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Physical Review

Application of the Modified Bloch Equation to the Low-Temperature Behavior of the Heisenberg Ferromagnet

Tomoyasu Tanaka, K. Moorjani, Tohru Morita

Published Jan 7, 1966• 1 citation
Publisher

Abstract

An equation for the normalized density matrix $\ensuremath{\rho}$ is given by $\ensuremath{-}\frac{\ensuremath{\partial}\ensuremath{\rho}}{\ensuremath{\partial}\ensuremath{\beta}}={(\mathcal{H}\ensuremath{-}〈\mathcal{H}〉)}_{\ensuremath{\rho}}$. The equation can be used to obtain a hierarchy of equations for the reduced density matrices as well as that for average values of dynamical quantities. In the present paper the latter set of equations is used to study the low-temperature behavior of the Heisenberg ferromagnet.

DOI
10.1103/physrev.141.323

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An equation for the normalized density matrix \ensuremathρ\ensuremath{\rho}\ensuremathρ is given by \ensuremath−\ensuremath∂\ensuremathρ\ensuremath∂\ensuremathβ=(H\ensuremath−〈H〉)\ensuremathρ\ensuremath{-}\frac{\ensuremath{\partial}\ensuremath{\rho}}{\ensuremath{\partial}\ensuremath{\beta}}={(\mathcal{H}\ensuremath{-}〈\mathcal{H}〉)}_{\ensuremath{\rho}}\ensuremath−\ensuremath∂\ensuremathβ\ensuremath∂\ensuremathρ​=(H\ensuremath−〈H〉)\ensuremathρ​. The equation can be used to obtain a hierarchy of equations for the reduced density matrices as well as that for average values of dynamical quantities. In the present paper the latter set of equations is used to study the low-temperature behavior of the Heisenberg ferromagnet.
The Green's function formalism proposed by Kondo and Yamaji for low-dimensional spin systems with S =1/2 has been extended to the cases of S >1/2. We apply this formalism to the study of one- and two-dimensional Heisenberg ferromagnets over the whole temperature range. It is shown that our theory can reproduce the correct results obtained by the high-temperature expansion method. On the other hand, the results at low temperatures are similar to those of the modified spin-wave theory, which is considered to predict the low-temperature properties of such systems rather correctly. The gross behaviors of the calculated thermodynamic quantities of ferromagnetic Heisenberg chains agree with those obtained by the exact diagonalization method.
The dynamical correlation function Gr{\mathit{G}}_{\mathbf{r}}Gr​(t)==〈Sr{\mathbf{S}}_{\mathbf{r}}Sr​(t)\ensuremath{\cdot}S0{\mathbf{S}}_{0}S0​(0)〉 and dynamical structure factor Sq{\mathit{S}}_{\mathbf{q}}Sq​(\ensuremath{\omega}) are calculated by the modified spin-wave theory for the low-dimensional quantum Heisenberg ferromagnets at low temperature. We use the Dyson-Maleev transformation, ideal spin-wave states, and the rotational averaging. Sq{\mathit{S}}_{\mathbf{q}}Sq​(\ensuremath{\omega}) satisfies the dynamic scaling relation. The explicit form of the scaling function is obtained. The classical limit of our results is compared with a molecular-dynamics calculation. The agreement is surprisingly good.
We investigate the thermodynamic properties of Heisenberg ferrimagnetic mixed-spin chains both numerically and analytically with particular emphasis on the combination of ferromagnetic and antiferromagnetic features. Employing a new density-matrix renormalization-group technique as well as a quantum Monte Carlo method, we reveal the overall thermal behaviour: at very low temperatures, the specific heat and the magnetic susceptibility multiplied by the temperature behave like and , respectively, whereas at intermediate temperatures, they exhibit a Schottky-like peak and a minimum, respectively. Developing the modified spin-wave theory, we complement the numerical findings and give precise estimates for the low-temperature behaviour.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
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.
The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid 3He^{3}\mathrm{He}3He in two dimensions.