The Strong Disorder Renormalisation Group in the age of Tensor Networks omer
The Strong Disorder Renormalisation Group in the age of Tensor Networks
Department of Physics and Centre for Scientific Computing
The University of Warwick
A. M. Goldsborough (Warwick)
06/06/2013 1 / 33
Motivation
We want a full quantum description of interacting many-body systems (e.g. Heisenberg, Hubbard etc.)
Including disorder in the interactions
We want to be able to probe the physical properties (e.g. energy, correlations)
Also entanglement of disordered quantum systems
A. M. Goldsborough (Warwick)
06/06/2013 2 / 33
Overview
What is a tensor network?
What is the strong disorder renormalisation group (SDRG)?
How can they be combined?
What can we learn from the tensor network SDRG?
A. M. Goldsborough (Warwick)
06/06/2013 3 / 33
A general wavefunction
One site:
σ
1
G. M. Crosswhite and D. Bacon, Phys. Rev. A, vol.78, 012356, 2008.
A. M. Goldsborough (Warwick)
06/06/2013 4 / 33
A general wavefunction
One site:
σ
1
Two sites:
2
G. M. Crosswhite and D. Bacon, Phys. Rev. A, vol.78, 012356, 2008.
A. M. Goldsborough (Warwick)
06/06/2013 5 / 33
An L-site wavefunction
2
...
σ
L
Problem: The Hilbert space grows exponentially in L .
That is: C
σ
1
...σ
L has d
L elements and quickly becomes impossible to calculate.
A. M. Goldsborough (Warwick)
06/06/2013 6 / 33
Simplifying C
The most obvious ansatz for C is a product of single sites:
σ
1
σ
2
σ
3
σ
L-1
σ
L
A. M. Goldsborough (Warwick)
06/06/2013 7 / 33
This doesn’t work for quantum systems
A simple example, 2 sites:
C
σ
1 = a b
, C
σ
2 = c d
⇒ C
σ
1 ⊗ C
σ
2 = ac ad bc bd
A product state:
|↑i |↑i ⇒ C
σ
1
σ
2 =
0 0
0 1
An entangled state:
⇒ a = c = 0 b = d = 1
√
2
( |↑i |↑i + |↓i |↓i ) ⇒ C
σ
1
σ
2 = √
2
No solution for a, b, c, d .
1 0
0 1
A. M. Goldsborough (Warwick)
06/06/2013 8 / 33
Schmidt decomposition of a general state
Start with a general state and decompose using a singular value decomposition (SVD):
σ
1
σ
2
σ
3
...
σ
L-2
SVD
σ
L-1
σ
L
σ
1 a
1
σ
2
σ
3
...
σ
L-2
σ
L-1
σ
L
A. M. Goldsborough (Warwick)
06/06/2013 9 / 33
The matrix product states (MPS)
Performing the SVD ( L − 1) times produces the matrix product state:
σ
1 a
1
σ
2 a
2
σ
3
σ
L-2 a
L-2
σ
L-1 a
L-1
σ
L
| Ψ i =
X X
σ
1
,...,σ
L a
1
,...,a
L − 1
M
σ a
1
1 M
σ a
1
2 a
2
. . . M
σ a
L − 1
L − 2 a
L − 1
M
σ a
L
L − 1
| σ
1
, . . . , σ
L i
The Hilbert space can be reduced by capping the size of the a indices, making large systems tractable whilst retaining entanglement information.
A. M. Goldsborough (Warwick)
06/06/2013 10 / 33
This can encode entangled states
The two-site entangled state again:
1
√
2
( |↑i |↑i + |↓i |↓i ) ⇒ C
σ
1
σ
2 =
1
√
2
Has an MPS representation:
1 0
0 1
M
σ
1
α
= M
σ
2
α
=
1
√
4
2
0 1
1 0
C
σ
1
σ
2 =
X
M
σ
1
α
M
σ
α
2
α
=
1
√
2
1
0
0
1
Thus we can use MPSs to model strongly-correlated quantum systems.
A. M. Goldsborough (Warwick)
06/06/2013 11 / 33
General tensor networks
In general a tensor network is a set of contracted tensors and can take any form.
σ
1
σ
2
σ
3
σ
4
σ
5
The structure of the network is key to its performance.
σ
5
A. M. Goldsborough (Warwick)
06/06/2013 12 / 33
The Heisenberg Hamiltonian
The spin-1/2 isotropic Heisenberg Hamiltonian:
H
Heis
=
L − 1
X
J i i =1 i
· S i +1
=
L − 1
X
J i i =1
S i z
S z i +1
+
1
2
S i
+
S
− i − 1
+ S i
−
S
+ i − 1 where the spin operators are Pauli matrices:
S
+
= S x
+ iS y
=
0 1
0 0
S
−
= S x
− iS y
=
0 0
1 0
S z
=
1
2
1
0
0
− 1
The interaction strength J i can be different for each pair of sites.
A. M. Goldsborough (Warwick)
06/06/2013 13 / 33
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.
E i
= −
3 J i
4
−
3
16 J i
( J
2 i − 1
+ J
2 i +1
)
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
A. M. Goldsborough (Warwick)
06/06/2013 14 / 33
Generalisation to FM/AFM chains
Westerberg et. al. generalised the SDRG to the random ferromagnetic/anti-ferromagnetic (FM/AFM) chain.
Combine the pair of spins with the largest energy gap to a new larger spin.
Find the energy gaps for the new spin.
renormalise
E. Westerberg, A. Furusaki, M.Sigrist, and P. A. Lee, Phys. Rev. Lett., vol.75, p.4302,
Dec. 1995.
A. M. Goldsborough (Warwick)
06/06/2013 15 / 33
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.
A. M. Goldsborough (Warwick)
06/06/2013 16 / 33
The matrix product operator (MPO)
The tensor network SDRG algorithm starts with a tensor network
Hamiltonian operator:
σ
1
σ
2
σ
3
...
σ
L-2
σ
L-1
σ
L
σ'
1 b
1
σ'
2 b
2
σ'
3
...
σ'
L-2 b
L-2
σ'
L-1 b
L-1
σ'
L
O =
X
σ
σ
0
1
1
,...σ
,...,σ b
1
L
0
L
,...,b
L − 1
W b
σ
1
1
,σ
0
1 W b
σ
2
1
,σ
0
2
,b
2
. . . W
σ
L − 1
,σ
0
L − 1 b
L − 2
,b
L − 1
W b
σ
L
,σ
0
L
L − 1
| σ
1
. . . σ
L i h σ
0
1
. . . σ
0
L
|
A. M. Goldsborough (Warwick)
06/06/2013 17 / 33
SDRG on matrix product operators
Diagonalise the two-site components of the MPO tensors with the largest gap.
Create isometric tensors w i 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)
06/06/2013 18 / 33
SDRG as a tree tensor network
A. M. Goldsborough (Warwick)
06/06/2013 19 / 33
SDRG as a tree tensor network
A. M. Goldsborough (Warwick)
06/06/2013 20 / 33
Correlation Functions
A. M. Goldsborough (Warwick)
06/06/2013 21 / 33
Correlation Functions for L=50
L=50, AFM 0 < J i
< 2 , 500 disorder realisations.
1
0.1
0.01
0.001
1
A. M. Goldsborough (Warwick)
| x
2
- x
1
|
10
50
06/06/2013 22 / 33
Entanglement Entropy
S
A | B
= − Tr ρ
A log
2
ρ
A
A
A. M. Goldsborough (Warwick)
06/06/2013 23 / 33
Entanglement Entropy
A A'
A. M. Goldsborough (Warwick)
06/06/2013 24 / 33
Entanglement Entropy for L=50
L=50, AFM 0 < J i
< 2
2 2
1.5
1
0.5
1.5
1
0.5
0
0
A. M. Goldsborough (Warwick)
10 20 30
Bipartition Position
40 50
0
06/06/2013 25 / 33
Tensor networks and binary tree graphs
The correlation in tensor networks is related to the holographic path distance D hol
:
C ( x
1
, x
2
) ∝ e
αD hol
( x
1
,x
2
)
The structure of the SDRG TTN is a random binary tree graph: x
G. Evenbly and G. Vidal, J. Stat. Phys. Vol.145, p.891, 2011.
A. M. Goldsborough (Warwick)
06/06/2013 26 / 33
Average D hol for distance along the chain (500 instances)
10
5
0
1
A. M. Goldsborough (Warwick)
| x
2
- x
1
|
10
50
06/06/2013 27 / 33
Correlation calculated using just the geometry
C ( x
1
, x
2
) = 1 .
9874 e
0 .
5714 D hol
( x
1
,x
2
)
1
0.1
0.01
0.001
1
A. M. Goldsborough (Warwick)
| x
2
- x
1
|
10
50
06/06/2013 28 / 33
Average D hol for L=1000 (500 instances)
0.1
0.01
0.001
0.0001
1e-05
1e-06
1
A. M. Goldsborough (Warwick)
10
| x
2
- x
1
|
100 1000
06/06/2013 29 / 33
Future work
Further analyse the links between the tensor network geometry and correlation.
Look into how disorder effects the entanglement of the system.
Investigate other interacting systems (e.g spin-1 Heisenberg,
Hubbard).
Extend the method to two dimensional systems.
A. M. Goldsborough (Warwick)
06/06/2013 30 / 33
Conclusions
The NRG method of Hikihara et. al. can be re-written as a tensor network.
The TTN gives an intuitive and efficient method of obtaining physical properties of the wavefunction.
The TTN allows a variational update of the wavefunction.
The geometry of the TTN captures the long range order of the system.
A. M. Goldsborough (Warwick)
06/06/2013 31 / 33
Thank you!
A. M. Goldsborough (Warwick)
06/06/2013 32 / 33
Complete trees
The average path length for an m-ary tree is:
A n
( r ) =
1
2 n − r
2 n
2 q + 1 + r
2 q − 1
− (3 + 2 n ) r
In the infinite limit: lim n →∞
A n
( r ) = 2 q m
2 r
+ 1 + m q m ( m − 1) where q = b log m r c , n is the number of levels and r = | x
2
− x
1
|
A. M. Goldsborough (Warwick)
06/06/2013 33 / 33
Complete binary trees
50 finite system infinite system
40
30
20
10
0
1 10
A. M. Goldsborough (Warwick)
100 1000 r
10000 1e+05 1e+06
06/06/2013 34 / 33
