Modular matrices as topological order parameter by a gauge-symmetry-preserved tensor renormalization approach

Abstract

Topological order has been proposed to go beyond Landau symmetry breaking theory for more than 20 years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in two-dimensional systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including $S$ and $T$ matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data ($S$ and $T$ matrices), without any nonuniversal contributions. Furthermore, it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, in which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.