Comb tensor networks

In this paper we propose a special type of a tree tensor network that has the geometry of a comb---a 1D backbone with finite 1D teeth projecting out from it. This tensor network is designed to provide an effective description of higher dimensional objects with special limited interactions, or, alternatively, one-dimensional systems composed of complicated zero-dimensional objects. We provide details on the best numerical procedures for the proposed network, including an algorithm for variational optimization of the wave-function as a comb tensor network, and the transformation of the comb into a matrix product state. We compare the complexity of using a comb versus alternative matrix product state representations using density matrix renormalization group (DMRG) algorithms. As an application, we study a spin-1 Heisenberg model system which has a comb geometry. In the case where the ends of the teeth are terminated by spin-1/2 spins, we find that Haldane edge states of the teeth along the backbone form a critical spin-1/2 chain, whose properties can be tuned by the coupling constant along the backbone. By adding next-nearest-neighbor interactions along the backbone, the comb can be brought into a gapped phase with a long-range dimerization along the backbone. The critical and dimerized phases are separated by a Kosterlitz-Thouless phase transition, the presence of which we confirm numerically. Finally, we show that when the teeth contain an odd number of spins and are not terminated by spin-1/2's, a special type of comb edge states emerge.