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

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 $d\le d_{space}$ 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.