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
| x2 - x1 |
10
20
30
40
50
LB
A. M. Goldsborough and R. A. Römer, Phys. Rev. B 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
1 / 36
Table of Contents
1 Strong Disorder Renormalization
2 Tensor Networks
3 Tensor Network SDRG
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
2 / 36
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)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
3 / 36
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)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
4 / 36
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)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
5 / 36
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)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
6 / 36
Table of Contents
1 Strong Disorder Renormalization
2 Tensor Networks
3 Tensor Network SDRG
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
7 / 36
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.
Want to measure physical properties and observables.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
8 / 36
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
9IWDS San Antonio 2014
9 / 36
Notation
Tensor networks are often represented using a graphical notation.
Vector:
V1
Vi =
V2
Matrix:
Mij =
M11 M12
M21 M22
Tensor:
Tijk...
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
10 / 36
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
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
=
9IWDS San Antonio 2014
11 / 36
A general wavefunction
One site:
σ1
G. M. Crosswhite and D. Bacon, Phys. Rev. A, vol.78, 012356, 2008.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
12 / 36
A general wavefunction
One site:
σ1
Two sites:
σ1 σ2
G. M. Crosswhite and D. Bacon, Phys. Rev. A, vol.78, 012356, 2008.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
13 / 36
An L-site wavefunction
|Ψi =
X
C σ1 ...σL |σ1 i ⊗ · · · ⊗ |σL i
σ1 ,...,σL
...
σ1 σ2
σL
Problem: The Hilbert space grows exponentially in L.
That is: C σ1 ...σL has dL elements and quickly becomes impossible to
calculate.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
14 / 36
Simplifying C
An obvious ansatz for C is a product of single sites to match the spin
basis:
σ1
A. M. Goldsborough (Warwick)
σ2
σ3
σL-1
Phys. Rev. B 89, 214203 (2014)
σL
9IWDS San Antonio 2014
15 / 36
This doesn’t work for entangled systems
A simple example, 2 sites:
a
c
ac ad
σ1
σ2
σ1
σ2
C =
, C =
⇒ C ⊗C =
b
d
bc bd
A product state:
|↑i |↑i ⇒ C σ1 σ2 =
0 0
a=c=0
⇒
0 1
b=d=1
An entangled state:
1
1
√ (|↑i |↑i + |↓i |↓i) ⇒ C σ1 σ2 = √
2
2
1 0
0 1
No solution for a, b, c, d.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
16 / 36
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
a1
σ1
a2
σ2
aL-2
σ3
σL-2
aL-1
σL-1
σL
U. Schollwöck, Ann. Phys. 326, 96 (2011).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
17 / 36
Encoding entangled states
MPSs can encode entangled states?
a
σ1
Maσ1
C σ1 σ2 =
=
X
a
A. M. Goldsborough (Warwick)
Maσ2
σ2
0 1
1 0
1 1 0
√
=
2 0 1
1
= √
4
2
Maσ1 Maσ2
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
18 / 36
Truncating the Hilbert space
The size of the tensors still scales exponentially with system size:
Truncate Hilbert space by capping the size of the a indices to χ. e.g.
Choose components that contribute most to the entanglement making
large systems tractable whilst retaining entanglement information.
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
19 / 36
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 |
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
20 / 36
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−
3
s+
3
sz3
+ J3 sz3 sz4
1






U. Schollwöck, Ann. Phys. 326, 96 (2011).
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
21 / 36
Table of Contents
1 Strong Disorder Renormalization
2 Tensor Networks
3 Tensor Network SDRG
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
22 / 36
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
=
=
A. M. Goldsborough and R. A. Römer, Phys. Rev. B 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
23 / 36
SDRG as a tree tensor network
Holographic
Dimension
Lattice
Dimension
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
24 / 36
Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realisations, χ = 10.
0
1
2
<< sx . sx >>
10
10
-2
tSDRG χ=10
5.81 exp[-0.63DTTN]
~| x2 - x1 |
10
-2
-4
10
0
10
AMG & RAR, PRB 89, 214203 (2014)
A. M. Goldsborough (Warwick)
1
10
2
| x2 - x1 |
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
25 / 36
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)
9IWDS San Antonio 2014
26 / 36
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 |
AMG & RAR, PRB 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
27 / 36
Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realisations, χ = 10.
0
1
2
<< sx . sx >>
10
10
-2
tSDRG χ=10
5.81 exp[-0.63DTTN]
~| x2 - x1 |
10
-2
-4
10
0
10
AMG & RAR, PRB 89, 214203 (2014)
A. M. Goldsborough (Warwick)
1
10
2
| x2 - x1 |
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
28 / 36
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'
AMG & RAR, PRB 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
29 / 36
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
AMG & RAR, PRB 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
30 / 36
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)
9IWDS San Antonio 2014
31 / 36
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
AMG & RAR, PRB 89, 214203 (2014)
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
32 / 36
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 posters!
A. M. Goldsborough (Warwick)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
33 / 36
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)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
34 / 36
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)
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
35 / 36
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
Phys. Rev. B 89, 214203 (2014)
9IWDS San Antonio 2014
36 / 36