The Strong Disorder Renormalisation Group and Tensor
Network Methods
Andrew M. Goldsborough and Rudolf A. Römer
Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, UK, CV4 7AL
MPO SDRG
Correlation Functions
The extended SDRG algorithm of Hikihara can be performed by tensor operations
• Disorder averaged correlation functions for XXX model with a flat probability of Ji
Abstract
We have developed a tensor network method of performing the numerical strong
disorder renormalisation group (SDRG) approach [1] on the random 1D spin-1/2
Heisenberg model. We use matrix product operators (MPOs) as a means of
describing the Hamiltonian. The coarse-graining is achieved by concentrating on
●
scale as [11]:
on an MPO as follows:
1. Identify coupling with the largest gap. Contract the two MPOs and fuse spin
indices:
the interaction with the largest energy gap and calculating the lowest eigenvectors
• Correlation functions in tensor networks scale as [12]:
,
of the two-site Hamiltonian. This isometric tensor can then be contracted with the
contract
MPO to create a new one site MPO tensor. The idea can be interpreted as a
fuse
where D(x1 ,x2 ) is the path distance through the tensor network that connects sites x1
multi-level coarse-graining of the Hamiltonian or as a multi-level tensor network
and x2 :
,
state with the form of a random binary tree tensor network (TTN). This
realization enables us to use the methods of TTNs [2] and MERAs (multi-scale
2. Diagonalise on-site element of the new MPO and keep the eigenvectors (vm)
entanglement renormalisation ansatz) [3] to perform a variational update and
corresponding to the lowest m eigenvalues (λm) such that only full
efficiently calculate expectation values and the entanglement entropy of the
eigenmultiplets are kept.
system.
3. Contract matrix of eigenvectors
Motivation
with two-site MPO to perform
=
the renormalisation. This creates
• Bring the strong disorder renormalisation group up to date with modern concepts in
an on-site block of the m lowest
many-body simulation.
eigenvalues:
• The size of the Hilbert space of quantum spin systems scales exponentially with
chain length L. Therefore exact diagonalisation (ED) techniques can only be used
1
for very small systems.
1
1
1
1
• Average path length can be fitted to correlations: A = 5.81 ± 0.93, α = 0.624 ±
0.012, giving scaling power -1.84 ± 0.04 for L=150, -2.04 ± 0.03 for L=500
• Tensor network methods give an intuitive representation of the coarse graining
mechanism and means of efficiently calculating properties of the system including
observables and entanglement.
Strong Disorder Renormalisation
• The principle of SDRG is to successively remove the spins that are most strongly
2
2
2
3
3
3
renormalise
2
2
3
3
and renormalises spins on boundary of block, creating another MPO tensor:
coupled.
• Ma et.al. [4] removed spin singlets from a random anti-ferromagnetic (AFM)
1
Heisenberg chain.
1
1
1
1
2
2
renormalise
2
2
2
renormalise
Entanglement Entropy
• Westerberg et.al. [5] combined spins to form an effective spin for the random
FM/AFM chain.
3
renormalise
3
3
3
3
• Peak entanglement entropy grows with rapidly with increasing disorder (ΔJ):
4. Diagonalise coupling terms on either side to update distribution of gaps and
repeat.
• The basis for our work is the extended SDRG method of Hikihara et.al. [1]. This
keeps multiple states to increase accuracy.
SDRG Tree Tensor Network
• The sets of eigenvectors in the MPO SDRG algorithm can be seen as isometric
(ww†=I) where::
• The system is made up of blocks (HB) and couplings between them (HC).
=
,
• The gap is defined as the energy difference between the highest energy
eigenmultiplets that are kept and lowest that are discarded when decimated.
• The whole SDRG algorithm can be represented as a tree tensor network (TTN):
• The algorithm proceeds as follows:
• As disorder increases, lower m DMRG fails to capture the entanglement entropy.
SDRG tends to overestimate the entanglement for lower values of disorder.
1. Find the coupling with the largest gap and combine the two blocks either side
to form one block:
2. Diagonalise the new block and keep the lowest m eigenvectors (vm) such that
only full eigenmultiplets are kept. Set the eigenvalues as the new block and
renormalise the couplings at either side with the eigenvectors:
3. Diagonalise the new couplings to update the gap distribution.
• For a clean XXX system (red) the entanglement shows a dimerised, but regular
Matrix Product Operator (MPO)
form. The disordered (black) system has higher peaks and lower troughs due to the
• Expectation values can be calculated efficiently by contracting only those tensors in
• Like the matrix product state [6,7], an operator on a lattice can be decomposed as a
matrix product operator (MPO) [8,7]:
disorder.
Future Work
the past causal cone of the operator [3].
• Entanglement or Von Neumann entropy between region A and B is defined as:
• Further investigate the correlations, entanglement and entanglement spectrum of
the disordered Heisenberg chain.
σ1
σ2
σ3
σL-2
σL-1
σL
Where ρA is the reduced density matrix for region A and sa are the singular values
obtained via Schmidt decomposition of the wavefunction into regions A and B. [2]
b1
σ'1
b2
σ'2
bL-2
σ'3
σ'L-2
bL-1
σ'L-1
• Like regular TTNs [2] and MERA [3], each isometry can be updated with respect to
their environment Υ:
• The reduced density matrix is made up of only those tensors that join blocks A and
B.
σ'L
• Below are the necessary contractions for < S3 . S15 > (left) and the reduced density
• The Heisenberg (XXX) Hamiltonian is:
=
+
+
matrix where A is half of the system (right).
• The Hamiltonian can be encoded as an MPO [9,10]:
• Further optimise and investigate how entanglement and correlation for the updated
1
1
A
2
A'
network.
Contact Details
2
• Email: a.goldsborough@warwick.ac.uk, r.roemer@warwick.ac.uk
References
3
[1] T. Hikihara, A. Furusaki, and M. Sigrist, Phys. Rev. B, 60, 12116 (1999).
[2] L Tagliacozzo, G. Evenbly, and G. Vidal, Phys Rev. B, 80, 235127 (2009).
[3] G. Evenbly, and G. Vidal, Phys. Rev. B, vol. 79, 144180 (2009).
[4] S. K. Ma, C. Dasgupta, and C. K. Hu, Phys. Rev. Lett., 43, 1434 (1979).
• Website: http://www.warwick.ac.uk/andrewgoldsborough/
3
[5] E. Westerberg, A. Furusaki, M.Sigrist, and P. A. Lee, Phys. Rev. Lett., 75, 4302 (1995).
[6] S. Ostlund, S. Rommer, Phys. Rev. Lett. 75, 3537 (1995).
[7] U. Schollwöck, Anals of Physics, 326, pp. 96–192 (2011).
[8] F. Verstraete, J.J. García-Ripoll, J.I. Cirac, Phys. Rev. Lett. 93, 207204 (2004).
[9] I.P. McCulloch, J. Stat. Mech.: Theor. Exp. P10014 (2007).
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[11] D. S. Fisher, Phys. Rev. B 50, 3799 (1994).
[12] G. Evenbly and G. Vidal, J. Stat. Phys. 145, 891 (2011).