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

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.

50 pages, 5 figures; published version, with note added and minor modifications

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• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids
• Non-Abelian topological order with SO(5)${}_1$ chiral edge states