New perspectives on the Ising model (original) (raw)
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The European Physical Journal B, 2006
In this paper, we study the Ising model with general spin S in presence of an external magnetic field by means of the equations of motion method and of the Green's function formalism. First, the model is shown to be isomorphic to a fermionic one constituted of 2S species of localized particles interacting via an intersite Coulomb interaction. Then, an exact solution is found, for any dimension, in terms of a finite, complete set of eigenoperators of the latter Hamiltonian and of the corresponding eigenenergies. This explicit knowledge makes possible writing exact expressions for the corresponding Green's function and correlation functions, which turn out to depend on a finite set of parameters to be self-consistently determined. Finally, we present an original procedure, based on algebraic constraints, to exactly fix these latter parameters in the case of dimension 1 and spin 3 2. For this latter case and, just for comparison, for the cases of dimension 1 and spin 1 2 [F. Mancini, Eur.
We find an exact general solution to the three-dimensional (3D) Ising model via an exact self-consistency equation for nearest-neighbors' correlations. It is derived by means of an exact solution to the recurrence equations for partial contractions of creation and annihilation operators for constrained spin bosons in a Holstein-Primakoff representation. In particular, we calculate analytically the total irreducible self-energy, the order parameter, the correlation functions, and the joined occupation probabilities of spin bosons. The developed regular microscopic quantum-field-theory method has a potential for a full solution of a long-standing and still open problem of 3D critical phenomena.
A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results
arXiv (Cornell University), 2015
The present article provides a tutorial review on how to treat the Ising and Ising-like models within the mean-field, effective-field and exact methods. The mean-field approach is illustrated on four particular examples of the lattice-statistical models: the spin-1/2 Ising model in a longitudinal field, the spin-1 Blume-Capel model in a longitudinal field, the mixed-spin Ising model in a longitudinal field and the spinS Ising model in a transverse field. The meanfield solutions of the spin-1 Blume-Capel model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A continuous quantum phase transition of the spinS Ising model driven by a transverse magnetic field is also explored within the mean-field method. The effective-field theory is elaborated within a single-and two-spin cluster approach in order to demonstrate an efficiency of this approximate method, which affords superior approximate results with respect to the mean-field results. The long-standing problem of this method concerned with a self-consistent determination of the free energy is also addressed in detail. More specifically, the effective-field theory is adapted for the spin-1/2 Ising model in a longitudinal field, the spinS Blume-Capel model in a longitudinal field and the spin-1/2 Ising model in a transverse field. The particular attention is paid to a comprehensive analysis of tricritical point, continuous and discontinuous phase transitions of the spinS Blume-Capel model. Exact results for the spin-1/2 Ising chain, spin-1 Blume-Capel chain and mixed-spin Ising chain in a longitudinal field are obtained using the transfer-matrix method, the crucial steps of which are also reviewed when deriving the exact solution of the spin-1/2 Ising model on a square lattice. The critical points of the spin-1/2 Ising model on several planar (square, honeycomb, triangular, kagomé, decorated honeycomb, etc.) lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. The mapping transformation technique
Equations-of-motion approach to the spin-1/2 Ising model on the Bethe lattice
Physical Review E, 2006
We exactly solve the ferromagnetic spin-1/2 Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green's function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green's functions and the correlation functions are written in terms of a finite set of parameters to be self-consistently determined. A procedure is developed, that allows us to exactly fix the unknown parameters in the case of a Bethe lattice with any coordination number z. Non-local correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities.
The Importance of the Ising Model
Progress of Theoretical Physics, 2012
Understanding the relationship which integrable (solvable) models, all of which possess very special symmetry properties, have with the generic nonintegrable models that are used to describe real experiments, which do not have the symmetry properties, is one of the most fundamental open questions in both statistical mechanics and quantum field theory. The importance of the two-dimensional Ising model in a magnetic field is that it is the simplest system where this relationship may be concretely studied. We here review the advances made in this study, and concentrate on the magnetic susceptibility which has revealed an unexpected natural boundary phenomenon. When this is combined with the Fermionic representations of conformal characters, it is suggested that the scaling theory, which smoothly connects the lattice with the correlation length scale, may be incomplete for H = 0.
Quantum correlations in the 1D spin-1/2 Ising model with added Dzyaloshinskii–Moriya interaction
Physica A: Statistical Mechanics and its Applications, 2014
We have considered the 1D spin-1/2 Ising model with added Dzyaloshinskii-Moriya (DM) interaction and presence of a uniform magnetic field. Using the mean-field fermionization approach the energy spectrum in an infinite chain is obtained. The quantum discord (QD) and concurrence between nearest neighbor (NN) spins at finite temperature are specified as a function of mean-field order parameters. A comparison between concurrence and QD is done and differences are obtained. The macroscopic thermodynamical witness is also used to detect quantum entanglement region in solids within our model. We believe our results are useful in the field of the quantum information processing.
Critical phenomena in 1D Ising model with arbitrary spin
EPJ Web of Conferences
The aim of this work was to study critical phenomena taking place in 1D Ising model with different exchange interactions signs and arbitrary spin values in a magnetic field. Exact analytical formulas for frustration fields, zero temperature magnetization and entropy at these fields are obtained. The general behavior of pair spin correlation function with the accounting of only interactions between nearest neighbors is examined.
Critical behavior of the classical spin-1 Ising model for magnetic systems
AIP Advances, 2022
In this work, the critical properties of the classical spin-1 Ising Hamiltonian applied to magnetic systems characterized by the first-neighbors biquadratic exchange, the anisotropy and the external magnetic field contributions are theoretically investigated. The first-neighbors bilinear exchange interaction is set equal to zero. For magnetic systems the bicubic exchange interaction must be set equal to zero as it would break the time-reversal invariance of the exchange Hamiltonian. To determine the critical behavior, the spin-1 Ising Hamiltonian is mapped onto the spin-1/2 Ising Hamiltonian by using the Griffith’s variable transformation. The critical surface of a 2D square magnetic lattice is determined in the parameter space as a function of the magnetic parameters and the phase transition occurring across it is quantitatively discussed by calculating, for each spin, the free energy and the magnetization. The free energy of the 2D square magnetic lattice, described via the three-...
A New Exact Method for Solving the Two-Dimensional Ising Model
The Journal of Physical Chemistry B, 1999
We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n. We have solved the matrix for n e 7 and n e 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n g 8 and is exact for n f ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows.
Ising Model with a Magnetic Field
Journal of Statistical Physics
The paper presents the low temperature expansion of the 2D Ising model in the presence of the magnetic field in powers of x=\exp (-J/(kT))x=exp(−J/(kT))andx = exp ( - J / ( k T ) ) andx=exp(−J/(kT))andz=\exp (B/(kT))z=exp(B/(kT))withfullpolynomialsinzuptoz = exp ( B / ( k T ) ) with full polynomials in z up toz=exp(B/(kT))withfullpolynomialsinzuptox^{88}x88andfullpolynomialsinx 88 and full polynomials inx88andfullpolynomialsinx^4x4uptox 4 up tox4uptoz^{-60}z−60,inthelattercasethepolynomialsareexplicitlygiven.Thenewresultpresentedinthepaperisanexpansionnotininversepowersofzbutinz - 60 , in the latter case the polynomials are explicitly given. The new result presented in the paper is an expansion not in inverse powers of z but inz−60,inthelattercasethepolynomialsareexplicitlygiven.Thenewresultpresentedinthepaperisanexpansionnotininversepowersofzbutin(z^2+x^8)^{-k}(z2+x8)−kwherethesubsequentcoefficients(polynomialsin( z 2 + x 8 ) - k where the subsequent coefficients (polynomials in(z2+x8)−kwherethesubsequentcoefficients(polynomialsinx^4x4)turnouttobedivisiblebyincreasingpowersofx 4 ) turn out to be divisible by increasing powers ofx4)turnouttobedivisiblebyincreasingpowersof(1-x^4)(1−x4).Thisresultgivesahintabouttheintriguing‘off−diagonal’correlationsintheIsingmodelmixingthe( 1 - x 4 ) . This result gives a hint about the intriguing ‘off-diagonal’ correlations in the Ising model mixing the(1−x4).Thisresultgivesahintabouttheintriguing‘off−diagonal’correlationsintheIsingmodelmixingtheB\ne 0B≠0contributionswiththeusuallowtemperatureB ≠ 0 contributions with the usual low temperatureB=0contributionswiththeusuallowtemperatureB=0$$ B = 0 expansion what may be useful on the road to find the full analytic expression for the partition function of the Ising model with non-vanishing magnetic field. The pape...