Berezinskii-Kosterlitz-Thouless Paired Phase in Coupled $XY$ Models

We study the effect of a linear tunneling coupling between two-dimensional systems, each separately exhibiting the topological Berezinskii-Kosterlitz-Thouless (BKT) transition. In the uncoupled limit, there are two phases: one where the one-body correlation functions are algebraically decaying and the other with exponential decay. When the linear coupling is turned on, a third BKT-paired phase emerges, in which one-body correlations are exponentially decaying, while two-body correlation functions exhibit power-law decay. We perform numerical simulations in the paradigmatic case of two coupled $XY$ models at finite temperature, finding evidences that for any finite value of the interlayer coupling, the BKT-paired phase is present. We provide a picture of the phase diagram using a renormalization group approach.

Figure 1

$\beta J\text{−}\beta K$ phase diagram, showing the paired phase $C$ appearing for finite values of $\beta K$. Left panel: Monte Carlo results ($L=64$) obtained from the fit of correlation functions $c(k)$ and $z(k)$ as described in the text. The lower line has been obtained using the PCC technique, as in the next panel, while the upper line guides the eye. Middle panel: Monte Carlo results ($L=64$) obtained through the PCC technique, allowing the Kasteleyn-Fortuin clusters to extend on a single layer only (upper red line) or on both layers (lower red line). Right panel: RG phase diagram; see main text.