Crossover and Critical Behavior in the Layered XY Model

Abstract

Motivated by the interplay between 2D and 3D scaling signatures observed in unconventional layered superconductors, we present a systematic Monte Carlo study of the three-dimensional classical XY model with anisotropic in-plane $J_\parallel$ and inter-plane $J_\perp$ couplings. Our study includes very small values of the system anisotropy $Δ=J_\perp /J_\parallel$ not studied before, and focuses on characterizing the crossover from quasi-2D topological scaling to genuine 3D critical behavior. The numerical results for the critical temperature unambiguously reveal a logarithmic scaling with $Δ$, directly related to the topological scaling in the 2D limit. Despite the 3D nature of the layered XY criticality, topological scaling signatures survive up to system sizes comparable to the crossover length $\ell_J$, which diverges at small $Δ$ with a scaling behavior reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition. This shows that genuine 3D symmetry-breaking behavior emerges only at exceedingly large system sizes when the anisotropy is very strong. Our results indicate that new experimental evidence is required to clarify the extent to which the critical signatures observed in layered strongly correlated materials are shaped by their pronounced anisotropy.

Figures (13)

• Figure 1: Critical temperature $T_c$ as a function of the coupling ratio ${\Delta = J_\perp / J_\parallel}$: The blue dots show our Monte Carlo results (errors smaller than markers) with the gray line being a guide to the eye. The red triangles show results from tanner1994dimensionalitats and the green cross shows the isotropic high-precision result from xu2019highprecision. The solid black line indicates the fit with Eq. \ref{['eq:crit_temp_delta']}, where $T_{BKT}$ is kept fixed to the literature value $0.89298(4)$hasenbusch2005twodimensional.
• Figure 2: Correlation length critical exponent $\nu$ as a function of the coupling ratio $\Delta = J_\perp / J_\parallel$: The blue dots represent extrapolation results from the Binder cumulant $b$ and the red triangles those from the rescaled stiffness $j$. In contrast, the average of the $5$ largest system sizes (without extrapolation) is indicated by the blue $\diagup$-shaded area for $b$ and by the red $\diagdown$-shaded area for $j$. The horizontal dashed line represents the isotropic high-precision result from xu2019highprecision.
• Figure 3: (a) Josephson length scale $\ell_J$ as a function of the coupling ratio ${\Delta = J_\perp / J_\parallel}$ for various temperatures $T$. We computed $\ell_J$ from Eq. \ref{['eq:josephson_criterion']} with a threshold $t=0.7$. For each temperature, the data points follow straight lines in $\log$-$\log$-scale, thus confirming the power-law scaling from Eq. \ref{['eq:josephson_length_scaling']}. (b) Curve collapse of layer-alignment $\Psi$ for temperature ${T \approx 0.899}$. We plot $\Psi$ over the rescaled system size ${\log [ L \Delta^{a (T)} ]}$ for ${L\geq 4}$, where the exponent ${a\approx0.575}$ was determined by minimization of the collapse error. (c) Anomalous dimension $\eta_{2D}$ over temperature $T$. The circles indicate curve collapse results from the layer-alignment $\Psi$ of the 3D model within a $1\sigma$ confidence interval. The triangles indicate high-precision results from maccari2017broadening from 2D simulations. The red dotted lines indicate the exact theory prediction $\eta_{2D}(T_{BKT})= 1/4$. All temperatures are normalized by ${T_{BKT} \approx 0.8929(4)}$hasenbusch2005twodimensional.
• Figure 4: Goodness-of-fit $\chi^2/\mathrm{DOF}$ for Binder cumulant as a function of the coupling ratio ${\Delta = J_\perp / J_\parallel}$: The blue dots correspond to the second-order finite-size scaling ansatz from Eq. \ref{['eq:binder_scaling']}, whereas the red triangles correspond to the BKT scaling. Both fits have been performed with the critical temperature and $\nu$ respectively $c$ unconstrained. Each scaling function has been expanded as a Taylor polynomial with $6$ free coefficients.
• Figure 5: Derivative of Binder cumulant $b$ with respect to inverse temperature $\beta = 1/T$ for (a) coupling ratio ${\Delta = 0.05}$ and (b) isotropic couplings ${\Delta = 1}$. Data points are indicated by circles and the associated fits are indicated by the solid lines. For clarity, here we only show results for system sizes $L=72,56,36,28,18,14$ (from top to bottom). The full extrapolation used to obtain the results displayed in Fig. \ref{['fig:nu_over_delta']} however uses more intermediate system sizes, as explained in the Supp. Mat. supplemental.