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
| x2 - x1 |
20
10
0
0
10
20
30
40
50
LB
A. M. Goldsborough and R. A. Römer, arXiv:1401.4874
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
1 / 22
1D Disordered Heisenberg chain
H=
L−1
X
Ji~si · ~si+1
i=1
Open boundary.
Analytic soultion when clean (Bethe Ansatz).
Coupling constants take a random value 0 < Ji < Jmax
Antiferromagnetic (AFM)
Use a box type distribution.
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
2 / 22
The strong disorder renormalisation 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.
Renormalise 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., vol.43, p.1434, Nov.1979
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
3 / 22
The strong disorder renormalisation group (SDRG)
Ground state is the random singlet phase
Mean spin-spin correlation function decays as a power-law (Fisher)
(−1)|i−j|
h~si · ~sj i ∝
|i − j|2
Mean entanglement entropy (SA,B = −TrρA log2 ρA ) scales
logarithmically with subsystem size (Refael and Moore)
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)
arXiv:1401.4874
DPG Dresden 2014
4 / 22
The numerical renormalisation group approach
Numerical renormalisation group (NRG) scheme for the FM/AFM
chain.
Keep multiple eigenvectors at each step to increase accuracy.
Combine the blocks with the largest gap into a new block.
Diagonalise the block keeping multiple eigenvectors.
diagonalise 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)
arXiv:1401.4874
DPG Dresden 2014
5 / 22
Tensor networks example - 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 , . . . , σL i
σ1 ,...,σL a1 ,...,aL−1
a1
σ1
a2
σ2
aL-2
σ3
σL-2
aL-1
σL-1
σL
The Hilbert space can be reduced by capping the size of the a indices,
making large systems tractable whilst retaining entanglement information.
U. Schollwöck, Ann. Phys. 326, 96 (2011).
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
6 / 22
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
A. M. Goldsborough (Warwick)
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 |
arXiv:1401.4874
DPG Dresden 2014
7 / 22
Heisenberg MPO Hamiltonian
Encode the Hamiltinian 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

1
Wb2 ,b3 Wb3 ,b4
0

=
0
0
0
J4 +
2 s4
J4 −
2 s4
0
0
0
0
0
0
0
0
J4 sz4
0
0
0
0
J3 + −
2 s3 s4
+
J3 − +
2 s3 s4
s−
4
s+
4
sz4
+ J3 sz3 sz4
1






U. Schollwöck, Ann. Phys. 326, 96 (2011).
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
8 / 22
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.
Contract to perform renormalization
=
=
arXiv:1401.4874
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
9 / 22
SDRG as a tree tensor network
Holographic
Dimension
Lattice
Dimension
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
10 / 22
Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realisations.
0
1
2
<< sx . sx >>
10
10
-2
tSDRG χ=10
5.81 exp[-0.63DTTN]
~| x2 - x1 |
10
-2
-4
10
0
10
1
10
2
| x2 - x1 |
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
11 / 22
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)
arXiv:1401.4874
DPG Dresden 2014
12 / 22
Correlation Functions
Only need to contract the tensors in the causal cone
Proposal:
h~si · ~sj i ∝ e−αhD(x1 ,x2 )i ∝ e−αβlog|x1 −x2 | ∝ |x1 − x2 |−a
20
DTTN ( x2 - x1 )
2.94 log | x2 - x1 | + 3.02
DTTN
15
10
5
0
1
10
100
| x2 - x1 |
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
13 / 22
Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realisations.
0
1
2
<< sx . sx >>
10
10
-2
tSDRG χ=10
5.81 exp[-0.63DTTN]
~| x2 - x1 |
10
-2
-4
10
0
10
1
10
2
| x2 - x1 |
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
14 / 22
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'
arXiv:1401.4874
DPG Dresden 2014
15 / 22
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)
arXiv:1401.4874
DPG Dresden 2014
16 / 22
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)
arXiv:1401.4874
DPG Dresden 2014
17 / 22
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)
arXiv:1401.4874
DPG Dresden 2014
18 / 22
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.
Please come to my poster: TT 80.21 on Wednesday at 15:00
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
19 / 22
Entanglement Entropy
L=30, AFM, 100 disorder realisations.
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)
arXiv:1401.4874
DPG Dresden 2014
20 / 22
Heisenberg MPO Hamiltonian
Encode the Hamiltinian into an MPO:

Wbi−1 ,bi
1
Ji +
2 si
Ji −
2 si
0
0
0
0
0
0
0
0
0

=
0
0
0

Ji szi 0

0
s−
i 
+
0
si 
0
szi 
0
1
1
1
2
2
3
3
4
4
5
5
U. Schollwöck, Ann. Phys. 326, 96 (2011).
A. M. Goldsborough (Warwick)
arXiv:1401.4874
DPG Dresden 2014
21 / 22
Heisenberg MPO Hamiltonian
Action of an isometry on two MPO tensors
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
arXiv:1401.4874
DPG Dresden 2014
22 / 22