Cross derivative of the Gibbs free energy: A universal and efficient method for phase transitions in classical spin models

Abstract

With an auxiliary weak external magnetic field, we reexamine the fundamental thermodynamic function, Gibbs free energy F (T, h), to study the phase transitions in the classical spin lattice models. A cross derivative, i.e. the second-order partial derivative of F (T, h) with respect to both temperature and field, is calculated to precisely locate the critical temperature, which also reveals the nature of a transition. The strategy is efficient and universal, as exemplified by the 5-state clock model, 2-dimensional (2D) and 3D Ising models, and the XY model, no matter a transition is trivial or exotic with complex excitations. More importantly, other conjugate pairs could also be integrated into a similar cross derivative if necessary, which would greatly enrich our vision and means to investigate phase transitions both theoretically and experimentally.

Clock model interpolation and symmetry breaking in O(2) models

“…Now we use the value of X to locate the Ising critical point and find the large-β BKT transitions for q = 5, 6. Notice that for q = 5, although both a small external field and a small deviation of q from an integer break the Z q symmetry to a Z 2 symmetry, the magnetic susceptibility with a weak external field does not have a peak around the large-β BKT transition, so it fails to predict the location of the phase transition [42,54], but here we always have an Ising critical point for fractional q. In Fig.…”

“…For q = 4.9999 (closer to 5) with h = 4 × 10 −5 , one can see that the small-β peak becomes higher than the large-β one, and the large-β peak is fading away, which is consistent with the results in Refs. [42,54] for the fivestate clock model. When the external field is decreased to h = 2 × 10 −5 , the large-β peak grows a much larger amount of height than the small-β one does and becomes higher than the small-β one.…”

“…But the crossover should go to the BKT transition point as the Z q symmetry is recovered. The cross derivative of the free energy should be able to capture this [54], but we will just focus on the magnetic susceptibility.…”

“…Calculation of the magnetic susceptibility in the presence of a weak external field is a universal method to detect critical points that can be easily implemented in the TRG. However, it does not show a peak to indicate the large-β BKT transition in the five-state clock model [42,54]. The study of the γ → ∞ limit with fractional q not only provides us a clear picture of what phases the symmetry-breaking term will drive the XY model to, paving the way to discussions for the full phase diagram at finite γ, but also brings us a new tool to detect BKT transitions in Z n models.…”

Moving from continuous to discrete symmetry in the 2D XY model

“…However, numerical studies suggest that the N = 5 transition may not be the same type as for N > 5, though may still be related [28][29][30]. The nature of the lower temperature transition is also believed to be of BKT-type based on central charge arguments [31][32][33]. While the upper critical temperature remains fairly constant as N → ∞, the lower critical temperature moves towards zero in that limit [34].…”