Vortex-Phonon Interaction in the Kosterlitz-Thouless Theory

TL;DR

This paper clarifies the nature of variables in the Kosterlitz-Thouless theory, revealing they are not true vortices or phonons, and resolves related paradoxes by relating them to canonical variables.

Contribution

It explicitly relates the canonical vortex-pair and phonon variables to the traditional variables in the Kosterlitz-Thouless theory, resolving longstanding paradoxes.

Findings

Identifies that the standard variables are not truly vortices or phonons.

Explains the absence of interaction between the variables.

Addresses the non-physical contributions of small vortex pairs.

Abstract

The "canonical" variables of the Kosterlitz-Thouless theory--fields $\Phi_0({\bf r})$ and $\phi({\bf r})$, generally believed to stand for vortices and phonons (or their XY equivalents, like spin waves, etc.) turn out to be neither vortices and phonons, nor, strictly speaking, {\it canonical} variables. The latter fact explains paradoxes of (i) absence of interaction between $\Phi_0$ and $\phi$, and (ii) non-physical contribution of small vortex pairs to long-range phase correlations. We resolve the paradoxes by explicitly relating $\Phi_0$ and $\phi$ to canonical vortex-pair and phonon variables.