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.

We propose a tensor network method for investigating strongly disordered systems that is based on an adaptation of entanglement renormalization [G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)]. This method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse-grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. Benchmark results from the disordered XXZ model demonstrates that this approach accurately captures ground state entanglement in disordered systems, even at long distances. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multi-scale entanglement renormalization ansatz for disordered systems.Comment: 14 pages, 21 figure

-We perform a matrix-product-state-based density matrix renormalisation group analysis of the phases for the disordered one-dimensional Bose-Hubbard model. For particle densities N/L = 1, 1/2 and 2 we show that it is possible to obtain a full phase diagram using only the entanglement properties, which come for free when performing an update. We confirm the presence of Mott insulating, superfluid and Bose glass phases when N/L = 1 and 1/2 (without the Mott insulator) as found in previous studies. For the N/L = 2 system we find a double-lobed superfluid phase with possible re-entrance.

We study the leaf-to-leaf distances on one-dimensionally ordered, full and complete m-ary tree graphs using a recursive approach. In our formulation, unlike in traditional graph theory approaches, leaves are ordered along a line emulating a one-dimensional lattice. We find explicit analytical formulas for the sum of all paths for arbitrary leaf separation r as well as the average distances and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.

We study the average leaf-to-leaf path lengths on ordered Catalan tree graphs with n nodes and show that these are equivalent to the average length of paths starting from the root node. We give an explicit analytic formula for the average leafto-leaf path length as a function of separation of the leaves and study its asymptotic properties. At the heart of our method is a strategy based on an abstract graph representation of generating functions. PACS numbers: