Abstract
In this paper, we revisit the q-state clock model for small systems. We present results for the thermodynamics of the q-state clock model for values from to for small square lattices of , with L ranging from to with free-boundary conditions. Energy, specific heat, entropy, and magnetization were measured. We found that the Berezinskii–Kosterlitz–Thouless (BKT)-like transition appears for , regardless of lattice size, while this transition at is lost for ; for , the BKT transition is never present. We present the phase diagram in terms of q that shows the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at the critical temperature for small systems, and the transition changes such that it is from the FM to the BKT phase for larger systems, while a second phase transition between the BKT and the PM phases occurs at . We also show that the magnetic phases are well characterized by the two-dimensional (2D) distribution of the magnetization values. We made use of this opportunity to carry out an information theory analysis of the time series obtained from Monte Carlo simulations. In particular, we calculated the phenomenological mutability and diversity functions. Diversity characterizes the phase transitions, but the phases are less detectable as q increases. Free boundary conditions were used to better mimic the reality of small systems (far from any thermodynamic limit). The role of size is discussed.
Highlights
The q-state clock model is the discrete version of the famous 2D XY model, which is probably the most extensively studied example showing the Berezinskii–Kosterlitz–Thouless (BKT) transition [1,2,3,4]in the presence of a frustrated quenched disordered phase [5,6,7,8]
These series of results are elaborated into a phase diagram of TC vs. q, where TC is determined from the C ( T ) curves
The results shown by the previous two figures, as well as similar ones for other intermediate values of q, show that the information method is able to recognize the transitions present in the clock model for low values of q when applied to the energy data vectors produced by the Monte Carlo (MC) simulations
Summary
The q-state clock model is the discrete version of the famous 2D XY model, which is probably the most extensively studied example showing the Berezinskii–Kosterlitz–Thouless (BKT) transition [1,2,3,4]. The nature of the phase transitions in the general clock model has been widely studied with different theoretical and numerical approaches. These studies have given mixed results for the characterization of transitions at the lower bound of q (for instance, see the summary of the related debates in [20]). We report the transition temperatures T1 for transitioning from the ferromagnetic (FM) to the P phase for very small q values, continuing to the transition from the FM to the BKT phase for larger values of q, as well as T2 for the transition from the BKT phase to the disordered paramagnetic (P) phase These series of results are elaborated into a phase diagram of TC vs q, where TC is determined from the C ( T ) curves. A presentation of the results and discussion are given in Section 3, and the last section is devoted to conclusions
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