Universality classes of the two thermal phase transitions

Determine the universality classes of the two temperature-driven phase transitions in the two-dimensional rotor model with short-range antiferromagnetic time-delayed interactions: the transition at which spatial and temporal quasi-long-range magnetization order breaks down, and the transition at which the Z_2 chiral symmetry is restored.

Background

The paper analyzes a two-dimensional rotor system with short-range antiferromagnetic retarded interactions and identifies three temperature regimes. At low temperature, spatial and temporal magnetization correlations exhibit quasi-long-range order while Z_2 chiral order persists. At an intermediate temperature, the magnetization quasi-long-range order is destroyed but chiral long-range order remains; at high temperature, all identified orders disappear.

These regimes imply two distinct thermal phase transitions: one involving the simultaneous loss of spatial and temporal quasi-long-range order and another involving restoration of the discrete Z_2 chiral symmetry. The paper reports the existence of these transitions but does not determine their universality classes.

References

Consequently, the system is expected to exhibit two phase transitions with increasing temperature. The first is defined by the concurrent breakdown of spatial and temporal quasi-long-range order, while the second corresponds to the restoration of $\mathbb{Z}_2$ chiral symmetry. Determining the universality classes of these transitions remains a subject for future study.

— Memory-driven Topological Defects and Unconventional Long-Range Order  (2609.02586 - Ding et al., 2 Sep 2026) in End Matter, Section "Stability of 2D non-equilibrium phases against thermal fluctuation"