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Interdisciplinary Information Sciences

One-Dimensional Chiral XY-Model at Finite Temperatures.

Yoshihiko Fukui, Tsuyoshi Horiguchi

Published Jan 1, 1995• 4 citations
Publisher

Abstract

We clarify the finite temperature properties of a chiral XY-model by calculating the free energy, the entropy, the specific heat, the correlation functions and so on, by using numerical solutions of an integral equation. The specific heat has a small peak in a lower-temperature side of the main peak for the system in which the period of ground-state configuration is 1 or 4. The information stored in the ground-state configuration is disturbed easily by the temperature when the configuration is not stable. The critical behavior of the correlation length is either of the XY type or of the Ising type according to whether or not the ground-state configuration has a long period.

DOI
10.4036/iis.1995.133

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We clarify the finite temperature properties of a chiral XY-model by calculating the free energy, the entropy, the specific heat, the correlation functions and so on, by using numerical solutions of an integral equation. The specific heat has a small peak in a lower-temperature side of the main peak for the system in which the period of ground-state configuration is 1 or 4. The information stored in the ground-state configuration is disturbed easily by the temperature when the configuration is not stable. The critical behavior of the correlation length is either of the XY type or of the Ising type according to whether or not the ground-state configuration has a long period.
Thermodynamic properties of the SU(nnn) Heisenberg model in one dimension is studied by means of high-temperature expansion for arbitrary nnn. The specific heat up to O[(βJ)23]O[(βJ)^{23}]O[(βJ)23] and the correlation function up to O[(βJ)18]O[(βJ)^{18}]O[(βJ)18] are derived with βJβJβJ being the antiferromagnetic exchange in units of temperature. It is found for n>2n>2n>2 that the specific heat shows a shoulder in the high-temperature side of a peak. The origin of this structure is clarified by deriving the temperature dependence of the correlation function. With decreasing temperature, the short-range correlation with two-site periodicity develops first, and then another correlation with nnn-site periodicity at lower temperature. This behavior is in contrast to that of the inverse square interaction model, where the specific heat shows a single peak according to the exact solution. Our algorithm has an advantage that neither computational time nor memory depends on the multiplicity nnn per site; the series coefficients are obtained as explicit functions of nnn.
We investigate ground state properties of a one-dimensional quantum chiral XY -model in a magnetic field. Numerical calculations are carried out by using a method of density matrix renormalization group. We also analyze behaviors of perturbation series for the ground state energy. For a special case that the natural cantedness is π⁄2, a value of the critical magnetic field for saturation of magnetization is shown to be given exactly in terms of the smallest zero of the Bessel function J0(x ). We obtain the ground state phase diagram.
The two-dimensional spin-1/2 XY model is investigated via an extensive quantum Monte Carlo simulation on square lattices as large as 128\ifmmode\times\else\texttimes\fi{}128. The transverse susceptibility and correlation length show precise thermodynamic scaling behaviors of a phase transition of the Kosterlitz-Thouless type at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.353(3). The correlation functions near the transition point exhibit a universal scaling behavior. Various critical exponents are determined and our results demonstrate that the universality class of the two-dimensional quantum system with continuous symmetry is the same as its classical analog. The specific heat exhibits a finite peak around kT/J=0.45, decreases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​, and falls off as T2{\mathit{T}}^{2}T2 near T=0, validating the spin-wave treatment. The ground-state energy is -0.5543(1)J per spin.
An antiferromagnetic classical XY model with the nearest-neighbor interaction 2 J and the next-nearest-neighbor interaction δ J on a one-dimensional lattice is investigated in the low temperature limit. We obtain the asymptotic behaviors at long distances, of the two-spin correlation function for δ>1/2 and of the chirality-chirality correlation function for δ=1. The behavior of the latter correlation function is shown to be similar to the one of the two-spin correlation function for the Ising model with the nearest-neighbor interaction J /2 on a one-dimensional lattice.
The authors present an exact solution for the one-dimensional Ising model in a random field. The distribution of field strengths is site independent, symmetric and has a three-peak structure. The free energy is obtained for all finite temperatures. The low-temperature behaviour is studied in detail. They find an expansion for the free energy in integer powers of temperature. The ground-state energy and the zero-point entropy are calculated explicitly. The specific heat is linear for low temperatures, in contradiction with mean-field theory. The origin of an additional exponential parameter is discussed.
The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.