Self-assembling tensor networks and holography in
disordered spin chains
Andrew M. Goldsborough, Rudolf A. Römer
Department of Physics and Centre for Scientific Computing
The University of Warwick
10
3
20
0
2.5
5
-2
1
10
| x2 - x1 |
100
=
tSDRG χ=10
5.81 exp[-0.63DTTN]
-2
-4
10
0
2
1.5
%failed
~| x2 - x1 |
10
SA,B
1
0
10
tSDRG χ=10
0.230 log2LB + 1.10
[log(2)/3] log2LB + 1.086
DTTN(x1 , x2)
2.94 ln |x2 - x1| + 3.02
2
<< sx . sx >>
DTTN
15
10
10
1
10
2
20
10
0
0
10
20
| x2 - x1 |
30
40
50
LB
A. M. Goldsborough, and R. A. Römer, Phys. Rev. B 89, 214203 (2014)
A. M. Goldsborough, S. A. Rautu, and R. A. Römer, arXiv:1406.4079 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Table of contents
1 Strong Disorder Renormalization
2 Tensor Networks
3 Tensor Network SDRG
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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Disorder and Condensed Matter Physics
Quenched disorder can dramatically effect observed phase
Smears phase transitions
New phases (L. Pollet, Comptes Rendus Physique 14, 712 (2013))
Localization (P. W. Anderson, Phys. Rev. 109, 1492 (1958))
Not integrable, but a lot gained from renormalization group
(F. Iglói, and C. Monthus,
Physics Reports 412, 277
(2005))
−4
x 10
8
14
7
12
6
10
5
∆µ
Numerically challenging
8
4
6
3
2
4
1
2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
U
Disordered Bose-Hubbard DMRG using ITensor
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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1D Disordered Heisenberg chain
H=
L−1
X
Ji~si · ~si+1
i=1
Paradigmatic model for quantum interacting system
Open boundary
Analytic solution when clean (Bethe Ansatz)
Coupling constants take a random value 0 < Ji < Jmax
Antiferromagnetic (AFM)
Use a box type distribution
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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The strong disorder renormalization group (SDRG)
The Ma, Dasgupta, Hu (MDH) algorithm for the random
anti-ferromagnetic (AFM) chain.
Remove the strongest coupled pair of spins and take the energy of the
singlet as the contribution to the total energy.
Renormalize the coupling between the neighbouring spins.
renormalize
Ji−1 Ji+1
J˜ =
Ji
S. K. Ma, C. Dasgupta, and C. K. Hu, Phys. Rev. Lett. 43, 1434 (1979)
A. M. Goldsborough (Warwick)
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The strong disorder renormalization group (SDRG)
Ground state is the random singlet phase
Mean spin-spin correlation function decays as a power-law†
h~si · ~sj i ∝
(−1)|i−j|
|i − j|2
Mean entanglement entropy (SA,B = −TrρA log2 ρA ) scales
logarithmically with subsystem size‡
SA,B ∼
†
‡
log2
log2 LB ' 0.231log2 LB
3
D. S. Fisher, Phys. Rev. B 50, 3799 (1994).
G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 (2004).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Some Applications of SDRG
Random AF spin-1/2 Heisenberg chain (S. Ma, C. Dasgupta, C. Hu.
PRL 43, 1434 (1979))
Random transverse field Ising model (D. Fisher PRL. 69, 534 (1992))
Quantum rotor model (E. Altman, Y. Kafri, A. Polkovnikov, G.
Refael. Physical Review B 81, 174528 (2010))
Random AF spin-1 Heisenberg chain with bilinear and biquadratic
interactions (V. Quinto, J. Hoyos, E. Miranda. ArXiv:1404.1924
(2014))
2 dimensional systems (O. Motrunich, S.-C. Mau, D. A. Huse, and D.
S. Fisher, Phys. Rev. B 61, 1160 (2000))
Recent reviews: G. Refael, and E. Altman, Comptes Rendus Physique
14, 725 (2013). F. Iglói, and C. Monthus, Physics Reports 412
277-431 (2005).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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The numerical renormalization group approach
Numerical renormalization group (NRG) scheme for the FM/AFM
chain.
Keep multiple eigenvectors at each step to increase accuracy.
Only keep full SU(2) blocks to retain spin-like energy spectrum.
Gap defined by the energy difference between the highest block
retained and lowest discarded.
renormalize
energy
T. Hikihara, A. Furusaki, and M. Sigrist, Phys. Rev. B 60,12116 (1999).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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The numerical renormalization group approach
Combine the blocks with the largest gap into a new block.
Diagonalize the block keeping multiple eigenvectors.
Express coupling Hamiltonians in terms of new blocks.
Diagonalize the couplings to update the distribution of gaps.
T. Hikihara, A. Furusaki, and M. Sigrist, Phys. Rev. B 60,12116 (1999).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Table of contents
1 Strong Disorder Renormalization
2 Tensor Networks
3 Tensor Network SDRG
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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Motivation
Want to model quantum many-body physics.
The Hilbert space grows exponentially with system size.
H ∼ dL
Want efficient simulation (polynomial in memory and time)
Want to be able to model various Hamiltonians (e.g. Bosons and
Fermions)
Want to avoid the sign problem of quantum Monte Carlo (M. Troyer,
U. Wiese, Phys. Rev. Lett. 94, 170201 (2005))
Want to measure physical properties and observables
R. Orús, Ann. Phys. 349, 117 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Notation
hΨ| Ĥ |Ψi =
X
X
X
X
X
σ1 ,...,σL σ10 ,...,σ 0 a1 ,...,aL a01 ,...,a0 b1 ,...,bL
L
L
σ ,σ 0
σ0
σ ,σ 0
σ0
2 2
∗σ1
∗σ2
1
2
A1,a
W1,b11 1 A1,a
0 Aa1 ,a2 Wb ,b Aa0 ,a0 . . .
1
1 2
1
1
0
0
σL−1 ,σL−1
σL−1
∗σL−1
. . . AaL−2
,aL−1 WbL−2 ,bL−1 Aa0
,a0
L−2
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
2
L−1
σ ,σ 0
σ0
L L
L
L
Aa∗σL−1
,1 WbL−1 ,1 Aa0
L−1 ,1
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Notation
Tensor networks are often represented using a graphical notation.
Vector:
V1
Vi =
V2
Matrix:
Mij =
M11 M12
M21 M22
Tensor:
Tijk...
S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. B 83, 115125 (2011)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Contraction
Connected lines denote a contraction.
Two vectors:
X
Vi Vi0 = S
=
i
Vector and matrix:
X
Vi Mij = Vj0
=
i
Two tensors:
X
Aij...k... Bi0 j 0 ...k... = Cij...i0 j 0 ...
k
=
S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. B 83, 115125 (2011)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Matrix product states (MPS)
The matrix product state takes the form:
X
X
σL−1
σL
|Ψi =
Maσ11 Maσ12a2 . . . MaL−2
aL−1 MaL−1 |σ1 i ⊗ · · · ⊗ |σL i
σ1 ,...,σL a1 ,...,aL−1
Basis of modern density-matrix renormalization group (DMRG) algroithms.
U. Schollwöck, Ann. Phys. 326, 96 (2011).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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The matrix product operator (MPO)
Operators can also be represented as the product of tensors:
O=
X
σ1 ,...σL
0
σ10 ,...,σL
b1 ,...,bL−1
σ ,σ10
Wb11
σ ,σ 0
σL−1 ,σ 0
σ ,σ 0
L L
L−1
Wb12,b22 . . . WbL−2 ,bL−1
WbL−1
|σ1 . . . σL i hσ10 . . . σL0 |
I. P. McCulloch, J. Stat. Mech. P10014 (2007)
A. M. Goldsborough (Warwick)
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Nearest neighbour interactions
The most simple one dimensional nearest-neighbour Hamiltonian with
OBCs takes the form:
L−1
X
H=
Ai Ai+1
i=1
which can be written as a MPO of the form:
[1]
W1,b1 = 1 A1 0 ,


1 Ai 0
[i]
Wbi−1 ,bi = 0 0 Ai  ,
0 0 1
 
0
[L]
WbL−1 ,1 = AL  .
1
1
1
2
2
3
3
G. M. Crosswhite and D. Bacon, Phys. Rev. A 78, 012356 (2008).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Contracting the MPO
Contracting the MPO along the virtual indices produces the desired
Hamiltonian.
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
H = A1 A2 + A2 A3 + A3 A4
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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On-site terms
On-site terms encode on-site potentials and magnetic fields. E.g.
H=
L−1
X
Ai Ai+1 +
i=1
L
X
Bi
i=1
where Bi are the on site terms.
[1]
W1,b1 = 1 A1 B1


1 Ai Bi
[i]
Wbi−1 ,bi = 0 0 Ai 
0 0 1
 
BL
[L]
WbL−1 ,1 = AL 
1
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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2
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3
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Heisenberg MPO Hamiltonian
Encode the Heisenberg Hamiltonian into an MPO:
H=
L−1
X
Ji~si · ~si+1
1
1
2
2
3
3
4
4
5
5
i=1

Wbi−1 ,bi
1
0

=
0
0
0
Ji +
2 si
Ji −
2 si
0
0
0
0
0
0
0
0

Ji szi 0

0
s−
i 
+
0
si 
0
szi 
0
1
I. P. McCulloch, J. Stat. Mech. P10014 (2007)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Heisenberg MPO Hamiltonian
Encode the Hamiltonian into an MPO:


Ji −
1 J2i s+
Ji szi 0
i
2 si
0

0
0
0
s−
i 

+

Wbi−1 ,bi = 0
0
0
0
si 
0
0
0
0
szi 
0
0
0
0
1
e.g. Wb2 ,b3 Wb3 ,b4
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Table of contents
1 Strong Disorder Renormalization
2 Tensor Networks
3 Tensor Network SDRG
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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SDRG on matrix product operators
Contract MPO tensors with largest gap
contract
fuse
Diagonalize on site component and keep the eigenvectors from the
lowest χ eigenvalues but keeping only full SU(2) blocks.
renormalize
energy
A. M. Goldsborough and R. A. Römer, Phys. Rev. B 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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SDRG on matrix product operators
Eigenvectors from diagonalization create an isometric tensor
=
Contract to perform renormalization on couplings
=
=
Diagonalize couplings to update distribution of gaps
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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SDRG on matrix product operators
One RG step
1
1
1
2
3
4
2
3
4
2
3
4
5
5
5
A. M. Goldsborough (Warwick)
renormalize
Phys. Rev. B 89, 214203 (2014)
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4
2
3
4
5
5
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SDRG on matrix product operators
One RG step
1
1
1
2
3
4
2
3
4
2
3
4
5
5
5
A. M. Goldsborough (Warwick)
renormalize
Phys. Rev. B 89, 214203 (2014)
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1
2
3
4
2
3
4
5
5
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SDRG on matrix product operators
One RG step
1
1
2
3
4
2
3
4
5
5
5
1
1
1
1
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
5
5
5
5
5
1
1
1
2
3
4
2
3
4
2
3
4
5
5
1
A. M. Goldsborough (Warwick)
renormalize
renormalize
Phys. Rev. B 89, 214203 (2014)
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SDRG as a tree tensor network
Holographic
Dimension
Lattice
Dimension
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Holography and the AdS/CFT correspondence
minimal surface
holographic dimension
S. Ryu, and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006)
B. Swingle, Phys. Rev. D 86, 065007 (2012)
G. Evenbly, and G. Vidal, J. Stat. Phys. 145, 891 (2011)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Geometry and tensor networks
minimal surface
geodesic
Entanglement scales with minimal surface separating regions A and B:
SA,B ∝ ∂A.
Correlations scale with the length of the geodesic connecting the sites:
C(x1 , x2 ) ∝ e−αD(x1 ,x2 )
G. Evenbly and G. Vidal, J. Stat. Phys., 145, 891, 2011.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Correlation Functions
Only need to contract the tensors in the causal cone
=
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realizations, χ = 10.
0
1
2
<< sx . sx >>
10
10
-2
tSDRG χ=10
10
-4
10
0
10
1
10
2
| x2 - x1 |
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Correlation Functions
SDRG prediction† of even-odd difference by a factor of 1.25.
Two-point correlation function is related to the average path length
joining the two points.
Proposal:
hh~si · ~sj ii ∝ he−αD(x1 ,x2 ) i ∼ e−αhD(x1 ,x2 )i ∝ e−alog|x2 −x1 | ∝ |x2 − x1 |a
20
DTTN ( x2 - x1 )
2.94 log | x2 - x1 | + 3.02
DTTN
15
10
5
0
†
1
10
100
| x2 - x1 |
J. Hoyos, A. Vieira, N. Laflorencie, E. Miranda, Phys. Rev. B 76, 174425 (2007).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realizations, χ = 10.
0
1
2
<< sx . sx >>
10
10
-2
tSDRG χ=10
1.67 < exp[DTTN] >
~| x2 - x1 |
~| x2 - x1 |
10
0.69
-2
-1.78
-4
10
0
10
1
10
2
| x2 - x1 |
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Entanglement Entropy
Von-Neumann entropy (SA|B = −TrρA log2 ρA ) proportional to the
number of bonds that need to be cut to separate A from B
A
A. M. Goldsborough (Warwick)
A'
Phys. Rev. B 89, 214203 (2014)
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Entanglement entropy for bipartitions (SA|B )
Average and average maximum entanglement entropy scale
logarithmically with system size
Need to increase bond dimension with length for DMRG to capture
the entanglement
3
SA|B
2.5
χ=4 average maximum
χ=4 average
0.358 log2L + 0.411
0.096 log2L + 0.671
2
1.5
1
10
20
30
40
50
60 70 80 90100
L
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Entanglement entropy for blocks (SA,B )
Rafael/Moore† : effective CFT with central charge c̃ = 1 · log2 and
SA,B '
log2
log2 LB ' 0.231logLB
3
3
SA,B
2.5
tSDRG χ=10
0.230 log2LB + 1.10
[log(2)/3] log2LB + 1.086
2
%failed
1.5
20
10
0
0
10
20
30
40
50
LB
†
G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 (2004).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Entanglement per bond
entanglement per bond the same for both bipartitions and blocks
S
= 0.47 ± 0.02
nA
L
SA,B, SA,B/nA
SA|B, SA|B/nA
0
1.5
10
20
10
20
30
40
50
30
40
50
1
0.5
2
1.5
1
0.5
0
LB
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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Conclusions
We have made a self constructing, inhomogeneous tensor network
based on SDRG.
We reproduce the power law mean correlation function and
logarithmic entanglement entropy scaling.
The correlation functions and entanglement entropy support the
notion of a holographic tensor network state.
Outlook: Random spin-1, spin-3/2 Heisenberg model.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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Ongoing work
ITensor DMRG for disordered systems (itensor.org)
Phase diagram for finite disordered Bose-Hubbard model using DMRG
Path lengths in regular m-ary tree graphs (arXiv:1406.4079)
Path lengths in Catalan tree graphs (in preparation)
Random spin-1, spin-3/2 Heisenberg model
Excited state DMRG for localized states
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Generalization to FM/AFM chains
Westerberg et. al. generalized the SDRG to the random
ferromagnetic/anti-ferromagnetic (FM/AFM) chain.
Decimation position determined by largest energy gap
|Ji |(Si + Si+1 )
Ji < 0
∆i =
Ji (|Si − Si+1 | + 1) Ji > 0
Combine the pair of spins with the largest energy gap to a new
composite spin
S̃ = |Si − sgn(Ji )Si+1 |
renormalize
E. Westerberg, A. Furusaki, M.Sigrist, and P. A. Lee, Phys. Rev. Lett. 75, 4302 (1995).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Generalization to FM/AFM chains
renormalize
Find the energy gaps for the new spin
˜ i−1 = ∆i−1 f (Si−1 , Si , Si+1 )
∆
˜ i+1 = ∆i+1 f (Si , Si+1 , Si+2 )
∆
If Si = Si+1 and J > 0 the pair form a singlet
2Ji−1 Ji+1
J˜ =
Si (Si + 1)
3Ji
renormalize
E. Westerberg, A. Furusaki, M.Sigrist, and P. A. Lee, Phys. Rev. Lett. 75, 4302 (1995).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Typical correlation
-2.035 r - 0.864
0.5
-2.284 r - 1.415
1
1
-5
2
0.5
-10
2
< log | < sx . sx > | >
tSDRG L=150 χ=4
tSDRG L=150 χ=10
tSDRG L=150 χ=50
tSDRG L=500 χ=20
< log10 | < sx . sx > | >
0
0
-20
-10
-30
0
10
5
| x2 - x1 |
A. M. Goldsborough (Warwick)
15
1/2
Phys. Rev. B 89, 214203 (2014)
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Entanglement Entropy
L=30, AFM, 100 disorder realizations.
2.5
SA|B
2
tSDRG χ=20 (average)
tSDRG χ=20 (average max)
vMPS χ=40
vMPS χ=20
vMPS χ=10
1.5
1
1 - ∆J / 2 < Ji < 1 + ∆J / 2
0.5
0
0.5
1
1.5
2
∆J
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Entanglement entropy and the area-law
Entanglement entropy is defined as
SA,B = −TrρA log2 ρA
where ρA is the reduced density matrix for region A.
General states satisfy a volume law
SA,B ∝ A
Ground states obey an area law
SA,B ∝ ∂A
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Geometry and tensor networks
MPS:
SA,B ∝ const.
C(x1 , x2 ) ∝ e−|x1 −x2 |/ξ
PEPS: SA,B ∝ L
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
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Holographic tensor networks
Multi-scale entanglement renormalization ansatz (MERA).
Dhol (x1 , x2 ) ∝ log|x1 − x2 |
CM ERA ∝ e−αDhol
∝ e−alog|x1 −x2 |
∝ |x1 − x2 |a
SA,B ∝ logL
Figure courtesy of G. Evenbly.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
Aachen 03/02/2015
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