Ising-Heisenberg Model Research Articles

Conditions for a ferromagnetic ground state of Ising and Heisenberg models. II. All Bravais lattices

For the isotropic Heisenberg Hamiltonian with ferromagnetic nearest- and antiferromagnetic next-nearest-neighbor interactions: $H={J}_{1}{\ensuremath{\Sigma}}_{(\mathrm{nn})}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n+{\ensuremath{\delta}}_{1}}+{J}_{2}{\ensuremath{\Sigma}}_{(\mathrm{nnn})}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n+{\ensuremath{\delta}}_{2}}$, ${J}_{1}<0$, ${J}_{2}>0$, necessary and sufficient conditions for a ferromagnetic ground state are obtained for all Bravais lattices with periodic boundaries and arbitrary spin $s$. For some lattices, e.g., the cubic lattices in two and three dimensions, these conditions are spin dependent. For the Ising and classical Heisenberg model, the threshold values for $\frac{|{J}_{1}|}{{J}_{2}}$ are calculated and compared with the quantum case. The results are also consistent with most of the experimental information.

Dual transformation in Abelian gauge theories

The dual transformation discovered in the two-dimensional Ising and planar Heisenberg models is applied to gauge theories in four dimensions. It is shown that after the dual transformation the Abelian Higgs model gives the same partition function as the relativistic hydrodynamics of Kalb and Ramond and of Nambu coupled to the Higgs scalar, and that these two theories have various dual relations. In our hydrodynamics a quantized vortex is created by the phase component of the Higgs scalar in a similar manner as the magnetic string of Nielsen and Olesen is created in the Abelian Higgs model.

Monte Carlo simulation of model spin systems

In this paper, applications of the Monte Carlo technique to estimate the static and dynamic properties of model spin systems are discussed. Finite-size effects and choice of boundary conditions in simulating different types of real systems are outlined. Various applications of the Monte Carlo simulations to one-, two- and three-dimensional Ising models and Heisenberg models are dealt with in some detail. Recent applications of the Monte Carlo method to spin glass systems and to estimate renormalisation group critical exponents are reviewed.

Correlation functions for the Heisenberg-Ising chain at T=0

The longitudinal and transverse space and time dependent spin correlation functions for the one-dimensional Heisenberg-Ising model are derived asymptotically in the weak-coupling-low-frequency-long-wavelength limit within the framework of ordinary many-body theory (field theory). By means of the Jordan-Wigner transformation and using methods developed previously for the Tomonaga model (1950), the zero-temperature correlation functions is expressed as functional integrals. The associated one-body problem is explicitly soluble in the asymptotic regions and the correlation functions assume the form of Gaussian integrals which are evaluated analytically in the weak-coupling-low-frequency-long-wavelength limit. Apart from a linear correction to the transverse exchange constant agreement with Luther and Peschel (1975) is obtained.

Critical behaviour of a compressible n-component model with cubic anisotropy

A generalisation of a model, in which an n-component order parameter Phi is coupled to an elastic continuum, to the case of cubic anisotropy is studied in a renormalisation group calculation. It is shown to all orders in epsilon =4-d that the inclusion of cubic anisotropy in the coupling terms yields a second-order transition if and only if all alpha i=2phi i-vd 0 (e.g. Ising and anisotropic XY and Heisenberg model) the absence of a stable fixed point is interpreted as a first-order transition. If the transition is close to second-order, renormalised exponents may be observed. The generalisation of the results to the case of lower symmetry and their relevance for other systems with non-analytic interaction is mentioned.

High-temperature analyticity in classical lattice systems

We prove analyticity of the correlation functions for classical lattice systems, including “continuous-spin” systems, at high temperatures and in strong external fields. For systems whose configuration spaces are homogeneous spaces for compact groups (e.g. Ising, plane rotator and classical Heisenberg models), improved estimates on the region of analyticity are obtained by generalizing an integral equation of Gruber and Merlini. Exponential cluster properties are also obtained for such systems with a finite-range interaction.

Calculation of critical exponents in two dimensions from quantum field theory in one dimension

We construct a relationship between the Baxter model in two dimensions and the Luttinger model in one, and use it to calculate critical exponents for the Baxter model from appropriate Luttinger-model correlation functions. An important part of this work involves the generalization of the Jordan-Wigner transformation to provide a representation for continuum spin operators. With this generalization, we are also able to calculate spin correlation functions for a continuum generalization of the spin-\textonehalf{} Heisenberg-Ising chain. We discuss the difference between the continuum and discrete lattice models, and with the help of a new scaling law, use previous results for the Baxter model to calculate new exponents for the Baxter and Heisenberg-Ising model on a lattice.

On the Critical Behavior of the Two-Dimensional Heisenberg-Ising Model

The singularity of the free energy (or sp~cific heat) is investigated for the twodimensional Heisenberg-Ising model by using the expansion theoryO on critical exponents and critical points. It is confirmed through the second order of the anisotropy parameter that the singularity of the specific heat remains logarithmic even when spin-flip terms exist (or even for a quantal case in the sense that our Hamiltonian corresponds to the quantal lattice gas modeF>). This conclusion will be useful in discussing the phase transition of the He monolayer on graphite through a spin model for it. 3> The Hamiltonian to discuss here is given by

One-Dimensional Anisotropic Heisenberg Model at Finite Temperatures

The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.

Magnetic Susceptibilities and Broad-Line PMR Spectra of Dichloro(pyridazine)copper(II) and Related Compounds

Abstract The magnetic susceptibilities of CuX2L (X: halide ion, L: 1,2,4-triazole, pyridazine, phthalazine, 4-amino-1,2,4-triazole) conform to the linear Ising model or Heisenberg model, indicating the presence of linear magnetic chains in the crystals. Fermi contact shift observed in the broad-line PMR spectrum indicates the diffusion of a positive hole from a copper atom to the molecule of heterocycles. The magnitude of the exchange integrals is related to the spin densities on the atoms of the heterocycles.

Thermodynamics of the Heisenberg-Ising model for | Δ | < 1 in one dimension

The thermodynamics of the one-dimensional Heisenberg-Ising model for | Δ | < 1 is reduced to a set of non-linear integral equations with finite unknowns.

Bogolyubov's statistical variational principle and the Green's function method applied to the Heisenberg-Ising model

Bogolyubov's statistical variational principle and the Green's function method applied to the Heisenberg-Ising model

Anomalous specific heat near the Curie temperature

Recent measurements of the specific heats of single crystals of magnetic salts are compared with theoretical calculations on the high temperature side of the Curie point for the Ising and classical Heisenberg models. It is found that there is an anomalous rise of specific heat when (T/Tc−1) similar, equals 10−3. This is interpreted as arising from coupling between the magnetic interaction and compressibility of the lattice.

High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.

Critical Indices for a System of Spins of Arbitrary Dimensionality Situated on a Lattice of Arbitrary Dimensionality

A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].

High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

High-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heinsenberg model are calculated for various lattices and anisotropies through order ${T}^{\ensuremath{-}8}$ (close-packed lattices) and ${T}^{\ensuremath{-}9}$ (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices $\ensuremath{\gamma}$ (susceptibility), $\ensuremath{\nu}$ (correlation range), and $\ensuremath{\alpha}$ (specific heat) as functions of anisotropy. Our results are consistent with the hypothesis that the critical indices change only when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended.

Dyson equation for magnons in the Heisenberg-Ising model

A Dyson equation of motion for magnons is derived. Contributions from kinematic effects are entirely included in the self-energy term. The existence of the solution for T ⩾ T C as well as T < T C is discussed.

Configurational Studies of the Ising and Classical Heisenberg Models

Recent developments in graph classification and enumeration on crystal lattices carried out by the Theoretical Group at King's College are described. As a result of this enumeration, several new terms have been added to the high-temperature specific-heat expansions for the Ising and classical Heisenberg models. The critical behavior is discussed. It has been possible to single out the types of graph which provide a dominant asymptotic contribution to the coefficients of these expansions. Hence the Curie temperature can be related to the statistical properties of self-avoiding walks on crystal lattices.

Critical Properties of Isotropically Interacting Classical Spins Constrained to a Plane

High-temperature expansions of the susceptibility and internal energy (specific heat) are presented for general lattice structure for a system of isotropically interacting unit vectors (or spins) which are constrained to lie in a plane. A phase transition (${T}_{c}g0$) is indicated for two-dimensional lattices; the expected result ${T}_{c}=0$ is found in one dimension, but only upon choosing a more suitable expansion parameter than $\frac{J}{\mathrm{kT}}$. Similarities with the corresponding expansions of the $S=\frac{1}{2}$ Ising and classical Heisenberg models are pointed out; in particular, it is found that certain critical properties of this planar model appear to be bounded on one side by the Ising model and on the other side by the Heisenberg model.