1D Three-state mean-field Potts model with first- and second-order phase transitions

Abstract

We analyze a three-state Potts model built over a ring, with coupling J0, and the fully connected graph, with coupling J. This model is an effective mean-field and can be solved exactly by using transfer-matrix method and Cardano formula. When J and J0 are both positive, the model has a first-order phase transition which turns out to be a smooth modification of the known phase transition of the traditional mean-field Potts model (J > 0 and J0 = 0), despite the connected correlation functions are now non zero. However, when J is positive and J0 negative, besides the first-order transition, there appears also a hidden (non stable) continuous transition. When J is negative the model does not own a phase transition but, interestingly, the dynamics induced by the mean-field equations leads to stable orbits of period 2 with a second-order phase transition and with the classical critical exponent β = 1/2, like in the Ising model.

The qualitative properties of 1D mixed-type Potts-SOS model with 1-spin and its dynamical behavior

“…It is worth mentioning that we followed the approach outlined by Wang et al [30] in our study. However, it is noteworthy that alternative results can be derived using the method proposed by Ostilli and Mukhamedov [31]. Simultaneously, we have computed the mean value of the spin, σ j , that corresponds to the given model.…”

“…In classical literature, the presence of an external field was taken into account when constructing the transfer matrix for 1D spin models. Ostilli and Mukhamedov [31] determined the free energy by deriving the transfer matrix for a 1D three-state Potts model in the presence of a q-component external field. However, under certain special circumstances, calculations were performed in the absence of the q-component external field (see [29,35] for more information).…”

“…Using the Kramers-Wannier transfer technique, which requires building a transfer matrix and determining its eigenvalues, we can solve the problem [16,29,35]. The matrix transfer approach is widely employed in statistical mechanics due to its broad range of applications, making it one of the most frequently utilized methods [29][30][31]. Taking into account the Hamiltonian provided by equation (2.1) and in the presence of an external magnetic field, the expression for the canonical partition function for this system can be formulated as follows:…”

“…This allows us to find explicit solutions by employing the Cardano formula, which provides the three roots. To investigate the model's thermodynamic limit, it suffices to analyze the eigenvalues of T[31]. It is evident that the eigenvalues of the matrix T in (3.5) coincide with the roots of equation(3.6).…”

Gibbs measures of Potts model on Cayley trees: a survey and applications

“…For future reading see [3], [5], [8], [10], [11], [17], [19], [23], [24], [27], [35], [52], [60], [64], [72]- [77], [79], [80], [96]- [112], [122], [128], [130].…”