Algebraically contractible topological tensor network states

TL;DR

This paper introduces a class of algebraically contractible tensor network states for topologically ordered systems, enabling exact contraction and efficient representation of lattice gauge theories with insights into entanglement and correlations.

Contribution

It generalizes algebraically contractible tensor networks from spin-1/2 to spin-S systems, providing a versatile framework for topological quantum states and lattice gauge theories.

Findings

Exact tensor network contraction independent of geometry

Perturbations reduce entanglement and induce correlations

Applicable to various lattice geometries like hexagonal and kagome

Abstract

We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with $Z_2$ topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin-S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices.