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Exchange Interaction Model のモンテカルロ・シミュレーション

Terumasa Koike, Shinji Kadowaki

Published Mar 31, 1995• 0 citations
OpenAlex

Abstract

A simple Monte Carlo method by Kadowaki and Ueda is applied to calculate the zero-field susceptibility of the one-dimensional spin-S exchange interaction model. Our numerical results for the one-dimensional spin-1/2 Heisenberg system are in good agreement with Takahashi's. For the spin-S exchange interaction model, we predicate that the exponent of the susceptibility as function of temperature, γ(T) is 2 as T→0. Therefore we support in the conjecture by Chen et al., γ(0)=2.

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A simple Monte Carlo method by Kadowaki and Ueda is applied to calculate the zero-field susceptibility of the one-dimensional spin-S exchange interaction model. Our numerical results for the one-dimensional spin-1/2 Heisenberg system are in good agreement with Takahashi's. For the spin-S exchange interaction model, we predicate that the exponent of the susceptibility as function of temperature, γ(T) is 2 as T→0. Therefore we support in the conjecture by Chen et al., γ(0)=2.
The phase transition for the spin systems described by the Hamiltonian H =-2 J ∑ < i , j > (ξ S i x S j x +η S i y S j y + S i z S j z )- g µ H ∑ i S i z is studied by means of the high-temperature series expansion of the zero field susceptibility. The first five terms in the susceptibility series are explicitly obtained for the two-dimensional square lattice in the cases of (i) 0≤ξ≤1, η=0 and (ii) 0≤ξ=η≤1. The critical temperature T c and the critical exponent γ (susceptibility) are obtained as functions of the spin S and the anisotropy parameter ξ. The critical temperature has a finite value in the XY model (ξ=1, η=0) as well as the Heisenberg model (ξ=η=1). The simple formulae expressing T c and γ valid for arbitrary S and ξ are proposed.
Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.
The magnetic susceptibility at absolute-zero temperature for the one-dimensional Hubbard model is studied exactly as a function of the concentration of electrons by using Lieb and Wu's theory for this system. Our analysis essentially follows Griffiths's method for the magnetic susceptibility of the one-dimensional Heisenberg antiferromagnet, and is a generalization of Takahashi's calculation to the systems with an arbitrary concentration of electrons. The ground-state energy and the magnitude of local moments at each site are also studied. Combined with the results on the susceptibility, they should suggest how the effect of the Coulomb interaction on the properties of the system at low temperatures changes with the concentration of electrons.
We use the diagram technique for spin operators to calculate Green's functions and observables of the spin-12\frac{1}{2}21​ quantum Heisenberg antiferromagnet on a square lattice. The first corrections to the self-energy and interaction are taken into account in the chain diagrams. The approximation reproduces main results of Takahashi's modified spin-wave theory [Phys. Rev. B 40, 2494 (1989)] and is applicable in a wider temperature range. The energy per spin calculated in this approximation is in good agreement with the Monte Carlo and small-cluster exact-diagonalization calculations in the range $0<~T\ensuremath{\lesssim}1.2J$ where J is the exchange constant. For the static uniform susceptibility the agreement is good for $T\ensuremath{\lesssim}0.6J$ and becomes somewhat worse for higher temperatures. Nevertheless the approximation is able to reproduce the maximum in the temperature dependence of the susceptibility near T=0.9J.T=0.9J.T=0.9J.
AbstractUsing the Monte Carlo method, critical behavior of the one-dimensional ferromagnetic Ising model has been investigated with allowance for the interaction of the second and third neighbors and four-particle interaction. The obtained results on the critical temperature were compared with the critical temperature of the quasi-one-dimensional Ising magnetic [(СН_3)_3NH] · FeCl_3 · 2H_2O and with the magnitude of the exchange interaction J/k _B = 17.4 K. Within the scope of the finite-dimensional scaling theory, the critical susceptibility exponent has been calculated. It has been shown that values of the susceptibility exponent for the one-dimensional Ising model with periodic boundary conditions are considerably less than the known values of the exponents for three-dimensional systems. The critical susceptibility exponent strongly depends on energy parameters; namely, it decreases with an increase in them.