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Phase Transitions

Modified 6-state and 8-state potts models in magnetic field

Felix A. Kassan-Ogly

Published Oct 1, 2000• 11 citations
Publisher

Abstract

The exact solutions for the modified 6-state and 8-state Potts models in an external magnetic field are obtained within the framework of the Kramers-Wannier transfer matrix. The explicit analytical expressions for thermodynamic quantities such as magnetization, magnetic susceptibility and specific heat are derived and analyzed at length as functions of temperature and magnetic field.

DOI
10.1080/01411590008228991

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The exact solutions for the modified 6-state and 8-state Potts models in an external magnetic field are obtained within the framework of the Kramers-Wannier transfer matrix. The explicit analytical expressions for thermodynamic quantities such as magnetization, magnetic susceptibility and specific heat are derived and analyzed at length as functions of temperature and magnetic field.
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 qqq-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→0T \to 0T→0 and T→∞T \to \inftyT→∞. The findings contribute to a deeper understanding of the model's thermodynamic characteristics and highlight its potential applications across various disciplines.
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.
We present a solution of the nonlinear renormalization group equations leading to the dominant and subdominant singular behaviors of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non-vanishing magnetic field. The solutions (i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and (ii) prove recently proposed relations among logarithmic exponents.
Abstract Results on the generalization of the Ising model to an arbitrary number of translations of a linear chain in an external magnetic field, taking into account various interactions between spins, are presented. An exact analytical expression has been derived for the largest eigenvalue of the Kramers–Wannier transfer matrix with a translation per two chain periods in an external magnetic field when the nearest and second neighbors are taken into account. Exact expressions have been established for the zero-temperature entropies and magnetizations at various magnitudes and signs of the exchange interactions and the magnetic field. Many of the zero-temperature entropies and zero-temperature magnetizations found are represented via the so-called mathematical ratios long known in the wonderful world of number mathematics (the golden ratio φ, the silver ratio δ, the supergolden ratio ψ, the plastic number ρ, and new (nameless) ones). A result whereby the frustrating entropies and magnetizations can be expressed via the limit of the ratio of certain number sequences without invoking the formalism of the Kramers–Wannier transfer matrix has been obtained.