Self-Assembling Tensor Networks and Holography in
Disordered Spin Chains (arXiv:1401.4874, accepted to PRB)
Andrew M. Goldsborough and Rudolf A. Römer
Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, UK, CV4 7AL
Abstract
Holographic
Dimension
We show that the numerical strong disorder renormalization group algorithm
(SDRG) of Hikihara et. al. [1] for the one-dimensional disordered Heisenberg
model naturally describes a tree tensor network (TTN) with an irregular structure
defined by the strength of the couplings. Employing the holographic
interpretation of the TTN in Hilbert space, we compute expectation values,
Lattice
Dimension
correlation functions and the entanglement entropy using the geometrical
properties of the TTN. We find that the disorder averaged spin-spin correlation
scales with the average path length through the tensor network while the
entanglement entropy scales with the minimal surface connecting two regions.
Furthermore, the entanglement entropy increases with both disorder and system
size, resulting in an area-law violation. Our results demonstrate the usefulness
of a self-assembling TTN approach to disordered systems and quantitatively
validate the connection between holography and quantum many-body systems.
Correlation Functions
Motivation
• To bring the strong disorder renormalization group up to date with modern
• Disorder averaged spin correlation functions for XXX model with a flat probability of Ji scale
concepts in many-body simulation.
as [8]:
• The size of the Hilbert space of quantum spin systems scales exponentially with
chain length L. Therefore exact diagonalization (ED) techniques can only be used for
• Correlation functions in holographic systems scale with the minimal path distance D(x1 ,x2 )
very small systems.
connecting sites x1 and x2 (top right) [9]:
• 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.
• (left) In the tensor network, D(x1 ,x2 ) is the number of tensors connecting the two sites. [10]
Strong Disorder Renormalization
• (below right) In the TTN, mean spin correlations fit the average path length (below left) with
α = 0.62 ± 0.02, giving scaling power -1.84 ± 0.04 for L=150, -2.04 ± 0.03 for L=500.
• The principle of SDRG is to successively remove the spins that are most strongly
coupled.
• Ma et.al. [2] removed spin singlets from a random anti-ferromagnetic (AFM)
Heisenberg chain.
renormalize
• The basis for our work is the extended SDRG method of Hikihara et.al. [1]. This
diagonalizes and keeps several eigenvectors to increase accuracy.
• The system is made up of blocks (HB ) and couplings between them (HC ).
• The blocks are combined in order of most strongly coupled, or largest gap, where
the gap is defined as the energy difference between the highest energy
eigenmultiplets that are kept and lowest that are discarded when decimated.
Matrix Product Operator (MPO)
Entanglement Entropy
• Entanglement or Von Neumann entropy between region A and B is defined as
• Like the matrix product state [3], an operator on a lattice can be decomposed as a
matrix product operator (MPO) [4,3]:
Where ρA is the reduced density matrix for region A. [3]
• In holographic systems, SA|B scales with the minimal surface separating the two regions (top
σ1
σ2
σ3
σL-2
σL-1
σL
right)[9], or the number of bonds cut (nA) in the tensor network (right). [10]
• (below) Away from boundaries, entanglement per bond S / nA = 0.47 ± 0.02 for both
bipartitions (split by one cut) and blocks (subsystem in the centre of the chain).
b1
σ'1
b2
σ'2
bL-2
σ'3
σ'L-2
bL-1
σ'L-1
• (below right) The logarithmic (log(2) log2(LB)/3) scaling of SA,B with block size (LB) implies an
σ'L
effective CFT with central charge log(2). [11]
• The Heisenberg (XXX) Hamiltonian is:
• The Hamiltonian can be encoded as an MPO [5,6]:
SDRG Tree Tensor Network
• The sets of eigenvectors from diagonalization can be seen as isometric tensors
(ww† =�) that map two MPO tensors to one:
=
=
Conclusions
• We have made an inhomogeneous tensor network that constructs itself based on the
• The whole SDRG algorithm can be represented as a tree tensor network (TTN)
(top centre).
• Expectation values can be calculated efficiently by contracting only those tensors in
the past causal cone of the operator. [7]
• Calculation of entanglement entropy requires contraction of only the tensors that
join regions A and B.
References
[1] T. Hikihara, A. Furusaki, and M. Sigrist, Phys. Rev. B, 60, 12116 (1999),
[2] S. K. Ma, C. Dasgupta, and C. K. Hu, Phys. Rev. Lett., 43, 1434 (1979),
[3] U. Schollwöck, Anals of Physics, 326, pp. 96–192 (2011),
coupling strengths of the strong disorder renormalization group (SDRG).
• We reproduce the mean power law correlation function and logarithmic entanglement
Contact Details
• Email: a.goldsborough@warwick.ac.uk, r.roemer@warwick.ac.uk
• Website: http://www.warwick.ac.uk/andrewgoldsborough/
• Preprint: http://arxiv.org/abs/1401.4874
entropy scaling found using other methods.
• The scaling of the correlation function with path length and entanglement entropy with
the size of the boundary of the region supports the notion of a holographic tensor
network state.
[4] F. Verstraete, J.J. García-Ripoll, J.I. Cirac, Phys. Rev. Lett. 93, 207204 (2004),
[5] I.P. McCulloch, J. Stat. Mech.: Theor. Exp. P10014 (2007),
[6] G. M. Crosswhite and D. Bacon, Phys. Rev. A, 78, 012356 (2008),
[7] G. Evenbly, and G. Vidal, Phys. Rev. B, vol. 79, 144180 (2009),
[8] D. S. Fisher, Phys. Rev. B 50, 3799 (1994),
[9] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006)
[10] G. Evenbly and G. Vidal, J. Stat. Phys. 145, 891 (2011),
[11] G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 (2004).