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

International Journal of Modern Physics B

CRITICAL TEMPERATURES OF THE ISING MODEL ON THE BETHE LATTICE FOR ARBITRARY VALUES OF SPIN

E. Jurčišinová, Marian Jurčisin

Published Aug 18, 2011• 5 citations
Publisher

Abstract

Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.

DOI
10.1142/s0217979211102137

Related papers

General solution of spin-1 Ising model in the effective field theory approximation: critical temperatures and spontaneous magnetization

E. Jurčišinová, Marian Jurčisin

The system of two polynomial equations is derived for direct simultaneous determination of the critical temperatures of the spin-1 Ising model on lattices with arbitrary values of the coordination number in the single-site cluster effective field theory approximation. Analysis of the critical temperatures allows to determine simple linear function 2z/3−1/2 that approximates the critical temperatures of the model with very high precision for all values of the coordination number z. It is demonstrated that the system of two equations for the critical temperatures of the model can be reduced to the corresponding single equation in the cases with z=3,4, and 5. In addition, the system of entirely algebraic polynomial equations for the determination of the spontaneous magnetization of the model simultaneously for all values of the coordination number is also derived and discussed.

Phase Transitions•Nov 7, 2021•3 citations

Approximate Accounting of Spin Correlations in the Ising Model

S. V. Semkin, Виктор Павлович Смагин

Abstract The expressions for average products of three neighboring spins in the Ising model have been found on lattices with coordination numbers 3 and 4 as functions of temperature and spontaneous magnetization. These expressions are used to compare the exact solution for the Ising model on a square lattice and the solutions found by approximate methods. A method of improving approximate methods is proposed for applying, in particular, to the Bethe approximation and to a change in the critical index of the temperature dependence of the spontaneous magnetization.

Physics of the Solid State•Sep 1, 2021•2 citations

Spontaneous Magnetization and Phase Diagrams of the Mixed Spin-1/2 and Spin-S Ising Model on the Bethe Lattice

Jozef Strečka, Cesur Ekiz

The effect of uniaxial single-ion anisotropy on magnetic and critical properties of the mixed spin-1/2 and spin-S (S > 1/2) Ising model on a three-coordinated Bethe lattice is rigorously examined with the help of star-triangle transformation and exact recursion relations.In particular, our attention is focused on the ferrimagnetic version of the model, which exhibits diverse temperature dependences of the total and both sublattice magnetizations.It is shown that the critical behavior of the mixed-spin Ising model on the Bethe lattice basically depends on whether the quantum spin number S is integer or half-odd-integer.

Acta Physica Polonica A•Apr 1, 2017•6 citations

MIXED SPIN-1 AND SPIN-$\frac{3}{2}$ BLUME–CAPEL ISING FERRIMAGNETIC SYSTEM ON THE BETHE LATTICE

Erhan Albayrak

The mixed spin-1 and spin-[Formula: see text] Blume–Capel Ising ferrimagnetic system for the central spin with spin-1 is studied on the Bethe lattice using the exact recursion equations. The exact expressions for the magnetization, the quadrupolar moment, the Curie temperature and the free energy are found and the phase diagrams are constructed on the Bethe lattice with the coordination numbers q = 3, 4 and 6 for the various values of the single-ion anisotropy constants dA = DA/J for spin-1 and dB = DB/J for spin-[Formula: see text]. The existence of a tricritical point is investigated for different values of q and the single-ion anisotropy constants. The phase diagrams in the (kTc/J, dA) plane for the central spin are obtained for two different cases; (1) dA = dB and (2) dA is varied for selected values of dB. The results are compared with those of other approximate methods.

International Journal of Modern Physics B•Mar 20, 2003•34 citations

Phase diagrams and the thermal variations of the order‐parameters in the mixed spin‐1 and spin‐$3 \over 2$ ising model on the Bethe lattice

Erhan Albayrak

Abstract The mixed spin‐1 and spin‐$3 \over 2$ Blume‐Capel Ising ferromagnetic system (J > 0) with two crystal‐field interactions, DA for spin‐1 and DB for spin‐$3 \over 2$, is studied on the Bethe lattice using the exact recursion equations for the central‐spin with spin‐$3 \over 2$. The exact expressions for the order‐parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. The phase diagrams on the $\bigg ({{kT_c}\over{J}}, \,{{D_A}\over{J}}\bigg )$ and $\bigg ({{kT_c}\over{J}}, \,{{D_B}\over{J}}\bigg )$ planes are studied for selected values of DB/J and DA/J, respectively, for the coordination numbers q = 3, 4, and 6. Besides the second‐order phase transition lines, the lines of the first‐order phase transitions terminating either at a tricritical point or at an isolated critical point are found. The thermal variations of the order‐parameters are studied with two formulations, both obtained on the Bethe lattice for the mixed spin‐1 and spin‐$3 \over 2$ Ising system using the exact recursion relations, but one with the central‐spin with spin‐1 and the other is for the central‐spin with spin‐$3 \over 2$ for q = 3, thus, it is shown that the choice of the central‐spin on the Bethe lattice has no effect on the phase diagrams.

physica status solidi (b)•Aug 22, 2003•15 citations

Ising Model with General Spin and Planar Rotator Model on the Bethe Lattice

Yoshiaki Tanaka, Norikiyo Uryû

In the Bethe lattice in which the Bethe approximation is exact, the Curie temperatures of the Ising model with spin \(S=\frac{1}{2}\), 1,…, \(\frac{5}{2}\) are obtained exactly by generalizing the method of Katsura and Takizawa for the system with \(S=\frac{1}{2}\). This method is applied to the planar rotator model on the infinite Cayley tree, and the critical temperature and the susceptibility are rigorously calculated. It is pointed out that the susceptibility diverges below the critical temperature, on the other hand the spontaneous magnetization vanishes as the ground state is degenerated infinitely.

Journal of the Physical Society of Japan•Apr 1, 1981•45 citations
Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.
The system of two polynomial equations is derived for direct simultaneous determination of the critical temperatures of the spin-1 Ising model on lattices with arbitrary values of the coordination number in the single-site cluster effective field theory approximation. Analysis of the critical temperatures allows to determine simple linear function 2z/3−1/2 that approximates the critical temperatures of the model with very high precision for all values of the coordination number z. It is demonstrated that the system of two equations for the critical temperatures of the model can be reduced to the corresponding single equation in the cases with z=3,4, and 5. In addition, the system of entirely algebraic polynomial equations for the determination of the spontaneous magnetization of the model simultaneously for all values of the coordination number is also derived and discussed.
Abstract The expressions for average products of three neighboring spins in the Ising model have been found on lattices with coordination numbers 3 and 4 as functions of temperature and spontaneous magnetization. These expressions are used to compare the exact solution for the Ising model on a square lattice and the solutions found by approximate methods. A method of improving approximate methods is proposed for applying, in particular, to the Bethe approximation and to a change in the critical index of the temperature dependence of the spontaneous magnetization.
The effect of uniaxial single-ion anisotropy on magnetic and critical properties of the mixed spin-1/2 and spin-S (S > 1/2) Ising model on a three-coordinated Bethe lattice is rigorously examined with the help of star-triangle transformation and exact recursion relations.In particular, our attention is focused on the ferrimagnetic version of the model, which exhibits diverse temperature dependences of the total and both sublattice magnetizations.It is shown that the critical behavior of the mixed-spin Ising model on the Bethe lattice basically depends on whether the quantum spin number S is integer or half-odd-integer.
MIXED SPIN-1 AND SPIN-32\frac{3}{2}23​ BLUME–CAPEL ISING FERRIMAGNETIC SYSTEM ON THE BETHE LATTICE
The mixed spin-1 and spin-[Formula: see text] Blume–Capel Ising ferrimagnetic system for the central spin with spin-1 is studied on the Bethe lattice using the exact recursion equations. The exact expressions for the magnetization, the quadrupolar moment, the Curie temperature and the free energy are found and the phase diagrams are constructed on the Bethe lattice with the coordination numbers q = 3, 4 and 6 for the various values of the single-ion anisotropy constants dA = DA/J for spin-1 and dB = DB/J for spin-[Formula: see text]. The existence of a tricritical point is investigated for different values of q and the single-ion anisotropy constants. The phase diagrams in the (kTc/J, dA) plane for the central spin are obtained for two different cases; (1) dA = dB and (2) dA is varied for selected values of dB. The results are compared with those of other approximate methods.
Phase diagrams and the thermal variations of the order‐parameters in the mixed spin‐1 and spin‐323 \over 223​ ising model on the Bethe lattice
Abstract The mixed spin‐1 and spin‐323 \over 223​ Blume‐Capel Ising ferromagnetic system (J > 0) with two crystal‐field interactions, DA for spin‐1 and DB for spin‐323 \over 223​, is studied on the Bethe lattice using the exact recursion equations for the central‐spin with spin‐323 \over 223​. The exact expressions for the order‐parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. The phase diagrams on the (kTcJ, DAJ)\bigg ({{kT_c}\over{J}}, \,{{D_A}\over{J}}\bigg )(JkTc​​,JDA​​) and (kTcJ, DBJ)\bigg ({{kT_c}\over{J}}, \,{{D_B}\over{J}}\bigg )(JkTc​​,JDB​​) planes are studied for selected values of DB/J and DA/J, respectively, for the coordination numbers q = 3, 4, and 6. Besides the second‐order phase transition lines, the lines of the first‐order phase transitions terminating either at a tricritical point or at an isolated critical point are found. The thermal variations of the order‐parameters are studied with two formulations, both obtained on the Bethe lattice for the mixed spin‐1 and spin‐323 \over 223​ Ising system using the exact recursion relations, but one with the central‐spin with spin‐1 and the other is for the central‐spin with spin‐323 \over 223​ for q = 3, thus, it is shown that the choice of the central‐spin on the Bethe lattice has no effect on the phase diagrams.
In the Bethe lattice in which the Bethe approximation is exact, the Curie temperatures of the Ising model with spin S=12S=\frac{1}{2}S=21​, 1,…, 52\frac{5}{2}25​ are obtained exactly by generalizing the method of Katsura and Takizawa for the system with S=12S=\frac{1}{2}S=21​. This method is applied to the planar rotator model on the infinite Cayley tree, and the critical temperature and the susceptibility are rigorously calculated. It is pointed out that the susceptibility diverges below the critical temperature, on the other hand the spontaneous magnetization vanishes as the ground state is degenerated infinitely.