Chiral conformal field theory for topological states and the anyon eigenbasis on the torus

TL;DR

This paper introduces a simplified chiral conformal field theory approach to construct and analyze topologically ordered states on the torus, providing explicit anyon eigenbases and characterizing their topological properties.

Contribution

The authors develop a pure chiral method for torus wave functions that simplifies the construction and yields the anyon eigenbasis, advancing the understanding of topological order in CFT-based models.

Findings

Constructed ground states for SO(n)$_1$ and SU(n)$_1$ chiral spin liquids.

Provided explicit modular $S$ and $T$ matrices for topological characterization.

Realized a complete wave function set for Kitaev's sixteenfold way.

Abstract

Model wave functions constructed from (1+1)D conformal field theory (CFT) have played a vital role in studying chiral topologically ordered systems. There usually exist multiple degenerate ground states when such states are placed on the torus. The common practice for dealing with this degeneracy within the CFT framework is to take a full correlator on the torus, which includes both holomorphic and antiholomorphic sectors, and decompose it into several conformal blocks. In this paper, we propose a pure chiral approach for the torus wave function construction. By utilizing the operator formalism, the wave functions are written as chiral correlators of holormorphic fields restricted to each individual topological sector. This method is not only conceptually much simpler, but also automatically provides us the anyon eigenbasis of the degenerate ground states (also known as the "minimally entangled states"). As concrete examples, we construct the full set of degenerate ground states for SO($n$)$_1$ and SU($n$)$_1$ chiral spin liquids on the torus, the former of which provide a complete wave function realization of Kitaev's sixteenfold way of anyon theories. We further characterize their topological orders by analytically computing the associated modular $S$ and $T$ matrices.