions on a 6x6 lattice (PBC)
interaction
† †( ) k kP k c c−=
| ( ) |totk
P P k=∑
pairing amplitude
Fermionic MERA
† † † †
,
[ ( )] 2free r s r s r s r rr s r
H c c c c c c c cγ λ= − + −∑ ∑pairing potential
chemical potential
3) scalability: free spinless fermions on a LxL lattice (PBC)with L up to 162
Benchmark results
MERA = quantum circuit + small causal cone
Conclusions
• 1D: quantum criticality
• 2D: frustrated antiferromagnetsinteracting fermions
• cost of simulations entanglement
What can be simulated efficiently?