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Journal of Applied Physics

Phase transitions in the classical X-Y and some related models (invited)

David Paul Landau

Published May 15, 1993• 9 citations
Publisher

Abstract

Although classical spin systems in two dimensions cannot show usual long range order, they may exhibit unusual phase transitions of several different types. We present an overview of the work which has been done to understand both the statics and dynamics of these systems and offer some challenges for the future.
DOI
10.1063/1.353480

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The aim of this contribution is to draw attention to some recent developments in the theory of critical phenomena and phase transitions that may be of interest to the magnetism community in that magnetic systems have provided a fertile testing ground for modern theoretical ideas concerning ordering and phase transitions while, on the other hand, many aspects of observed magnetic phase behavior have required sophisticated theory for their proper elucidation. The new works in question have been published or will appear shortly: accordingly, only an outline is presented here together with references to the main literature sources which, in turn, include fuller references to earlier work not explicitly mentioned. Four linked topics are addressed: (i) finite-size effects at first-order transitions; (ii) asymptotic degeneracy in transfer matrices, which is intimately related to long-range order; (iii) some consequences of the conformal covariance of fluctuations at a critical point in bulk and in finite systems, especially in two spatial dimensions; (iv) the discovery of ‘‘exactly soluble’’ two-dimensional models exhibiting two infinite classes of distinct critical and multicritical points.
We study phase transitions and the nature of order in a class of classical generalized O(N) nonlinear σ models (NLS) constructed by minimally coupling pure NLS with additional degrees of freedom in the form of (i) Ising ferromagnetic spins, (ii) an advective Stokesian velocity, and (iii) multiplicative noises. In examples (i) and (ii), and also (iii) with the associated multiplicative noise being not sufficiently long-ranged, we show that the models may display a class of unusual phase transitions between stiff and soft phases, where the effective spin stiffness respectively diverges and vanishes in the long wavelength limit at two dimensions (2D), unlike in pure NLS. In the stiff phase, in the thermodynamic limit the variance of the transverse spin (or, the Goldstone mode) fluctuations are found to scale with the system size L in 2D as lnlnL with a model-dependent amplitude, which is markedly weaker than the well-known lnL dependence of the variance of the broken symmetry modes in models that display quasi-long-range order in 2D. Equivalently, for N=2 at 2D the equal-time spin-spin correlations decay in powers of inverse logarithm of the spatial separation with model-dependent exponents. These transitions are controlled by the model parameters those couple the O(N) spins with the additional variables. In the presence of long-range noises in example (iii), true long-range order may set in 2D, depending upon the specific details of the underlying dynamics. Our results should be useful in understanding phase transitions in equilibrium and nonequilibrium low-dimensional systems with continuous symmetries in general.
The X-Y model in one- and two-dimensions is investigated by means of the path integral. The low temperature properties are shown to be described by the phase mode. In particu-lar, the quantum model in the limit S--'fco is found to be well expressed by the planer model. The spin-correlation functions are calculated explicitly. As a result it is found that at zero temperature the quasi long-range order in one-dimension and the long-range order in two-dimen-sions appear. The spontaneous magnetization is explicitly given at zero temperature in two-dimensions. At finite temperatures the dynamical behaviour is the same as that studied by Nelson and Fisher. The divergence of the susceptibility at T=O in one-dimension is shown to be different from that of the classical model. Considerable progress has been made recently 111 understanding the nature of the phase transition in the two-dimensional planer model without long-range or-cler.l)~81 It has been revealed that below the transition temperature there exists a topological order in which bound pairs of vortex and antivortex are formed, the