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Physical review. B, Condensed matter

Fluctuation and susceptibility for the spin van der Waals model and the bounds of Falk and Bruch

I. M. Kim, M. Howard Lee

Published Oct 1, 1981• 22 citations
Publisher

Abstract

The transverse components of the susceptibility and fluctuation for the spin van der Waals model are obtained for high- and low-temperature $\mathrm{XY}$- and Ising-like regimes. The difference between the susceptibility and fluctuation, which arises because of noncommutativity, is related to certain correlation functions of ${S}_{z}$. In the $\mathrm{XY}$-like regime, the difference vanishes since the principal long-range order is $〈{S}_{x}〉$. But in the Ising-like regime, the difference persists since the principal long-range order now is $〈{S}_{z}〉$. Our results are compared with the bounds of Falk and Bruch on the ratio of the susceptibility to the fluctuation. In the critical region as well as in the high-temperature region the upper and lower bounds merge. In these regions, the two correlation functions become identical in the manner indicated by Falk and Bruch. In the low-temperature region, the bounds do not merge. Here we find that in the Ising-like regime of the model the susceptibility and fluctuation are properly different, but in the $\mathrm{XY}$-like regime the two functions nevertheless become identical. Thus, the susceptibility and fluctuation can evidently be the same without the bounds necessarily merging.

DOI
10.1103/physrevb.24.3961

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The transverse components of the susceptibility and fluctuation for the spin van der Waals model are obtained for high- and low-temperature XY\mathrm{XY}XY- and Ising-like regimes. The difference between the susceptibility and fluctuation, which arises because of noncommutativity, is related to certain correlation functions of Sz{S}_{z}Sz​. In the XY\mathrm{XY}XY-like regime, the difference vanishes since the principal long-range order is 〈Sx〉〈{S}_{x}〉〈Sx​〉. But in the Ising-like regime, the difference persists since the principal long-range order now is 〈Sz〉〈{S}_{z}〉〈Sz​〉. Our results are compared with the bounds of Falk and Bruch on the ratio of the susceptibility to the fluctuation. In the critical region as well as in the high-temperature region the upper and lower bounds merge. In these regions, the two correlation functions become identical in the manner indicated by Falk and Bruch. In the low-temperature region, the bounds do not merge. Here we find that in the Ising-like regime of the model the susceptibility and fluctuation are properly different, but in the XY\mathrm{XY}XY-like regime the two functions nevertheless become identical. Thus, the susceptibility and fluctuation can evidently be the same without the bounds necessarily merging.
For the S=12S=\frac{1}{2}S=21​, XY\mathrm{XY}XY model of a quantum lattice fluid or a ferromagnet the conventional order parameter does not commute with the Hamiltonian. As a result, the mean-square fluctuation of the order parameter and the isothermal susceptibility are not related in the usual way the general fluctuation theorem. For the above model, arguments are here presented to support the idea that as $T\ensuremath{\rightarrow}{T}_{c}$ the quantum effect due to the noncommutation becomes masked and the two quantities have the same critical behavior. This work is consistent with the exact results of Falk and Bruch who defined a certain moment of the spectral density and used inequalities to establish that if the moment \ensuremath{\rightarrow} 0 as $T\ensuremath{\rightarrow}{T}_{c}$, then the susceptibility-fluctuation ratio becomes unity thus ensuring coinciding critical behavior. The latter result applies to a large class of models including the one considered here.
The time-dependent autocorrelation function has been derived for the xxx component of the total spin for the S=12S=\frac{1}{2}S=21​ constant-coupling anisotropic Heisenberg model, i.e., the Van der Waals system. For T>TcT>{T}_{c}T>Tc​, the time correlation is shown to be Gaussian for both XY\mathrm{XY}XY-like and Ising-like regimes of the model. For T<TcT<{T}_{c}T<Tc​, the correlation function is still Gaussian if the model is XY\mathrm{XY}XY-like; but it is oscillatory if the model is Ising-like. Critical slow down appears only with the Ising-like system for this time-correlation function.
Bounds are presented relating zero-field static isothermal magnetic susceptibilities to the mean-square fluctuations of corresponding magnetization variables. The lower bounds contain the first frequency moment of a spectral density. When this moment $\overline{\ensuremath{\omega}}$ approaches zero, the upper and lower bounds merge, and the susceptibility is determined by the mean-square fluctuation. In particular, if the susceptibility diverges at a temperature Tc{T}_{c}Tc​, and if the expectation of the double commutator appearing in $\overline{\ensuremath{\omega}}$ is finite at and near Tc{T}_{c}Tc​, then the fluctuation and the susceptibility diverge in the same manner, and their critical exponents will be identical.
The application of the method of collective variables to study the behavior of non-universal characteristics of a three-dimensional Ising-like system in the critical region has been illustrated by an example of the correlation function and the susceptibility. An analytic procedure for the calculation of those characteristics has been developed in the quartic-distribution approximation for order parameter fluctuations. The asymptotics of the correlation function at large distances obtained for the critical temperature (T = Tc) is shown to differ qualitatively from that in the T =/ Tc case because of the presence of the critical regime region for all fluctuation modes.
We calculate the two-dimensional probability for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase is a well-defined ring function and we calculate an upper bound for . Even so, this is consistent with the susceptibility per spin, , being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable , where L is the system size and J is the coupling constant. We show that has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in at the Kosterlitz - Thouless - Berezinskii transition.
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.