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Journal of Physics C Solid State Physics

One-dimensional X-Y model with random coupling constants. I. Thermodynamics

Edward R. Smith

Published Jul 1, 1970• 53 citations
Publisher

Abstract

The one-dimensional X-Y model in a z axis magnetic field with random coupling constants is studied. The free energy is calculated for the case in which the squares of the coupling constants have a generalized Poisson distribution. The effect of randomness in the coupling constants on the low-temperature thermodynamics is studied. In particular, it is found that a phase transition which occurs at zero temperature in the non-random system disappears in the random system. The thermodynamic quantities which are singular in the non- random system have derivatives of all orders in the random system.

DOI
10.1088/0022-3719/3/7/001

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The one-dimensional X-Y model in a z axis magnetic field with random coupling constants is studied. The free energy is calculated for the case in which the squares of the coupling constants have a generalized Poisson distribution. The effect of randomness in the coupling constants on the low-temperature thermodynamics is studied. In particular, it is found that a phase transition which occurs at zero temperature in the non-random system disappears in the random system. The thermodynamic quantities which are singular in the non- random system have derivatives of all orders in the random system.
We analyse the one dimensional X-Y model with random coupling constants in a z-axis magnetic field. For the non-random system at zero temperature, there is a thermodynamic singularity at finite field, corresponding to a change in the type of spin ordering. Some thermodynamic quantities diverge and the correlation length becomes infinite. For the random system however, we can show that all the thermodynamic quantities are smoothly varying and the correlation length is bounded above at the critical point.
We investigate the temperature dependence of the finite-field magnetization of the S = 1/2 bondalternating XY model in the Random magnetic field along the z-direction having the Lorentzian distribution.The random-averaged free energy can be exactly calculated, which enables us to obtain the thermodynamical quantities.The temperature dependence of the finite-field magnetization shows various behaviors depending on the parameters δ and Γ, where δ and Γ are the bond-alternation parameter and the Lorentzian distribution parameter, respectively.
The one-dimensional Ising model with random exchange energy is formulated in terms of a homogeneous integral equation. Assuming the input distribution has a narrow width proportional to ${N}^{\ensuremath{-}1}$, the integral equation is solved by perturbation in ${N}^{\ensuremath{-}1}$. The shift in the free energy of the system up to the order of ${N}^{\ensuremath{-}2}$ is given. It is found that for a symmetrical distribution, the shift due to the randomness is second order in ${N}^{\ensuremath{-}1}$ and negative, depending only upon the variance of the input distribution. The first-order shift for the asymmetric distribution comes entirely from the asymmetry. After a shift in energy is made to account for this asymmetry, the effective shift is identical to that for the symmetrical case. The shifts of all the thermodynamic properties of the system are also given. The randomness is found to decrease the magnetization for all temperature and applied field. However, shifts in magnetic susceptibility and specific heat are oscillatory in sign.
Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures (TTT) greater than zero; the Lorentzian-squared terms vanish at T=0T=0T=0. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.
Thermodynamic properties of one-dimensional Ising chains with random nearest neighbor exchange interaction J are studied exactly in the case of distributions of the type P(J) = A|J|ν for Jmin ≤J ≤Jmax and P(J) = 0, otherwise. The results are compared with those for a periodic chain. We find that the specific heat C ∼Tν+1 for Jmin = 0 and kT ≪Jmax. The dependence of spin-spin correlation function on distance and temperature is discussed. The absence of the peak in the zero field χ⊥ as a function of temperature when ν = 0 is noted in contrast to the case of the periodic chain. The effect of randomness on the low temperature behaviour of these thermodynamic quantities is different. For instance whereas C becomes analytic at T = 0, χ|| either remains singular (ferromagnetic) or becomes analytic (antiferromagnetic). The effect of random xy interaction (∑iJi⊥(SixSi + 1x + SiySi + 1y)) on the high temperature behaviour of χ|| and χ⊥ is discussed.
A Hamiltonian dynamics is defined for the XY\mathrm{XY}XY model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte Carlo results in the explored temperature region. The behavior of the magnetization and energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature ${T}_{\mathrm{KT}}\ensuremath{\approx}0.855.$