Self-assembling tensor networks and holography in disordered spin chains

Abstract

We show that the numerical strong disorder renormalization group algorithm of Hikihara et al. [Phys. Rev. B 60, 12116 (1999)] 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, 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.

Leaf-to-leaf distances and their moments in finite and infinite ordered m -ary tree graphs

“…1. At each internal vertex we place an isometric tensor [12,25] with initially random entries and so-called bond dimension x = 4. Using as proxy a spin-1/2 Heisenberg model H = Yl!,=\ V ' V +i.…”

“…A two-point correlation function (?V | ■ sX 2 ) is calcu lated [12,22] for all pairs of sites and averaged for all points separated by |jc2j c i |. The results are given in Fig.…”

“…Such trees are no longer com plete, but nevertheless have many applications in the sciences [1,12], Let us again com pute the average leaf-to-leaf distance C( K\r) for a given r, when all possible pairs o f leaves o f separation r and all possible trees o f L -1 internal nodes are considered. Here H denotes the random character o f trees under consideration.…”

“…Last, we numerically study the case of incomplete random trees, which is closest related to the tree tensor networks considered in Ref. [12], Let us start by considering the complete binary tree shown in Fig. 1(a).…”

Superfluid density and quasi-long-range order in the one-dimensional disordered Bose–Hubbard model

“…In the present work we apply a tree tensor network (TTN) method [37][38][39][40][41] (specifically our recently introduced gauge-adaptive TTN technique [42]), a natural ansatz for the simulation of one-dimensional quantum many-body systems with PBC, to the disordered Bose-Hubbard model. We compute the superfluid stiffness and correlation functions, avoiding the detrimental boundary effects arising in simulations with open boundary conditions.…”