Two-spin Correlation Function Research Articles

New method for the Ising model phase diagram

An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.

Theory of field dependent correlation functions for light scattering in magnetic semiconductors

Magnetic field dependent correlation functions for antiferromagnets near the spin flop-paramagnetic phase boundary have been calculated using an RPA technique. The exponents describing the field dependence of the magnetization, the staggered magnetization and the correlation length, which are obtained from these calculations, are compared to those obtained using renormalization groups results. Here, we use our recent results relating the symmetry forbidden one-phonon Raman scattering in magnetic semiconductors to the two-spin correlation function, to calculate the field dependence of the Raman integrated intensity in EuTe. Light scattering experiments are suggested where the exponents related to the spin flop-paramagnetic phase boundary, Hc(T), and the field dependence of the staggered magnetization can be measured from the one phonon–one spin Raman spectrum in EuTe.

Correlation near the free surface of an Ising ferromagnet

The two-spin and four-spin correlation functions involving one and two pairs of nearest neighbours respectively are evaluated near the free surface of an Ising ferromagnet and the departure from the bulk value is found. These correlation functions could be termed the energy density and the energy density-energy density correlation functions. Near the critical temperature Tc the energy density in the surface layer is found to behave as t2ln t with t= mod T-Tc mod . This contrasts with the bulk energy density which has a t ln t singularity. Correlations between a surface and a bulk spin are evaluated above and below Tc. Correlations between surface spins when the coupling between the surface spins is different from those in the bulk are examined.

Two-spin dynamic autocorrelations and energy diffusion in the Heisenberg paramagnet

We establish the kinetic equation for the two-spin time-dependent correlation function in the Weiss limit. Considering this equation in the limit of small wavenumbers, we obtain an explicit expression for the energy diffusion coefficient. We also discuss some difficulties connected with overcounting problems of the Weiss approximation.

Quantum-field-theory calculation of the two-dimensional Ising model correlation function

The equivalence between the two-dimensional Ising model and a free fermion field theory is used to rederive various results for the two-spin correlation function in the critical region.

Magnetostriction and the two-spin correlation function in EuO

The longitudinal and transverse magnetostrictions along the [100] direction of EuO were measured at temperatures between the Curie point ${T}_{C}=69.25 \mathrm{and} 245$ K in magnetic fields $H$ up to 145 kOe. The volume magnetostriction was obtained from these measurements. The data indicate that above ${T}_{C}$ the isotropic magnetostriction (independent of the direction of $\stackrel{\ensuremath{\rightarrow}}{H}$ is much larger than the anisotropic magnetostriction. The isotropic magnetostriction is attributed to the volume dependence of the dominant nearest-neighbor exchange constant $J$. The results are compared with the theory of Callen and Callen. Assuming only nearest-neighbor exchange interactions, this theory relates the isotropic magnetostriction to the $H$ dependence of the two-spin correlation function $〈{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}〉$ for nearest neighbors. An alternative theoretical approach, based on thermodynamic and statistical-mechanical arguments is presented and is shown to lead to equivalent results. Based on these theoretical results for the magnetostriction, numerical predictions are obtained using high-temperature series expansions. These numerical results point to the inadequacy of the molecular-field approximation, particularly near ${T}_{C}$. The overall agreement between theory (evaluated using series expansions) and experiment is very good. However, there are small discrepancies near ${T}_{C}$, consistent with other data which show that for EuO the model of a Heisenberg ferromagnet with nearest-neighbor exchange interactions is good, but not exact. Susceptibility and magnetization data (obtained for the purpose of interpreting the magnetostriction data) are also compared with results of series expansions for zero and finite fields. The zero-field susceptibility is in good agreement with theory, except near ${T}_{C}$ where small deviations appear. The field variation of the susceptibility $\ensuremath{\chi}$ in intermediate fields, for which $\frac{\ensuremath{\chi}(H)}{\ensuremath{\chi}(0)}\ensuremath{\gtrsim}0.9$, is in good agreement with predictions based on "double series" in powers of $\frac{J}{kT}$ and $\frac{{\ensuremath{\mu}}_{B}H}{kT}$.

Moments of two-time spin-pair correlation functions for a one-dimensional anisotropic Heisenberg model at infinite temperature

The first ten moments of the infinite-temperature space and frequency dependent two-spin correlation functions, ƒ x r(ω) and ƒ z r(ω) are obtained for the one-dimensional anisotropic Heisenberg model for r = 0 and r = 1. These are compared with those previously known.

High-temperature series for classical n-vector models with general anisotropy

A compact parametrization is derived for the high-temperature series expansions of the free energy and two-spin correlation function of classical n-vector models with completely general anisotropic pair interactions. The number of independent coefficients needed for general anisotropy is much less than required in conventional expansions. The series for the free energy of the Heisenberg model (n=3) on three cubic lattices is presented to tenth order as an example.

Correlation function and transition temperature bounds for bond-disordered Ising systems

For spin 1/2 Ising models with ferromagnetic pair interactions Jij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q) is less than or equal to the value ( sigma p sigma q)J for the corresponding non-random system in which each Jij is replaced by its average value Jij. The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi or=0 and delta ( sigma p sigma q)/ delta Jrs>or=0 imply that delta 2( sigma p sigma q)/ delta Jrs2<or=0.

Magnetostriction and the two-spin correlation function in MnF 2

The longitudinal magnetostriction along the [001] axis of MnF 2 was measured at temperatures 64 < T < 300 K in magnetic fields, H, up to 130 kOe. This magnetostriction is proportional to H 2 at low H, exhibits a λ anomaly near the Néel temperature T N , and shows the field-induced transition from the antiferromagnetic phase to the paramagnetic phase for T just below T N . The results are well described by a model which relates the magnetostriction to the two-spin correlation function.

Spin correlations in a Heisenberg system within the paramagnetic region

The two-spin correlation function for a Heisenberg system is investigated, in the paramagnetic region, using a new approach through which one is able to reveal the dynamics more clearly than previously. The conventional mode-mode coupling theory is obtained as a lowest-order approximation; this entails ignoring all third- and higher-order irreducible spin correlations. It is argued that the mechanism for sloppy spin waves is lost in the mode-mode theory and that this is the reason for some of the most noticable discrepancies between theory and experiments.

Two-spin correlation function for a Heisenberg magnet: Application to magnetostriction

The ratio between the magnetic-field-induced change in the two-spin correlation function and the square of the magnetization for Heisenberg ferromagnets and antiferromagnets is calculated as a function of temperature. The results explain magnetostriction data in EuO and MnF 2.

First-order approximation for the time-dependent Ising model

Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent Ising models in two and three dimensions. This constitutes the first-order approximation which improves the mean-field theory (the zeroth-order approximation) by incorporating specific short-range correlations into the distribution function. We first calculate the single-spin expectation value. Although the dynamic equation may need to be solved by using a numerical method, its approximate solution in the pretransitional region can be studied in detail. We then calculate the two-spin correlation function. This requires yet another extension of Bethe's method. Again approximate solutions can be obtained in the pretransitional region. Our results for the static pair correlation functions are equivalent to that of Elliott and Marshall, but our derivation is simpler. We also calculate the time-dependent correlation function for the magnetization, and reproduce the fluctuation-dissipation relation.

Two scale factor universality in the spherical model

A universal relation, proposed by Stauffer, Ferer and Wortis, between the scale factors in the scaling functions for the free energy and two-spin correlation function, is shown to be valid for the spherical model with general pair interactions, provided an appropriate definition of the correlation length is adopted. The hypothesis remains valid for the spherical model and numerically for the three-dimensional Ising model, when extended to include the scale factor for the magnetic field.

Upper bound on correlation functions of Ising ferromagnet

We use Thompson's argument to show that, for pair interactions, the two-spin correlation function is bounded above by a solution of the non-linear Debye-Hückel equation. For many-spin interactions, Griffith's third inequality is strengthened.

Two-spin correlation function of spin [formula omitted] Ising model on a bethe lattice

The two-spin correlation function of the Ising model on a Bethe lattice as a function of temperature and magnetic field. It is found that the coherence length is finite at finite temperatures.

Sixth frequency moment of two-spin correlation functions in a linear chain of spins at high temperature

We have computed for an arbitrary spin the sixth frequency moments of the two-spin correlation function for an anisotropic Heisenberg Hamiltonian in a linear chain of spins at high temperature, and we compare our results with those previously known.

Experimental study of high-temperature spin dynamics in one-dimensional Heisenberg paramagnets

From room-temperature measurements of proton spin-lattice relaxation rate in (${\mathrm{C}{\mathrm{H}}_{3})}_{4}$NMn${\mathrm{Cl}}_{3}$ (TMMC) and CsMn${\mathrm{Cl}}_{3}$ \ifmmode\cdot\else\textperiodcentered\fi{} 2${\mathrm{H}}_{2}$O (CMC) as a function of magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ the shape of the low-frequency spectral density of the spin fluctuations is directly obtained. For $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ parallel to the chain axis the spectral density contains a diffusive term which diverges as ${\ensuremath{\omega}}^{\ensuremath{-}\frac{1}{2}}$. The experimental results for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}\ensuremath{\parallel}\stackrel{\ensuremath{\rightarrow}}{\mathrm{c}}$ are in good agreement with a theoretical calculation in TMMC in which the two-spin correlation function is assumed to decay at long times as ${t}^{\ensuremath{-}\frac{1}{2}}$. The lack of frequency dependence of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ perpendicular to the chain axis indicates a difference in the cutoff time of the two-spin correlation function for the spin components parallel and perpendicular with respect to the external magnetic field.

Bounds for Ising systems

Ising systems are approximated by a generalized mean-field approximation in which only some of the interactions are replaced by their expectation values. If all the interactions are positive then it is shown that a self-consistent solution exists for the magnetization in this model and that it gives an upper bound for the true magnetization. Using this mean-field approximation, a generalization of the random-phase approximation is obtained. This is shown to give an upper bound for two-spin correlation functions.

Critical Behavior of Magnets with Dipolar Interactions. II. Feynman-Graph Expansion for Ferromagnets near Four Dimensions

The Feynman-graph-expansion approach of Wilson is used to study the critical behavior for $0lt=(\frac{T}{{T}_{c}})\ensuremath{-}1\ensuremath{\ll}\frac{G}{J{a}^{d}}$ and $H=0$ of an isotropic ferromagnet in $d=4\ensuremath{-}\ensuremath{\epsilon}(\ensuremath{\epsilon}g0)$ dimensions with exchange and dipolar interactions between $d$-component spins. [Here $G=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{2}$ measures the dipole-dipole interaction strength, $J$ is the short-range exchange parameter, and $a$ is the lattice spacing.] The susceptibility and the two-spin correlation functions are calculated to first order in $\ensuremath{\epsilon}$, and agree with previous work, based on the renormalization-group approach. In addition the correlation function for transverse-spin fluctuations at ${T}_{c}$ is investigated, yielding the critical exponent $\ensuremath{\eta}\ensuremath{\approx}\frac{20{\ensuremath{\epsilon}}^{2}}{867}$ [whereas for short-range exchange forces one has $\ensuremath{\eta}\ensuremath{\approx}\frac{3{\ensuremath{\epsilon}}^{2}}{{12}^{2}}$]. The limiting angular dependence of the four-spin correlation function is obtained.