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Progress of Theoretical Physics

Critical Behavior of Classical n-Vector Model near Zero Temperature

S. Hikami

Published Apr 1, 1976• 3 citations
Publisher

Abstract

n-vector model for classical spin system is investigated for 0<d≤2 near zero transition temperature. The critical exponent for the susceptibility is obtained and the critical amplitude is studied in 1/n expansion for d<dc. One- and two-dimensional systems are considered concerning the critical values of dc and nc, where critical temperature may become finite.

DOI
10.1143/ptp.55.1108

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Europhysics Letters (EPL)•Nov 20, 1995•60 citations
n-vector model for classical spin system is investigated for 0<d≤2 near zero transition temperature. The critical exponent for the susceptibility is obtained and the critical amplitude is studied in 1/n expansion for d<dc. One- and two-dimensional systems are considered concerning the critical values of dc and nc, where critical temperature may become finite.
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.
A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in T/∣J∣\sqrt{T/|J|}T/∣J∣​ about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.
Critical behavior of Ising spin-glass with $\ifmmode\pm\else\textpm\fi{}J$ distribution is discussed. The results are based on the high-temperature series expansion of Edwards-Anderson susceptibility in powers of w(=tanh2JkT)w (=\frac{{tanh}^{2}J}{\mathrm{kT}})w(=kTtanh2J​) for 2D, 3D, and 4D "cubic" lattices to orders 19, 17, and 15, respectively. The lengths of the series are comparable to the best available for the pure Ising model. While 3D and 4D systems exhibit transitions at finite temperatures, in 2D a zero-temperature transition with a power-law divergence (in inverse temperature) for the susceptibility is found. The estimates of critical temperatures and exponents are presented.
Critical phenomena for the for the annealed n-vector model are studied in 1/n expansion by assuming that the exchange interactions between spins are fluctuating variables obeying the Gaussian distribution. It is shown that for an overall fluctuation of the interaction between spins the susceptibility never diverges at any finite temperature except for n→∞ (spherical limit) and n=1 (Ising) cases. On the other hand, if the fluctuation is restricted to only nearest-neighbor pairs, the susceptibility is proved to diverge at an appropriate transition temperature in the limit n→∞. The critical exponent γ in this case is shown to be the same as the spherical model value. Also the dependence of transition temperature on the strength of fluctuation is discussed.