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arXiv:2603.19351

Crossover and Critical Behavior in the Layered XY Model

Roman Kracht, Andrea Trombettoni, Ilaria Maccari, Nicolò Defenu

Published Mar 19, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.19351

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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∥J_\parallelJ∥​ and inter-plane J⊥J_\perpJ⊥​ couplings. Our study includes very small values of the system anisotropy Δ=J⊥/J∥Δ=J_\perp /J_\parallelΔ=J⊥​/J∥​ 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 ΔΔΔ, 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 ℓJ\ell_JℓJ​, which diverges at small ΔΔΔ 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.
Early theoretical studies of the two-dimensional XY model with four-fold anisotropy (XY h 4 ) predicted that any finite anisotropy should completely suppress Kosterlitz–Thouless (KT) transition of the isotropic system and replace it with a second-order phase transition. This prediction is inconsistent with Monte Carlo (MC) simulations, which suggested KT-like behavior at weak anisotropy. To resolve this discrepancy, we performed large-scale MC simulations to systematically examine the XY h 4 model across a wide range of anisotropic strengths. Our results indicate that while weak anisotropy can mimic KT behavior in small systems, a detailed finite-size scaling analysis of larger systems reveals that the temperature derivative of Binder ratio and the correlation length ratio displays scaling behaviors characteristic of a second-order transition even at the weak h 4 values. The critical exponent ν is non-universal and varies continuously with h 4 from weak to strong anisotropy. These findings confirm that XY h 4 model exhibits a second-order phase transition for weak finite anisotropy, effectively reconciling theoretical predictions with numerical observations.
Scaling equations are obtained for thermally excited vortex loops in the anisotropic three-dimensional (3D) XY model with interplane/intraplane coupling ratio ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$=${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}2}$. For high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors, ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}2}$, related to the Ginzburg-Landau masses, is in a strong anisotropy regime, ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$0.5, and a length scale r0{\mathit{r}}_{0}r0​=${\ensuremath{\gamma}}_{0}$a0{\mathit{a}}_{0}a0​ naturally arises (a0{\mathit{a}}_{0}a0​= lattice constant). (i) For loop major axes a>r0{\mathit{r}}_{0}r0​, the dominant excitations are 3D elliptical vortex loops cutting multiple planes, with a\ensuremath{\sim}${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$(T)=a0{\mathit{a}}_{0}a0​\ensuremath{\Vert}\ensuremath{\varepsilon}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ where \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}\ensuremath{\equiv}\ensuremath{\Vert}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​)/Tc{\mathit{T}}_{\mathit{c}}Tc​\ensuremath{\Vert}, and \ensuremath{\nu}=0.67. The vorticity segment components (\ensuremath{\mu}=x^,y^,z^) are ${\mathit{J}}_{\mathrm{\ensuremath{\mu}}}$(r)=0,\ifmmode\pm\else\textpm\fi{}1, and interact via a Biot-Savart-like law. The renormalized anisotropy ${\ensuremath{\gamma}}_{\mathit{l}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}1, asymptotically isotropic, as l=ln(a/a0{\mathit{a}}_{0}a0​)\ensuremath{\rightarrow}\ensuremath{\infty}: anisotropy is irrelevant. (ii) For loop scales r0{\mathit{r}}_{0}r0​>a>2a0{\mathit{a}}_{0}a0​ the dominant excitations are quasi-2D rectangular loops with short sides Jz{\mathit{J}}_{\mathit{z}}Jz​(r)=\ifmmode\pm\else\textpm\fi{}1 cutting single planes, that are thus effectively decoupled at finite scales r0{\mathit{r}}_{0}r0​. The Jz{\mathit{J}}_{\mathit{z}}Jz​=\ifmmode\pm\else\textpm\fi{}1 vortex components, of in-plane separation \ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{W}}$mN\ensuremath{\Vert}r0{\mathit{r}}_{0}r0​, interact via a logarithmic [ln(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/r0{\mathit{r}}_{0}r0​)] plus linear [Q0{\mathit{Q}}_{0}Q0​(\ensuremath{\Vert}R${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\parallel}}}$\ensuremath{\Vert}/a0{\mathit{a}}_{0}a0​-1)] potential. As ${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}0, the coefficient Q0{\mathit{Q}}_{0}Q0​\ensuremath{\rightarrow}0, and the Kosterlitz-Thouless limit is recovered. (iii)The 3D transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ versus ${\mathit{K}}_{\mathrm{\ensuremath{\perp}}}$/${\mathit{K}}_{\mathrm{\ensuremath{\parallel}}}$, calculated for strong anisotropies from ``2D'' (3D) scaling for scales r0{\mathit{r}}_{0}r0​ (>r0{\mathit{r}}_{0}r0​), matches existing Monte Carlo data well. Critical regions \ensuremath{\Vert}${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$\ensuremath{\Vert}\ensuremath{\le}${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1/\ensuremath{\nu}}$ are estimated. Contact is made with ideas of an intrinsic critical current arising from linear \ensuremath{\sim}${\ensuremath{\gamma}}_{0}^{\mathrm{\ensuremath{-}}1}$${\mathit{R}}_{\mathrm{\ensuremath{\parallel}}}$ vortex segment effective potentials.
We report Monte Carlo simulations of Villain's periodic Gaussian XY model on L2{\mathit{L}}^{2}L2\ifmmode\times\else\texttimes\fi{}N lattices of film geometry (L\ensuremath{\gg}N) with up to N=16 layers, employing the single-cluster update algorithm combined with improved estimators for measurements. The boundary conditions are periodic within each layer and free at the bottom and top layer. Based on data for the specific heat, the spin-spin correlation function, and the susceptibility in the high-temperature phase we study the crossover from three- to two-dimensional behavior as criticality is approached. For the transition temperatures, determined from Kosterlitz-Thouless fits to the correlation length and susceptibility, we observe a pronounced scaling behavior with N. The associated critical exponent, however, deviates from theoretical expectations. More qualitatively, we further discuss the distribution and shapes of vortex loops in the crossover region.
The critical behavior of vortices in layered systems is studied. The vortex interactions that we use approximate those of vortices in a layered superconductor. We do a mathematically rigorous, real-space renormalization-group study on our model to derive the recursion relations. Terms not found by other studies on layered systems have been derived. It is found that one of the new terms contributes to layer decoupling just above the transition temperature, behavior which is consistent with Monte Carlo studies of this system. By analylzing the correlation length, we are able to study the dependence of the transition temperature and of the size of the three-dimensional (3D) critical region on the strength of the interlayer coupling. The size of the 3D critical region above the transition is found to depend on the interlayer coupling in a way which is different than that predicted by others. We study our results in the context of the high-temperature superconductors and find that they are in good accord with experiments on these materials.
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.