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INTRODUCTION TO THE MODERN THEORY OF PHASE TRANSITIONS

Iozef Sznajd

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

The evolution of the description of the phase transitions from singularities in the thermodynamical potentials via symmetry breaking to the divergence of the scale-length is presented. The origin of the power laws at the critical state is discussed. The power laws occurring in several phenomena are reviewed. The modem theory of the critical phenomena based on the concepts of scaling and renormalization, on a very elementary level, is introduced.

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The evolution of the description of the phase transitions from singularities in the thermodynamical potentials via symmetry breaking to the divergence of the scale-length is presented. The origin of the power laws at the critical state is discussed. The power laws occurring in several phenomena are reviewed. The modem theory of the critical phenomena based on the concepts of scaling and renormalization, on a very elementary level, is introduced.
Power laws are distinctive features of critical phenomena and can be ascribed to diverging length-scales. One of the most important contributions of the studies of critical phenomena is the shift in point of view to a length-scale based analysis and classifying terms of a Hamiltonian as relevant or irrelevant rather than classifying them as numerically weak or strong. The idea of universality transcends the domain of critical phenomena. Critical points are classified by the number of relevant variables required to describe them. A universality class would be described by the exponents and also the amplitude ratios of the various singulax quantities on both sides of the critical point. In a correlation based approach, we need to compute the critical correlations of all possible combinations of the degrees of freedom. Scaling is the rule for criticality, originating from a diverging length scale, while anomalous dimension is seen for fluctuation dominated cases.
A theory for tricritical phenomena proposed by Wegner and the author is described on an elementary level. The emphasis of the discussion is on (i) the differences between critical and tricritical phase transitions, (ii) the crossover phenomena that occur near tricritical points, and (iii) the determination of tricritical exponents, scaling fields and scaling functions. The results have been obtained by renormalization‐group techniques and a scaling field approach to crossover phenomena. The elements of these techniques are briefly explained.
This brief overview is designed to introduce some of the advances that have occurred in our understanding of phase transitions and critical phenomena. The presentation is organized around three simple questions: (i) What are the basic phenomena under consideration? (ii) Why do we care? (iii) What do we actually do? To answer the third question, the author shall briefly review scaling, universality, and renormalization, three of the many important themes which have served to provide the framework of much of our current understanding of critical phenomena. The style is that of a colloquium, not that of a mini-review article.
Abstract In finite systems singularities, such as discontinuities or divergences, associated with phase transitions are rounded. The extrapolation of finite size calculations to the singular thermodynamic limit makes use of scaling laws, introduced by Fisher, which involve the size of the system. The use and the theoretical basis of these laws are briefly reviewed here; some difficulties associated with the analytic calculation of the scaling functions are discussed. The correct treatment of zero‐momentum modes leads to a modified renormalization group picture; above the upper critical dimension (u.c.d.) finite size scaling does not hold but the properties of the finite system are easily calculable; below the u.c.d. the scaling functions are obtained in a systematic expansion in fractional powers of ϵ = 4–d. The rounding of first order transitions is also briefly considered; in spite of the lack of universality of the bulk transition itself, the rounding is universal because the correlation length of the finite system is large.
Abstract Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.
We attempt to present in a systematic fashion a theory of critical behaviour in terms of simple probabilistic concepts. This approach arose out of renormalization group treatments of critical phenomena in phase transitions, but it is relevant also for turbulence, certain geophysical situations, and in general for any statistical problem in which scaling laws are expected to play a significant role. The presentation here reflects the recent emphasis on phase transitions but the formulation is sufficiently general to be easily adapted for a different context. Familiarity with probability theory at an elementary level is the only prerequisite for following this article.