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

TL;DR

This paper analyzes a one-dimensional $q$-state modified Potts model in an external field using the transfer-matrix framework. By constructing the transfer matrix and identifying its largest eigenvalue, it derives a closed-form per-site free energy $f(eta,h,J,q)=-rac{1}{eta} ext{ln}igl(rac{1+(q-1)e^{2h+2Jeta}}{e^{h+Jeta}}igr)$ and exact expressions for key thermodynamic functions: entropy, magnetization, susceptibility, and specific heat capacity, each obtained from derivatives of $f$ with respect to the natural variables. The work combines cavity-method insights with the transfer-matrix approach and provides numerical plots and asymptotic analyses as $T o 0$ and $T o abla o \

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.

Figures (9)

• Figure 1: (Color online) 3D plots of the free energy function \ref{['FEF-q1']} for the range $\beta \in [0.001, 30]$ and $h \in [-3, 3]$: (a) for $J = -12, q = 16$; (b) for $J = 0.95, q = 16$.
• Figure 2: (Color online) Graphs of free energy given in \ref{['FEF-q1']} (left) for $h=-3,J=5.15$ (right) for $h=4,J=-3$.
• Figure 3: (Color online) Three-dimensional (3D) plots of the entropy function \ref{['Entropy1a']} within the range $T\in [0.001,2]$ and $h \in [-3,18]$: (a) for $J = -2, q = 17$ (b) for $J =8, q =22$ within the range $T\in [0.001,22]$ and $h \in [-3,22]$
• Figure 4: (Color online) Entropy function plots \ref{['Entropy1a']} as a function of temperature $T$: (left) for $h = -3$, $J = 5.3$, $q = 7$; (right) for $h = 2$, $J = -5.3$, $q = 17$, 22.
• Figure 5: (Color online) Three-dimensional plots of the magnetization function \ref{['Mag-Formula1']} are presented for the following parameter ranges: (a) $J = -2, q = 17$ with $T \in [0.001, 30]$ and $h \in [-3, 3]$; (b) $J = 8, q = 22$ with $T \in [0.001, 40]$ and $h \in [-8, 8]$.

Theorems & Definitions (1)

• Remark 3.1