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Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States

Heidar Moradi, Xiao-Gang Wen

TL;DR

A way-universal wave-function overlap-to extract universal topological data from generic ground states of gapped systems in any dimensions to provide a much more powerful alternative to the topological entanglement entropy and allow for more efficient numerical implementations.

Abstract

We propose a way -- universal wave function overlap -- to extract universal topological data from generic ground states of gapped systems in any dimensions. Those extracted topological data should fully characterize the topological orders with gapped or gapless boundary. For non-chiral topological orders in 2+1D, this universal topological data consist of two matrices, $S$ and $T$, which generate a projective representation of $SL(2,\mathbb Z)$ on the degenerate ground state Hilbert space on a torus. For topological orders with gapped boundary in higher dimensions, this data constitutes a projective representation of the mapping class group $MCG(M^d)$ of closed spatial manifold $M^d$. For a set of simple models and perturbations in two dimensions, we show that these quantities are protected to all orders in perturbation theory.

Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States

TL;DR

A way-universal wave-function overlap-to extract universal topological data from generic ground states of gapped systems in any dimensions to provide a much more powerful alternative to the topological entanglement entropy and allow for more efficient numerical implementations.

Abstract

We propose a way -- universal wave function overlap -- to extract universal topological data from generic ground states of gapped systems in any dimensions. Those extracted topological data should fully characterize the topological orders with gapped or gapless boundary. For non-chiral topological orders in 2+1D, this universal topological data consist of two matrices, and , which generate a projective representation of on the degenerate ground state Hilbert space on a torus. For topological orders with gapped boundary in higher dimensions, this data constitutes a projective representation of the mapping class group of closed spatial manifold . For a set of simple models and perturbations in two dimensions, we show that these quantities are protected to all orders in perturbation theory.

Paper Structure

This paper contains 1 section, 28 equations, 3 figures.

Table of Contents

  1. Cumulant Expansion

Figures (3)

  • Figure 1: (a) Lattice under consideration, with the spins living on the links. (b) Tensor network for $\mathbb Z_N$ gauge theory. The lattice is chosen with the orientation shown. The tensors live on the lattice sites and the dots represent the physics indices. (c) Symmetry of the $\mathbb Z_N$ tensor.
  • Figure 2: Definition of $S$ and $T$ transformations. The $S$ transformation corresponds to rotating configurations $90$ degrees, while $T$ corresponds to a shear transformation. Note that this transformation does not leave the space of closed loop configurations invariant.
  • Figure 3: In the string-net basis, a modular $S$ transformation flips the topological sectors $(\alpha,\beta)\rightarrow (\beta,-\alpha \mod N)$, while a $T$ transformation has the effect $(\alpha,\beta)\rightarrow (\alpha,\alpha+\beta\text{ mod } N)$.