Speaker
Description
The Mermin-Wagner theorem states that in a two-dimensional $XY$ (or planar rotator) model with nearest-neighbor interactions the continuous symmetry cannot be broken and thus no standard phase transition can occur. Nevertheless, the model is well known to show a so-called Berezinskii-Kosterlitz-Thouless (BKT) phase transition due to the presence of topological excitations, called vortices and antivortices. At the BKT transition the vortices and antivortices bind in pairs, which results in an algebraic decay of the correlation function within the quasi-long-range-ordered BKT phase below the BKT transition temperature $T_{\rm BKT}$.
The standard planar rotator model can be generalized by including (pseudo)nematic higher-order coupling terms, which give rise to further, fractional vortex excitations. The presence of both integer and fractional vortices can lead to a richer critical behavior and such models have been demonstrated to show up to three different phase transitions as a function of temperature [1]. Here, we propose related models that can display even an arbitrary number of phase transitions belonging to different universality classes. The models are based on the standard two-dimensional planar rotator model, which is generalized by including $n$ higher-order nematic terms with exponentially increasing order $q^k$, $k = 1,\dots,n$, and linearly increasing interaction strength.
By employing Monte Carlo simulation, we demonstrate that under certain conditions the number of phase transitions in such models is equal to the number of terms in the generalized Hamiltonian and thus it can be predetermined by construction [2]. The proposed models produce the desirable number of phase transitions by solely varying the temperature. With decreasing temperature, the system passes through a sequence of different phases with gradually decreasing symmetries. A finite-size scaling analysis shows that the corresponding phase transitions all start at higher temperatures with the (no spontaneous symmetry breaking) BKT transition and may proceed through a sequence of discrete $\mathbb{Z}_q$ symmetry-breaking transitions between different nematic phases down to the lowest-temperature ferromagnetic phase. More specifically, for the higher-order nematic terms that exponentially increase with the basis $q = 2$ and $4$ the remaining transitions below $T_{\rm BKT}$ are found to belong to the Ising, for $q = 3$ to the three-state Potts, and for $q \geq 5$ to the BKT universality classes.
Acknowledgements
This research was funded by the projects VEGA 1/0695/23 and APVV-20-0150.
References
[1] F.C. Poderoso et al., “New Ordered Phases in a Class of Generalized XY Models” Physical Review Letters, vol. 106, no. 6. American Physical Society (APS), Feb. 10, 2011. https://doi.org/10.1103/physrevlett.106.067202
[2] M. Žukovič, “Generalized XY Models with Arbitrary Number of Phase Transitions,” Entropy, vol. 26, no. 11. MDPI AG, p. 893, Oct. 23, 2024. https://doi.org/10.3390/e26110893
