A dichotomy theory for the height functions of the BKT transition

TL;DR

This paper investigates the height functions related to the BKT transition, revealing a phase dichotomy with exponential decay in localized phases and logarithmic variance growth in delocalized phases, including at the transition point.

Contribution

It establishes a phase dichotomy for the height functions at the BKT transition, showing the effective temperature jumps and delocalization at the transition point.

Findings

Exponential decay of two-point functions in localized phases.

Logarithmic growth of variance in delocalized phases.

Delocalization includes the transition point, with a universal temperature gap.

Abstract

This text considers the discrete height functions associated with the Berezinskii--Kosterlitz--Thouless transition (BKT) at slope zero. Our main results are as follows. * Sharpness: If the model is localised, then the two-point function (covariance) decays exponentially fast in the distance between the points. * Effective temperature gap: If the model is delocalised, then the variance grows at least as $c\log n$, where $n$ is the distance to the boundary and $c>0$ a universal constant not depending on the temperature. Thus, the effective temperature must jump from $0$ to at least $c$ at the transition point; values in the interval $(0,c)$ are forbidden. * Delocalisation at the transition point: The delocalised phase includes the transition point, in the sense that it is a closed set in the phase diagram in the appropriate topology. These results contribute to the understanding of the regime at and around the transition point which remained largely unexplored. In a follow-up paper, the sharpness derived here is used to establish that the localisation-delocalisation transition is equivalent to the BKT transition in the dual XY and Villain models.