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On the analytical evaluation of the magnetization of ferromagnetic lattices

Yamen Hamdouni

Published Feb 11, 2019• 0 citations
OpenAlex

Abstract

We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-$\frac{1}{2}$ Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field $h$ exceeds its critical value $h_c=J$, where $J$ is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value, $\frac{1}{2}$, following a power series with terms involving the power laws $T^{1/2}$, $T^{3/2}$, $T^{5/2}$ ..etc. When $h<h_c$, the zero temperature magnetization per spin turns out to be equal to $\frac{1}{\pi}\arcsin(\frac{h}{h_c})$. In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form $T$, $T^3$, $T^{5}$,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.

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We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-12\frac{1}{2}21​ Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field hhh exceeds its critical value hc=Jh_c=Jhc​=J, where JJJ is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value, 12\frac{1}{2}21​, following a power series with terms involving the power laws T1/2T^{1/2}T1/2, T3/2T^{3/2}T3/2, T5/2T^{5/2}T5/2 ..etc. When h<hch<h_ch<hc​, the zero temperature magnetization per spin turns out to be equal to 1πarcsin⁡(hhc)\frac{1}{\pi}\arcsin(\frac{h}{h_c})π1​arcsin(hc​h​). In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form TTT, T3T^3T3, T5T^{5}T5,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.
Abstract We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where h c is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter the expansion is numerically observed to be convergent in the full range . To that end we express the free energy at mean magnetization per site as a series in whose coefficients can be similarly recursively computed in closed form. This series converges for all . The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in . It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange (0<Js<J0<{J}_{s}<J0<Js​<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below Tc{T}_{c}Tc​, thus providing a possible explanation for previous experiments. For sufficiently strong negative Js{J}_{s}Js​, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.
The renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=${C}_{\ensuremath{\chi}}^{(0)}$(JS2{\mathrm{JS}}^{2}JS2/T) [1+${C}_{\ensuremath{\chi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$(T/JS3{\mathrm{JS}}^{3}JS3)1/2{)}^{1/2})1/2. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=${C}_{\ensuremath{\xi}}^{(0)}$(JS2{\mathrm{JS}}^{2}JS2/T) [1+${C}_{\ensuremath{\xi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where a is the lattice spacing. The values of the coefficients ${C}_{\ensuremath{\chi}}^{(j)}$ and ${C}_{\ensuremath{\xi}}^{(j)}$ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.
For the ferromagnetic Ising model the low-temperature series expansion with temperature grouping polynomials is studied. We show that certain roots of these polynomials converge to the critical field Hc{H}_{c}Hc​, and in favorable cases we can determine the critical field quite accurately. Knowing the critical field Hc{H}_{c}Hc​, one can determine the asymptotic behavior of the temperature grouping polynomials numerically. The essential feature is a power-law behavior. Hence, the low-temperature critical indices ${\ensuremath{\alpha}}^{\ensuremath{'}}$, $\ensuremath{\beta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ can be determined. The values are in general agreement with those found by Pad\'e analysis. A critique of the accuracy of the method and its possibilities is given.