Topological Properties of Tensor Network States From Their Local Gauge and Local Symmetry Structures

TL;DR

This paper develops methods to determine the topological properties of tensor network states by analyzing their local gauge and symmetry structures, enabling the calculation of key topological features like ground state degeneracy and quasiparticle statistics.

Contribution

It introduces a framework to compute topological properties of tensor network states from their local gauge and symmetry structures, extending the understanding of topological order in many-body quantum systems.

Findings

Method to calculate string and brane operators from tensor networks

Ability to determine ground state degeneracy

Analysis of quasiparticle statistics

Abstract

Tensor network states are capable of describing many-body systems with complex quantum entanglement, including systems with non-trivial topological order. In this paper, we study methods to calculate the topological properties of a tensor network state from the tensors that form the state. Motivated by the concepts of gauge group and projective symmetry group in the slave-particle/projective construction, and by the low-dimensional gauge-like symmetries of some exactly solvable Hamiltonians, we study the $d$-dimensional gauge structure and the $d$-dimensional symmetry structure of a tensor network state, where $d\leq d_{space}$ with $d_{space}$ the dimension of space. The $d$-dimensional gauge structure and $d$-dimensional symmetry structure allow us to calculate the string operators and $d$-brane operators of the tensor network state. This in turn allows us to calculate many topological properties of the tensor network state, such as ground state degeneracy and quasiparticle statistics.