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arXiv:hep-lat/0512022

Properties of phase transitions of higher order

Wolfhard Janke E, Desmond A. Johnston, Ralph Kenna B

Published Dec 15, 2005• 0 citations
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Abstract

There is only limited experimental evidence for the existence in nature of phase transitions of Ehrenfest order greater than two. However, there is no physical reason for their non-existence, and such transitions certainly exist in a number of theoretical models in statistical physics and lattice field theory. Here, higher-order transitions are analysed through the medium of partition function zeros. Results concerning the distributions of zeros are derived as are scaling relations between some of the critical exponents.

DOI
10.48550/arxiv.hep-lat/0512022

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There is only limited experimental evidence for the existence in nature of phase transitions of Ehrenfest order greater than two. However, there is no physical reason for their non-existence, and such transitions certainly exist in a number of theoretical models in statistical physics and lattice field theory. Here, higher-order transitions are analysed through the medium of partition function zeros. Results concerning the distributions of zeros are derived as are scaling relations between some of the critical exponents.
There is only limited experimental evidence for the existence in nature of phase transitions of Ehrenfest order greater than two.However, there is no physical reason for their non-existence, and such transitions certainly exist in a number of theoretical models in statistical physics and lattice field theory.Here, higher-order transitions are analysed through the medium of partition function zeros.Results concerning the distributions of zeros are derived as are scaling relations between some of the critical exponents.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
By the early 1960's advances in statistical physics had established the existence of universality classes for systems with second-order phase transitions and characterized these by critical exponents which are different to the classical ones. There followed the discovery of (now famous) scaling relations between the power-law critical exponents describing second-order criticality. These scaling relations are of fundamental importance and now form a cornerstone of statistical mechanics. In certain circumstances, such scaling behaviour is modified by multiplicative logarithmic corrections. These are also characterized by critical exponents, analogous to the standard ones. Recently scaling relations between these logarithmic exponents have been established. Here, the theories associated with these advances are presented and expanded and the status of investigations into logarithmic corrections in a variety of models is reviewed.
By the early 1960's advances in statistical physics had established the existence of universality classes for systems with second-order phase transitions and characterized these by critical exponents which are different to the classical ones. There followed the discovery of (now famous) scaling relations between the power-law critical exponents describing second-order criticality. These scaling relations are of fundamental importance and now form a cornerstone of statistical mechanics. In certain circumstances, such scaling behaviour is modified by multiplicative logarithmic corrections. These are also characterized by critical exponents, analogous to the standard ones. Recently scaling relations between these logarithmic exponents have been established. Here, the theories associated with these advances are presented and expanded and the status of investigations into logarithmic corrections in a variety of models is reviewed.
Many important concepts used in the study of complex systems have their origin in the lattice statistical mechanics of cooperative phase transitions. The classic order-disorder transitions of Potts/Ising models illustrate emergent phenomena such as spontaneous symmetry breaking, scale-free critical behaviour and power-law singularities. However, a class of lattice statistics models show that rather more complex behaviour can be obtained, under conditions which are suffciently general that they might also occur in less idealised models. A 5-state lattice statistics model is analysed to explore behaviour in and around a so-called phase. The massless phase, characterised by power law decay of correlations, appears as an intermediate regime between high temperature disorder and low-temperature ordering. The critical exponents (and fractal spatial distributions) occur over a finite range of temperatures. The state of the massless phase is characterised by a topological ordering with analogues of spin-waves being the dominant perturbations. The transition from the ordered to massless is analysed using new exact series expansions obtained by the finite lattice method. The results are also compared to Monte Carlo simulations. The results are compared to other studies in the small number of cases where such series expansions exist for comparable models, including the 6-state clock model. One of the properties that makes the massless phase dificult to study is the weak nature of the associated Kosterlitz-Thouless transition with exponentially-weak behaviour rather than power-law behaviour. As a further diffculty, the limiting behaviour is confined to a narrow regime, outside which one sees an apparent cross-over to power law behaviour. This suggests that behaviour analogous to the phases will be diffcult to characterise as one moves beyond idealised lattice systems.