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

TL;DR

This work tests whether a BKT paired phase can arise in bilayer XY systems. Using Monte Carlo simulations and analytical arguments, it shows that the previously proposed BKT paired phase is absent in the ferromagnetic interlayer model, and introduces a new paired-phase gradient-coupled model that exhibits a genuine BKT paired phase with two distinct transitions. The study reveals that the paired-spin anomalous dimension $η_p$ varies continuously along the BKT boundary, a feature suggesting physics beyond standard BKT universality and hinting at renormalization effects from spin-wave dynamics. These results illuminate how interlayer coupling types shape critical behavior and point to possible extensions to higher dimensions and experimental realizations.

Abstract

Intriguing phases may emerge when two-dimensional systems are coupled in a bilayer configuration. In particular, a Berezinskii-Kosterlitz-Thouless (BKT) paired superfluid phase was predicted and claimed to be numerically observed in a coupled $XY$ model with ferromagnetic interlayer interactions, as reported in [\href{https://doi.org/10.1103/PhysRevLett.123.100601}{Phys. Rev. Lett. 123, 100601 (2019)}]. However, both our Monte Carlo simulations and analytical analysis show that this model does not exhibit a BKT paired phase. We then propose a new model incorporating paired-phase gradient interlayer interactions to realize the BKT paired phase. Moreover, we observe that the anomalous magnetic dimension varies along the phase transition line between the disordered normal phase and the BKT paired phase. This finding requires an understanding beyond the conventional phase transition theory.

Figures (10)

• Figure 1: Phase diagram of (a) $H_{\textrm{single}}$ and (b) $H_{\textrm{pair}}$. The solid lines with data points on them are phase boundaries. The black dotted vertical lines correspond to the intervals used in Fig. \ref{['fig:model_1']} and Fig. \ref{['fig:model_2']}. "SF" denotes the usual superfluid phase, "SF$_2$" denotes the superfluid phase with two superfluids, "Disorder" denotes the disordered normal phase, and "PSF" denotes the BKT paired phase. The dashed orange line in (a) represents an additional phase transition claimed in PhysRevLett.123.100601PhysRevB.111.094415, which is not observed in our calculations. The schematic figures on the right half panel illustrate the key characteristics of the SF and PSF phases. In the SF phase, the vortices of the single-layer spins in each layer, as well as those of the paired spins, are tightly bound. The two vortices within a pair have the same sign due to the ferromagnetic interlayer interactions. In the PSF phase, the single-layer spins remain disordered, while the paired spins form bound vortices, giving rise to a superfluid of paired spins.
• Figure 2: The numerical results for $H_{\textrm{single}}$ along the black dotted vertical line in Fig. \ref{['fig:PD']}(a) with $K = 1$ are presented. The correlation length ratios $\xi_a/L$ for single-layer spins and $\xi_p/L$ for paired spins as functions of $J$ are shown in (a) and (b), respectively. In the corresponding inset, $J(L)$ is plotted against $1/\ln^2(L/L_0)$ for interpolation to estimate the critical point of the BKT transition, and the different colored lines represent different values of the correlation length ratio used for interpolation. The gray lines in the insets indicate the results from least squares fitting, which are consistent with the interpolation results. The gray line in the main figure indicates the transition point obtained by considering the ratios of two types of spins, while the dashed line represents another transition point reported in PhysRevLett.123.100601PhysRevB.111.094415. The magnetization ratio $R_{M,4}$ is plotted in (c), and its inset shows the correlation-function ratio $R_{g,4}(r)$ at $J=0.8$.
• Figure 3: The numerical results for $H_{\textrm{pair}}$ along the black dotted vertical line in Fig. \ref{['fig:PD']}(b) with $K = 0.9$ are presented. The correlation length ratios $\xi_a/L$ for single-layer spins and $\xi_p/L$ for paired spins as functions of $J$ are shown in (a) and (b), respectively. The gray lines in (a) and (b) represent the estimated transition points for single-layer spins and paired spins, respectively. The correlation functions $g_a(r)$ and $g_p(r)$ at $J = 0.5$, effectively at the paired BKT point $J_{c1}=0.499(2)$, are plotted in (c) and (d). It is clearly shown that the two-point correlation function $g_a(r)$ within a single layer decays exponentially fast, while the paired correlation $g_p(r)$ decays algebraically. Note that $g_a$ is significantly smaller than $10^{-4}$ for $r \approx 50$ while $g_p \approx 0.2$ for $r = 128$.
• Figure 4: The log-log plot of the squared magnetization density $\langle M_p^2 \rangle$ for paired spins versus system size $L$ at various critical points along the phase boundary between the BKT paired phase and the disordered phase. The approximately straight lines with different slopes clearly indicate that the paired anomalous magnetic exponent $\eta_p$ varies along the phase boundary. This is in contrast with the naive expectation from the universality that it should be a constant, raising an open question on the underlying mechanism. The inset displays $\eta_p$ versus increasing $K$ along the phase boundary.
• Figure 5: The ratio of two types of correlation functions $R_{g,2}$ at $K=0$ and $J=1.2$ (QLRO phase) for the $H_{\textrm{single}}$ model. The straight-line behavior and collapse at $R_{g,2} = 1$ indicate that $g_p(r) = g_a^2(r)$.