Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2502.00444

Exact solution of the three-state generalized double-chain Potts model

Pavel Khrapov, G. E. Skvortsov

Published Feb 1, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2502.00444

Related papers

Exact solution of generalized triple Ising chains with multi-spin interactions

Pavel Khrapov, Nikita Volkov

We obtain the exact physical characteristics of the triple-chain Ising model on a torus with all possible multispin interactions invariant with respect to rotation by the angle $2π/ 3$. The exact value of the partition function in a finite cyclically closed strip of length $L$, as well as the free energy, internal energy, specific heat, magnetization, susceptibility, and entropy in the thermodynamic limit at $L \to \infty$ are found by the transfer-matrix method for the model. The spectrum of the transfer-matrix and the structure of its eigenvectors are found. For two special cases - for the model with multispin interactions of even number of spins and for the model with some interactions of two, three, four and six spins, simplified expressions of the mentioned physical characteristics are obtained; in the thermodynamic limit they are expressed through the logarithm of the root of the quadratic equation. For the model with multispin interactions of an even number of spins, a kind of pair correlations in the thermodynamic limit is found, and it is shown that the magnetization at zero magnetic field is equal to zero; the structure of the ground states of the system is found and examples of their projections of seven-dimensional space onto three-dimensional space and examples of configurations corresponding to these ground states are given. The correlation length is shown and its graphs are given. As special cases, we consider the planar triangular model with all possible interactions, including, perhaps, different triple interactions inside neighboring triangles, and the planar model with nearest-neighbor, next nearest-neighbor, and plaquette interactions. For them the main exact physical characteristics have been found. This allowed us to obtain them for the planar gonihedric model as well.

arXiv:2406.10683•Jun 15, 2024•0 citations

Exact Solution and Correlation Functions of Generalized Double Ising Chains

Pavel Khrapov, Stepan Shchurenkov

In this paper the exact solution and correlation functions for a double-chain Ising model with multi-spin interactions and symmetric Hamiltonian density are obtained. The study employs the transfer matrix method to derive fundamental thermodynamic characteristics of the system. The main results include exact expressions for the partition function, free energy, internal energy, specific heat capacity, magnetization, susceptibility, and entropy in a strip of finite length and in the thermodynamic limit. The work provides explicit formulas for the eigenvalues and shows structure of eigenvectors of the transfer matrix. The expression for magnetization in the thermodynamic limit using components of normalized eigenvector corresponding to the maximum eigenvalue is obtained. A detailed analysis is conducted for a special case of interactions involving all kinds of two- and four-spin interactions. This gives the simplified formula for free energy, it is calculated using the root of quadratic equation. Using special relation between parameters the solution for gonihedric model on the plane is obtained. The research reveals properties of the system, including specific features of ground states and phase diagram characteristics. Particular attention is given to the behavior of physical quantities near frustration points and the investigation of spin correlation functions. Plots of physical characteristics, including inverse correlation length, illustrating the obtained results are constructed.

arXiv:2511.00451•Nov 1, 2025•0 citations

Double potts chain and exact results for some two-dimensional spin models

M. A. Yurishchev

An exact analytical solution for the q -state Potts model on a 2×∞ ladder with arbitrary two-, three-, and four-site interactions in a unit cell in presented is a closed form. This solution is used to show that the finite-size internal energy equation [6] yields an accurate value of the critical temperature for the triangular Potts lattice with three-site interactions in alternate triangular faces. It is argued that the above equation is exact at least for self-dual models on isotopic strips.

arXiv:cond-mat/0111401•Journal of Experimental and Theoretical Physics•Nov 1, 2001•1 citation

One-dimensional 3-state and 4-state standard Potts models in magnetic field

Felix A. Kassan-Ogly

The exact solutions for 3-state and 4-state standard Potts models in 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 analysed at length as functions of temperature and magnetic field.

Phase Transitions•Feb 1, 2000•10 citations

Exact solution and pair correlation functions for a generalized three-chain Ising tube with multispin interactions

Pavel Khrapov, Nikita Volkov

We obtain an exact solution for a generalized three-chain Ising tube (TCGIT) of length $L$ with toroidal boundary conditions and the most general $C_3$-invariant Hamiltonian on an elementary prism, containing 20 independent coupling constants, including an external magnetic field. Using an $8\times 8$ transfer matrix, we derive the exact partition function of the finite system and obtain the free energy, internal energy, specific heat, magnetization, magnetic susceptibility, and entropy in the thermodynamic limit $L\to\infty$. In the general case, $λ_{\max}$ is determined by a quartic equation, whereas in the principal special case with even-spin interactions (PSC) the spectrum simplifies substantially: the characteristic polynomial factorizes, and $λ_{\max}$ is given by the root of a quadratic equation. For mirror-symmetric subfamilies, we derive explicit formulas for the pair correlation functions and express the magnetization in terms of the components of the eigenvector associated with $λ_{\max}$; in the even-spin case with $h=0$, the magnetization vanishes. Important special cases include the width-three planar model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, including the entropy limit $S(T\to0^+)=(\ln 2)/3$ for $k\ge 0$ and $S(T\to0^+)=0$ for $k<0$, as well as the width-three planar triangular model with distinct nearest-neighbor couplings, three-spin interactions involving neighboring triangles, and an external field.

arXiv:2605.17600•May 17, 2026•0 citations

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

Felix A. Kassan-Ogly

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.

Phase Transitions•Oct 1, 2000•11 citations
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 obtain the exact physical characteristics of the triple-chain Ising model on a torus with all possible multispin interactions invariant with respect to rotation by the angle 2π/32π/ 32π/3. The exact value of the partition function in a finite cyclically closed strip of length LLL, as well as the free energy, internal energy, specific heat, magnetization, susceptibility, and entropy in the thermodynamic limit at L→∞L \to \inftyL→∞ are found by the transfer-matrix method for the model. The spectrum of the transfer-matrix and the structure of its eigenvectors are found. For two special cases - for the model with multispin interactions of even number of spins and for the model with some interactions of two, three, four and six spins, simplified expressions of the mentioned physical characteristics are obtained; in the thermodynamic limit they are expressed through the logarithm of the root of the quadratic equation. For the model with multispin interactions of an even number of spins, a kind of pair correlations in the thermodynamic limit is found, and it is shown that the magnetization at zero magnetic field is equal to zero; the structure of the ground states of the system is found and examples of their projections of seven-dimensional space onto three-dimensional space and examples of configurations corresponding to these ground states are given. The correlation length is shown and its graphs are given. As special cases, we consider the planar triangular model with all possible interactions, including, perhaps, different triple interactions inside neighboring triangles, and the planar model with nearest-neighbor, next nearest-neighbor, and plaquette interactions. For them the main exact physical characteristics have been found. This allowed us to obtain them for the planar gonihedric model as well.
In this paper the exact solution and correlation functions for a double-chain Ising model with multi-spin interactions and symmetric Hamiltonian density are obtained. The study employs the transfer matrix method to derive fundamental thermodynamic characteristics of the system. The main results include exact expressions for the partition function, free energy, internal energy, specific heat capacity, magnetization, susceptibility, and entropy in a strip of finite length and in the thermodynamic limit. The work provides explicit formulas for the eigenvalues and shows structure of eigenvectors of the transfer matrix. The expression for magnetization in the thermodynamic limit using components of normalized eigenvector corresponding to the maximum eigenvalue is obtained. A detailed analysis is conducted for a special case of interactions involving all kinds of two- and four-spin interactions. This gives the simplified formula for free energy, it is calculated using the root of quadratic equation. Using special relation between parameters the solution for gonihedric model on the plane is obtained. The research reveals properties of the system, including specific features of ground states and phase diagram characteristics. Particular attention is given to the behavior of physical quantities near frustration points and the investigation of spin correlation functions. Plots of physical characteristics, including inverse correlation length, illustrating the obtained results are constructed.
We obtain an exact solution for a generalized three-chain Ising tube (TCGIT) of length LLL with toroidal boundary conditions and the most general C3C_3C3​-invariant Hamiltonian on an elementary prism, containing 20 independent coupling constants, including an external magnetic field. Using an 8×88\times 88×8 transfer matrix, we derive the exact partition function of the finite system and obtain the free energy, internal energy, specific heat, magnetization, magnetic susceptibility, and entropy in the thermodynamic limit L→∞L\to\inftyL→∞. In the general case, λmax⁡λ_{\max}λmax​ is determined by a quartic equation, whereas in the principal special case with even-spin interactions (PSC) the spectrum simplifies substantially: the characteristic polynomial factorizes, and λmax⁡λ_{\max}λmax​ is given by the root of a quadratic equation. For mirror-symmetric subfamilies, we derive explicit formulas for the pair correlation functions and express the magnetization in terms of the components of the eigenvector associated with λmax⁡λ_{\max}λmax​; in the even-spin case with h=0h=0h=0, the magnetization vanishes. Important special cases include the width-three planar model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, including the entropy limit S(T→0+)=(ln⁡2)/3S(T\to0^+)=(\ln 2)/3S(T→0+)=(ln2)/3 for k≥0k\ge 0k≥0 and S(T→0+)=0S(T\to0^+)=0S(T→0+)=0 for k<0k<0k<0, as well as the width-three planar triangular model with distinct nearest-neighbor couplings, three-spin interactions involving neighboring triangles, and an external field.