Entropy and Mutability for the q-State Clock Model in Small Systems

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 $$q=2$$ to $$q=20$$ for small square lattices of $$L\times L$$, with L ranging from $$L=3$$ to $$L=64$$ with free-boundary conditions. Energy, specific heat, entropy, and magnetization were measured. We found that the Berezinskii–Kosterlitz–Thouless (BKT)-like transition appears for $$q>5$$, regardless of lattice size, while this transition at $$q=5$$ is lost for $$L<10$$; for $$q\le 4$$, 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 $$T_1$$ 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 $$T_2$$. 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.