Arbitrary number of thermally induced phase transitions in different universality classes in $XY$ models with higher-order terms

TL;DR

This work extends the planar XY model by adding $n_t-1$ higher-order nematic couplings ${\cos}(q^k\phi)$ with exponentially increasing order and increasing strength, enabling an arbitrary number of temperature-driven phase transitions. Using Metropolis Monte Carlo and finite-size scaling with reweighting on square lattices, the authors map rich phase diagrams: a single BKT transition from the paramagnetic phase to the highest-order nematic phase, followed by $n_t-2$ non-BKT transitions to progressively lower-order nematic and ferromagnetic phases. For $q\le4$ these lower-temperature transitions fall into Ising ($q=2,4$) or three-states Potts ($q=3$) universality, mirroring discrete Clock-model behavior, while $q=5$ exhibits additional split transitions and nonuniversal critical behavior due to competing terms, including domain-structured ordering. The results illuminate how higher-order couplings and symmetry breaking shape multi-stage criticality and suggest intriguing extensions to 3D, where true long-range order and possibly different universality classes may emerge.

Abstract

We propose generalized variants of the $XY$ model capable of exhibiting an arbitrary number of phase transitions only by varying temperature. They are constructed by supplementing the magnetic coupling with $n_t-1$ nematic terms of exponentially increasing order with the base $q=2,3,4$ and $5$, and increasing interaction strength. It is found that for $q=2,3$ and $4$ with sufficiently large coupling strength of the final term, the models exhibit a number of phase transitions equal to the number of the terms in the generalized Hamiltonian. Starting from the paramagnetic phase, the system transitions through the cascade of $n_t-1$ nematic phases of the orders $q^{k}$, $k=n_t-1,n_t-2,\hdots,1$, that are characterized by $q^{k}$ preferential spin directions symmetrically disposed around the circle, to the ferromagnetic (FM) phase at the lowest temperatures. Besides the BKT transition from the paramagnetic phase, all the remaining transitions have a non-BKT nature: depending on the value of $q$ they belong to either the Ising ($q=2$ and $4$) or the three-states Potts ($q=3$) universality class. For $q=5$, due to the interplay between different terms, the phase transitions between the ordered phases observed for $q<5$ split into two and the number of the ordered phases increases to $2n_t-1$. These phases are characterized by a domain structure with the gradually increasing short-range FM ordering within domains that extends to different kinds of FM ordering in the last two low-temperature phases. The respective transitions do not seem to obey any universality.

Figures (10)

• Figure 1: (Color online) Temperature dependencies of (a-c) the specific heat (d-f) the generalized magnetizations and (g-i) the generalized susceptibilities, for $n_t=3$ and (a,d,g) $q=2$, (b,e,h) $q=3$, and (c,f,i) $q=4$. The coupling constants are set to: $J_{1}=0.1$, $J_{2}=0.3$ and $J_{4}=1$ for $q=2$, $J_{1}=0.25$, $J_{3}=0.5$ and $J_{9}=1$ for $q=3$, and $J_{1}=0.25$, $J_{4}=0.5$ and $J_{16}=1$ for $q=4$. Background colors highlight approximate regions occupied by the phases I, II and III. Spin angle distributions at selected temperatures in the respective phases are demonstrated in the insets of panels (d-f).
• Figure 2: (Color online) Temperature dependencies of (a) the specific heat (b) the generalized magnetizations and (c) the generalized susceptibilities, for $n_t=3$ and $q=5$. The coupling constants are set to $J_{1}=0.3$, $J_{5}=0.6$ and $J_{25}=1$.
• Figure 3:
• Figure 4: (Color online) Spatial correlation function decay in the respective phases, represented by temperature $T$, for $n_t=3$ and $q=5$. $\xi$ denotes the estimated correlation length for the exponential (exp.) decay regime (phases IIa, IIb and III) and $\eta$ is the estimated algebraic (alg.) decay exponent (phases Ia and Ib).
• Figure 5: (Color online) Potential function, ${\mathcal{H}^{i,j}}$, and its contributions, ${\mathcal{H}_{1}^{i,j}}$, ${\mathcal{H}_{5}^{i,j}}$ and ${\mathcal{H}_{25}^{i,j}}$, coming from the $n_t=3$ terms as functions of the phase difference $\Delta \phi$.