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Le Journal de Physique Colloques

ZERO TEMPERATURE CRITICAL BEHAVIOUR OF THE ONE DIMENSIONAL X-Y MODEL WITH RANDOM COUPLING CONSTANTS

E.R. Smith

Published Feb 1, 1971• 1 citation
Publisher

Abstract

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.

DOI
10.1051/jphyscol:19711360

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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.
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 study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.
We have investigated the dynamics of the isotropic N-vector model with long-range exchange couplings in random magnetic fields using a 1/N expansion. The leading order is exactly solved, showing the existence of a ferromagnetic phase separated from the disordered paramagnetic phase by a line. The critical behaviour of the system has been examined in the next-to-leading order of the 1/N expansion, showing that the critical exponents are by no means related to the ones of the pure system in d-2 dimensions. The ordered phase has been also investigated in the next-to-leading order, revealing a typical Goldstone behaviour of the non time-persistent part of the transverse fluctuations. For the longitudinal fluctuations, two different types of coexistence singularities emerge, one from the non time-persistent (as in the pure systems), vanishing with the temperature, and a more divergent one from the time-persistent part of the correlation.
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.
The spin-spin correlation functions of the random one-dimensional Ising model are calculated exactly and found to have a probability distribution. It is then possible to define two characteristic lengths, one to describe the behaviour of the mean and the other of the standard deviation of the probability distribution. The existence of these two characteristic lengths then suggests that the thermodynamic behaviour of the system is considerably more complex than for the corresponding nonrandom system.