Dynamical Properties of Potts Model with Invisible States

Abstract

Abstract. We study dynamic behavior of Potts model with invisible states near the firstorder phase transition temperature. This model is regarded as a standard model to analyse nature of phase transition. We can control the energy barrier between the ordered state and disordered state without changing the symmetry which breaks at the transition point. We focus on melting process starting from the perfect ordered state. We calculate time-dependency of the order parameter, density of invisible state, and internal energy. All of them show two-step relaxation behavior. We also analyze the relationship between the characteristic melting time and characteristic scale of the energy barrier by changing the number of invisible states. We find that characteristic melting time increases as the energy barrier enlarges in this model. This model is regarded as a fundamental model to analyze dynamic behavior near the first-order phase transition point. IntroductionFrustration causes many interesting static and dynamic behavior which are not observed in unfrustrated systems because of peculiar density of states [1][2][3][4][5][6][7][8][9][10]. In two-dimensional frustrated systems, there have been found many nontrivial phase transitions such as order by disorder [11,12], reentrant phase transition [13][14][15][16][17], topological phase transition [18], and novel type of first-order phase transition. Recently, strange first-order phase transitions have been found in two-dimensional frustrated continuous spin systems [19][20][21][22].In [19], the authors studied equilibrium properties of the classical Heisenberg model on triangular lattice with nearest neighbor ferromagnetic interaction J 1 and third-nearest neighbor antiferromagnetic interaction J 3 . They found that a first-order phase transition with threefold symmetry breaking occurs at finite temperature. This looks a strange phase transition, since phase transition with threefold symmetry breaking is often second-order phase transition on two-dimensional lattice e.g. the three-state ferromagnetic Potts model [23]. After this study, similar nature of first-order phase transition have been found by a number of researchers [20][21][22]. Stoudenmire et al. found a first-order phase transition with breaking of threefold symmetry in J 1 − J 3 model with biquadratic interaction on triangular lattice. Okumura et al. also found

Classical phase transitions in a one-dimensional short-range spin model

“…The possibility of classical, equilibrium phase transitions below the lower critical dimension is important at a fundamental level as well as for potential manifestations in real-world systems. Onsager's solution of the 2D Ising model [25] was only recently confirmed experimentally [26] and theoretical investigations have shown that adding invisible states can alter the type of phase transition present in such models [16,17,19]. Other recent theoretical and experimental developments include the establishment of a link between complex fields and quantum coherence times [20,21], opening up new ways to access complex fields physically [27].…”

“…Fixing z 1 = z 2 = 1 in Eq. (17) we arrive at the equation for the coordinates of the partition function zeros in the complex y−plane at given pair of (q, r). It is most convenient to display Fisher zeros in the complex t = y −1 plane.…”

“…The Potts model with invisible states was introduced nearly a decade ago in Refs. [16,17]. It differs from the ordinary Potts model [18] in that spins in an invisible state do not interact with their neighbours but they do contribute to entropy.…”

Ising model with invisible states on scale-free networks

“…Thus introducing invisible states does not change the interaction energy, but rather the number of configurations, or equivalentlythe entropy. This model was originally suggested to explain why the phase transition with the q−fold symmetry breaking undergoes a different order than predicted theoretically [7,8,9]. Analysis of this model on different lattices has been a subject of intensive analytic [10,11,12,13,14,15,16] and numerical [7,8,9] studies.…”

Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states

“…To do so we consider the Potts model with invisible states which was introduced a few years ago [19,20] in order to explain some questions about the order of a phase transition where Z(3) symmetry is broken. It differs from the ordinary Potts model [21,22] by adding non-interacting states; if a spin is in one such state, it is "invisible" to its neighbours.…”