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arXiv:2504.17616

One-dimensional $q$-state modified Potts model and its thermodynamic functions

Hasan Akın

Published Apr 24, 2025• 0 citations
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Abstract

Since its introduction, the Potts model has gained widespread popularity across various fields due to its diverse applications. Even minor advancements in this model continue to captivate scientists worldwide, and small modifications often intrigue researchers from different disciplines. This paper investigates a one-dimensional \(q\)-state modified Potts model influenced by an external magnetic field. By leveraging the transfer matrix method, exact expressions are derived for key thermodynamic quantities, including free energy, entropy, magnetization, susceptibility, and specific heat capacity. Numerical analyses explore how these thermodynamic functions vary with relevant parameters, offering insights into the system's behavior. Additionally, the asymptotic properties of these quantities are examined in the limiting cases \(T \to 0\) and \(T \to \infty\). The findings contribute to a deeper understanding of the model's thermodynamic characteristics and highlight its potential applications across various disciplines.

DOI
10.48550/arxiv.2504.17616

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One-dimensional qqq-state modified Potts model and its thermodynamic functions
Since its introduction, the Potts model has gained widespread popularity across various fields due to its diverse applications. Even minor advancements in this model continue to captivate scientists worldwide, and small modifications often intrigue researchers from different disciplines. This paper investigates a one-dimensional (q)-state modified Potts model influenced by an external magnetic field. By leveraging the transfer matrix method, exact expressions are derived for key thermodynamic quantities, including free energy, entropy, magnetization, susceptibility, and specific heat capacity. Numerical analyses explore how these thermodynamic functions vary with relevant parameters, offering insights into the system's behavior. Additionally, the asymptotic properties of these quantities are examined in the limiting cases (T \to 0) and (T \to \infty). The findings contribute to a deeper understanding of the model's thermodynamic characteristics and highlight its potential applications across various disciplines.
The q-state Potts model on a diamond chain has mathematical significance in analyzing phase transitions and critical behaviors in diverse fields, including statistical physics, condensed matter physics, and materials science. By focusing on the three-state Potts model on a diamond chain, we reveal rich and analytically solvable behaviors without phase transitions at finite temperatures. Upon investigating thermodynamic properties such as internal energy, entropy, specific heat, and correlation length, we observe sharp changes near zero temperature. Magnetic properties, including magnetization and magnetic susceptibility, display distinct behaviors that provide insights into spin configurations in different phases. However, the Potts model lacks genuine phase transitions at finite temperatures, in line with the Peierls argument for one-dimensional systems. Nonetheless, in the general case of an arbitrary q state, magnetic properties such as correlation length, magnetization, and magnetic susceptibility exhibit intriguing remnants of a zero-temperature phase transition at finite temperatures. Furthermore, residual entropy uncovers unusual frustrated regions at zero-temperature phase transitions. This feature leads to the peculiar thermodynamic properties of phase boundaries, including a sharp entropy change resembling a first-order discontinuity without an entropy jump, and pronounced peaks in second-order derivatives of free energy, suggestive of a second-order phase transition divergence but without singularities. This unusual behavior is also observed in the correlation length at the pseudocritical temperature, which could potentially be misleading as a divergence.
Here we consider a one-dimensional q-state Potts model with an external magnetic field and an anisotropic interaction that selects neighboring sites that are in the spin state 1. The present model exhibits unusual behavior in the low-temperature region, where we observe an anomalous vigorous change in the entropy for a given temperature. There is a steep behavior at a given temperature in entropy as a function of temperature, quite similar to first-order discontinuity, but there is no jump in the entropy. Similarly, second derivative quantities like specific heat and magnetic susceptibility also exhibit strong acute peaks similar to second-order phase transition divergence, but once again there is no singularity at this point. Correlation length also confirms this anomalous behavior at the same given temperature, showing a strong and sharp peak which easily one may confuse with a divergence. The temperature where this anomalous feature occurs we call the pseudocritical temperature. We have analyzed physical quantities, like correlation length, entropy, magnetization, specific heat, magnetic susceptibility, and distant pair correlation functions. Furthermore, we analyze the pseudocritical exponents that satisfy a class of universality previously identified in the literature for other one-dimensional models; these pseudocritical exponents are for correlation length ν=1, specific heat α=3, and magnetic susceptibility μ=3.
An exact analytical solution of generalized three-state double-chain Potts model with multi-spin interactions which are invariant under cyclic shift of all spin values is obtained. The partition function in a finite cyclically closed strip of length L, as well as the free energy, internal energy, entropy and heat capacity in thermodynamic limit are calculated using transfer-matrix method. Partial magnetization and susceptibility are suggested as the generalization of usual physical characteristics of a system. Proposed model can be interpreted as a generalized version of standard Potts model (which has Hamiltonian expressed through Kronecker symbols) and clock model (with Hamiltonian expressed through cosines). Considering a particular example of the model with plenty of forces, model's ground states are found, figures of its thermodynamic characteristics and discussed their behaviour at low temperature are shown.