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

Journal of Mathematical Physics

Crystal field effect in an Ising model

Ibha Chatterjee

Published Jul 1, 1984• 10 citations
Publisher

Abstract

The crystal field effect has been studied in the linear Ising model which can be solved exactly. For spins S= 1/2 , 1, (3)/(2) it is easy to diagonalize the transfer matrix analytically, but for spins S> (3)/(2) , the transfer matrices are diagonalized numerically. The numerical results are accurate to seven decimal places and can be treated as exact for all practical purposes. Crystal field has no effect on spin- 1/2 systems, its effect has been studied in systems with spins S> (1)/(2) . The ferromagnetic as well as the antiferromagnetic sysceptibilities have been computed for the systems with and without the crystal field. It has been found that, for small crystal fields, the susceptibility behavior is not much different from that in the absence of crystal field. But for large crystal fields, not only the antiferromagnetic susceptibilities but the ferromagnetic susceptibilities also start showing maxima which appear for integer spins only.

DOI
10.1063/1.526406

Related papers

Crystal-field effect on the transverse susceptibility in spin-SIsing chains

Ibha Chatterjee

The crystal-field effect in the one-dimensional Ising model for spins S=1 to (5/2) has been studied in linear-response theory by considering its effect on the susceptibility in transverse directions. For S=1 analytical expressions are obtained, and for higher spins numerical calculations are performed for various values of the crystal-field parameter (D) and the exchange parameter (J). It has been found that the susceptibility maxima which appear for D<2J, do not occur for D>2J.

Physical review. B, Condensed matter•Dec 1, 1986•4 citations

Antiferromagnetic triangular Ising model: Critical behavior of the ground state

Henk W. J. Blöte, M. P. Nightingale

The critical ground state of the antiferromagnetic triangular Ising model is studied with finite-size scaling, with the use of exact techniques and numerical transfer-matrix methods. Exact solution of the model in zero field yields a conformal anomaly equal to 1, and values of critical exponents. For nonzero fields, dominant eigenvalues of the transfer matrix are computed for systems with linear dimensions up to 27 with the use of an efficient coding of spin states as a system of nonintersecting strings. The results are in perfect agreement with the Gaussian-model--Coulomb-gas scenario proposed by Nienhuis et al. for this model. Quantitative agreement is found with the predictions for exponents of spin-wave and vortex operators. Furthermore, we locate the field-induced Kosterlitz-Thouless transition to a long-range-ordered state at a field ${\mathit{H}}_{\mathit{c}}$=0.266\ifmmode\pm\else\textpm\fi{}0.010.

Physical review. B, Condensed matter•Jun 1, 1993•64 citations

Investigation of Two-Dimensional Triangular Ising Lattice in the Absence and Presence of a Magnetic Field

S. Ranjbar

An example of a two-dimensional crystal in which each lattice point interacts with six neighbors is the so-called \triangular Ising lattice. In this work the transfer matrix method is used to nd the thermodynamic properties of the triangular Ising model with a limited number of rows, in the absence or presence of a magnetic eld. In the absence of a magnetic eld the results have been used to nd an analytical expression for the partition function of a model with any number of rows, from which the thermodynamic properties of the triangular planar lattice model are obtained as a polynomial in terms of x = e 2j tanh 2j=2 cosh2j, where J = jkT is the interaction energy of the two nearest neighbor spins. The degree of the polynomial depends on the number of rows in the model. The results are compared with the exact values. The calculated free energy is found to be very accurate, but the calculated internal energy and heat capacity show some small deviations from the exact values. In the presence of a magnetic eld the solution to the matrices is too complicated for a general analytical expression to be proposed for a model with any number of rows. However, our exact solution for the model with n = 4, 5, and 7 reveals an important point: that such results are independent of n when the eld-spin interaction energy is almost equal to or larger than that of the spin-spin interaction.

Chinese Journal of Physics•Aug 1, 2003•2 citations

Thermodynamic properties of the spin S=3/2 quantum ferromagnetic Blume-Capel model in a transverse crystal field

D. C. Carvalho, J. A. Plascak

The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.

Physical review. E•Mar 24, 2021•15 citations

On the existence of an analytic solution to the 1-D Ising model with nearest and next-nearest neighbor interactions in the presence of a magnetic field

Farid Taherkhani, Ebrahim Daryaei, Hadi Abroshan +3 more

To solve the controversy, regarding the existence of an analytic solution to the 1-D Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions in the presence of a magnetic field, we apply the transfer matrix method to solve the 1-D Ising model in the presence of a magnetic field, taking both NN and NNN interactions into account. We show that it is possible to write a transfer matrix only if the number of sites is even. Even in such a case, it is impossible to diagonalize the transfer matrix in an analytic form. Therefore, we employ a numerical method to obtain the eigenvalues of the transfer matrix. Moreover, the heat capacity, magnetization, and magnetic susceptibility versus temperature for different values of the competition factor (the ratio of NNN to NN interactions) are shown.

Phase Transitions•Dec 15, 2010•6 citations

Integrability and non-integrability in the Ising model

Michael Assis

The Ising model at magnetic field H=0 is one of the most important exactly solved models in statistical mechanics, solved on the square lattice in 1944 by Onsager, and later by others on the triangular lattice; its magnetic susceptibility at H=0 continues to be an unsolved aspect of the model, however. The susceptibility can either be viewed as a sum over all correlation functions of the integrable model at H=0, or else as a second derivative of the free energy of the non-integrable Ising model in a field. Therefore, its analytic properties, though derived from the non-integrable model, can be studied through series expansion of the integrable correlation functions. We begin this process by analyzing the first four terms in the form factor expansion of the diagonal correlation functions, and after summing over the diagonal form factor expansion, the first four terms of the diagonal susceptibility expansion. We have been able to reduce the form factor and susceptibility expansion terms, given as multi-dimensional integrals, to closed-form functions in all cases. Under the limit of H going to infinity with the interaction energy E going to minus infinity, the isotropic Ising model becomes a hard particle lattice gas model, with exclusion of nearest neighbor lattice sites. In this limit, the triangular lattice Ising model becomes the exactly solved hard hexagon model, while the square lattice becomes the non-integrable hard squares model. We study in detail these models for finite lattice sizes in order to understand the differences between integrable and non-integrable models. We consider the partition functions zeros and their density for different boundary conditions, and find notable differences in the density which is attributed to an extra factorization in the transfer matrices of hard hexagons which is absent in hard squares. We also study the special point at fugacity z=-1 of hard squares where all eigenvalues of the transfer matrix are equimodular and where the grand partition function's value depends on boundary conditions. | 300 pages

•Jan 1, 2014•0 citations
The crystal field effect has been studied in the linear Ising model which can be solved exactly. For spins S= 1/2 , 1, (3)/(2) it is easy to diagonalize the transfer matrix analytically, but for spins S> (3)/(2) , the transfer matrices are diagonalized numerically. The numerical results are accurate to seven decimal places and can be treated as exact for all practical purposes. Crystal field has no effect on spin- 1/2 systems, its effect has been studied in systems with spins S> (1)/(2) . The ferromagnetic as well as the antiferromagnetic sysceptibilities have been computed for the systems with and without the crystal field. It has been found that, for small crystal fields, the susceptibility behavior is not much different from that in the absence of crystal field. But for large crystal fields, not only the antiferromagnetic susceptibilities but the ferromagnetic susceptibilities also start showing maxima which appear for integer spins only.
The critical ground state of the antiferromagnetic triangular Ising model is studied with finite-size scaling, with the use of exact techniques and numerical transfer-matrix methods. Exact solution of the model in zero field yields a conformal anomaly equal to 1, and values of critical exponents. For nonzero fields, dominant eigenvalues of the transfer matrix are computed for systems with linear dimensions up to 27 with the use of an efficient coding of spin states as a system of nonintersecting strings. The results are in perfect agreement with the Gaussian-model--Coulomb-gas scenario proposed by Nienhuis et al. for this model. Quantitative agreement is found with the predictions for exponents of spin-wave and vortex operators. Furthermore, we locate the field-induced Kosterlitz-Thouless transition to a long-range-ordered state at a field Hc{\mathit{H}}_{\mathit{c}}Hc​=0.266\ifmmode\pm\else\textpm\fi{}0.010.
An example of a two-dimensional crystal in which each lattice point interacts with six neighbors is the so-called \triangular Ising lattice. In this work the transfer matrix method is used to nd the thermodynamic properties of the triangular Ising model with a limited number of rows, in the absence or presence of a magnetic eld. In the absence of a magnetic eld the results have been used to nd an analytical expression for the partition function of a model with any number of rows, from which the thermodynamic properties of the triangular planar lattice model are obtained as a polynomial in terms of x = e 2j tanh 2j=2 cosh2j, where J = jkT is the interaction energy of the two nearest neighbor spins. The degree of the polynomial depends on the number of rows in the model. The results are compared with the exact values. The calculated free energy is found to be very accurate, but the calculated internal energy and heat capacity show some small deviations from the exact values. In the presence of a magnetic eld the solution to the matrices is too complicated for a general analytical expression to be proposed for a model with any number of rows. However, our exact solution for the model with n = 4, 5, and 7 reveals an important point: that such results are independent of n when the eld-spin interaction energy is almost equal to or larger than that of the spin-spin interaction.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.
To solve the controversy, regarding the existence of an analytic solution to the 1-D Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions in the presence of a magnetic field, we apply the transfer matrix method to solve the 1-D Ising model in the presence of a magnetic field, taking both NN and NNN interactions into account. We show that it is possible to write a transfer matrix only if the number of sites is even. Even in such a case, it is impossible to diagonalize the transfer matrix in an analytic form. Therefore, we employ a numerical method to obtain the eigenvalues of the transfer matrix. Moreover, the heat capacity, magnetization, and magnetic susceptibility versus temperature for different values of the competition factor (the ratio of NNN to NN interactions) are shown.
The Ising model at magnetic field H=0 is one of the most important exactly solved models in statistical mechanics, solved on the square lattice in 1944 by Onsager, and later by others on the triangular lattice; its magnetic susceptibility at H=0 continues to be an unsolved aspect of the model, however. The susceptibility can either be viewed as a sum over all correlation functions of the integrable model at H=0, or else as a second derivative of the free energy of the non-integrable Ising model in a field. Therefore, its analytic properties, though derived from the non-integrable model, can be studied through series expansion of the integrable correlation functions. We begin this process by analyzing the first four terms in the form factor expansion of the diagonal correlation functions, and after summing over the diagonal form factor expansion, the first four terms of the diagonal susceptibility expansion. We have been able to reduce the form factor and susceptibility expansion terms, given as multi-dimensional integrals, to closed-form functions in all cases. Under the limit of H going to infinity with the interaction energy E going to minus infinity, the isotropic Ising model becomes a hard particle lattice gas model, with exclusion of nearest neighbor lattice sites. In this limit, the triangular lattice Ising model becomes the exactly solved hard hexagon model, while the square lattice becomes the non-integrable hard squares model. We study in detail these models for finite lattice sizes in order to understand the differences between integrable and non-integrable models. We consider the partition functions zeros and their density for different boundary conditions, and find notable differences in the density which is attributed to an extra factorization in the transfer matrices of hard hexagons which is absent in hard squares. We also study the special point at fugacity z=-1 of hard squares where all eigenvalues of the transfer matrix are equimodular and where the grand partition function's value depends on boundary conditions. | 300 pages