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Modern Physics Letters A

HIGHER GENUS WAVEFUNCTION OF ANYON SYSTEM

Jnanadeva Maharana, Sudhakar Panda

Published Jan 20, 1991• 0 citations
Publisher

Abstract

A method is proposed for constructing the wavefunction of anyons on Riemann surfaces of arbitrary genus. This has been carried out in the framework of computing the correlation function of the chiral vertex operators, involving free scalar fields, with appropriate external momenta. Our technique, as a check, reproduces the already known anyonic wavefunction on the sphere as well as that on the torus.

DOI
10.1142/s0217732391000129

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A method is proposed for constructing the wavefunction of anyons on Riemann surfaces of arbitrary genus. This has been carried out in the framework of computing the correlation function of the chiral vertex operators, involving free scalar fields, with appropriate external momenta. Our technique, as a check, reproduces the already known anyonic wavefunction on the sphere as well as that on the torus.
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.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.