Boundary states of a bulk gapped ground state in $2$-d quantum spin systems

TL;DR

This paper defines boundary states for 2D gapped quantum spin systems using operator algebra, linking boundary categories to the bulk's braided tensor category via the Drinfeld center under certain conditions.

Contribution

It introduces a rigorous mathematical framework for boundary states and establishes a categorical equivalence with the bulk braided tensor category.

Findings

Boundary states are characterized within an operator algebraic framework.

A $C^*$-tensor category $ ilde{rak{M}}$ is constructed from boundary states.

The Drinfeld center of $ ilde{rak{M}}$ is shown to be equivalent to the bulk braided category.

Abstract

We introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of $2$-d quantum spin systems. With approximate Haag duality at the boundary, we derive a $C^*$-tensor category $\tilde{\mathcal{M}}$ out of such boundary state. Under a non-triviality condition of the braiding in the bulk, we show that the Drinfeld center (with an asymptotic constraint) of $\tilde{\mathcal{M}}$ is equivalent to the bulk braided $C^*$-tensor category derived in [14].