The Strong Disorder Renormalisation Group and
Tensor Networks
Andrew M. Goldsborough , Rudolf A. Römer
Department of Physics and Centre for Scientific Computing
The University of Warwick
A. M. Goldsborough (Warwick)
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Tensor Networks, Holography and Disorder
Andrew M. Goldsborough , Rudolf A. Römer
Department of Physics and Centre for Scientific Computing
The University of Warwick
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A general wavefunction
One site:
σ1
G. M. Crosswhite and D. Bacon, Phys. Rev. A, vol.78, 012356, 2008.
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A general wavefunction
One site:
σ1
Two sites:
σ1 σ2
G. M. Crosswhite and D. Bacon, Phys. Rev. A, vol.78, 012356, 2008.
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An L-site wavefunction
...
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.
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Simplifying C
The most obvious ansatz for C is a product of single sites:
σ1
A. M. Goldsborough (Warwick)
σ2
σ3
σL-1
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σL
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This doesn’t work for quantum 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.
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Encoding entangled states
How about the product of matrices?
a
σ1
Maσ1
C σ1 σ2 =
=
X
Maσ2
σ2
0 1
1 0
1 1 0
=√
2 0 1
1
= √
4
2
Maσ1 Maσ2
a
This is a matrix product state (MPS).
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Matrix product states (MPS)
The matrix product state can be generalised to any length chain:
a1
σ1
|Ψi =
X
a2
σ2
X
σ1 ,...,σL a1 ,...,aL−1
aL-2
σ3
σL-2
aL-1
σL-1
σL
σ
L−1
σL
Maσ11 Maσ12a2 . . . MaL−2
aL−1 MaL−1 |σ1 , . . . , σL i
The Hilbert space can be reduced by capping the size of the a indices,
making large systems tractable whilst retaining entanglement information.
U. Schollwöck, Anals of Physics, vol.326, pp.96192, Jan.2011
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General tensor networks
In general a tensor network is a set of contracted tensors and can take any
form.
σ1
σ2
σ3
σ4
σ5
σ5
The structure of the network is key to its performance.
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The matrix product operator (MPO)
Not just wavefunctions can be represented as the product of tensors:
σ1
σ'1
O=
X
σ1 ,...σL
0
σ10 ,...,σL
b1 ,...,bL−1
σ3 ... σL-2
σ2
b1
σ ,σ10
Wb11
A. M. Goldsborough (Warwick)
σ'2
b2
σ ,σ 0
σ'3 ... σ'L-2
σL-1
bL-2
σL−1 ,σ 0
σ'L-1
σL
bL-1
σ'L
σ ,σ 0
L L
L−1
Wb12,b22 . . . WbL−2 ,bL−1
WbL−1
|σ1 . . . σL i hσ10 . . . σL0 |
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The AdS/CFT correspondence
minimal surface
holographic dimension
S. Ryu, and T. Takayanagi, Phys. Rev. Lett., vol.96, 181602, May 2006
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AdS/CFT and MERA
Multi-scale entanglement renormalisation ansatz (MERA)
G. Evenbly, and G. Vidal, J. Stat. Phys. 145, p.891918, 2011.
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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.
renormalise
S. K. Ma, C. Dasgupta, and C. K. Hu, Phys. Rev. Lett., vol.43, p.1434, Nov.1979
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The numerical renormalisation group approach
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, vol. 60, p. 12116, 1999.
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SDRG on matrix product operators
Diagonalise the two-site components of the MPO tensors with the
largest gap.
Create isometric tensors wi from the set of eigenvectors.
Contract the isometry to perform the renormalisation.
renormalise
contract
where the isometries have the following properties:
=
A. M. Goldsborough (Warwick)
,
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=
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SDRG as a tree tensor network
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Correlation Functions
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Average holographic distance for SDRG
Average path length for L=150 over 2000 disorder realisations.
20
DTTN ( x2 - x1 )
2.941 ln | x2 - x1 | + 3.016
DTTN
15
10
5
0
1
10
100
| x2 - x1 |
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Correlation functions
L=150, AFM 0 < Ji < 2, 2000 disorder realisations.
10
0
1
2
<< Sx . Sx >>
SDRG m=10, 2000 seeds
5.81 exp[-0.624DTTN]
10
10
-2
-4
10
0
10
1
10
2
| x2 - x1 |
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Entanglement Entropy
SA|B = −TrρA log2 ρA
A
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Entanglement Entropy
A
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A'
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Entanglement Entropy for L=30
L=30, AFM 0 < Ji < 2
3
SDRG m = 20, disorder
DMRG m = 20, no disorder
2.5
SA|B
2
1.5
1
0.5
0
0
A. M. Goldsborough (Warwick)
5
10
15
x
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20
25
30
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Entanglement Entropy for L=30
L=30, AFM, 100 disorder realisations.
2.5
SA|B
2
SDRG m = 20 (average)
SDRG m = 20 (average max)
DMRG m = 40
DMRG m = 20
DMRG m = 10
1.5
1
1 - ∆J/2 < Ji < 1 + ∆J/2
0.5
0
0.5
1
1.5
2
∆J
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Conclusions
Tensor networks give an efficint means of modelling quantum
many-body systems.
The numerical strong disorder renormalisation group naturally
self-assembles a tree tensor network.
The holographic geometry of the SDRG TTN enables it to capture
correlation and entanglement.
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Thank you!
M. C. Escher, Circle Limit III, 1959.
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