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On the theory of phase transitions

Ulrich R. Müller, Gemma de la Flor

Published Nov 25, 2024• 1 citation
Publisher

Abstract

Abstract Whether a compound experiences a phase transition is a matter of thermodynamics. According to the Landau theory, the Gibbs free energy can be expanded to a Taylor series with respect to an order parameter η. For a continuous phase transition, only even powers of η may appear in the Taylor series. For a temperature-driven phase transition, η is proportional to [(Tc − T)/Tc]β, T being the temperature, Tc the critical temperature, and β the critical exponent. Landau theory shows discrepancies with experimental findings, especially when there are short-range atomic interactions. An improvement has been achieved with the renormalization-group theory, where fluctuations are considered.

DOI
10.1093/oso/9780192858320.003.0022

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Abstract Whether a compound experiences a phase transition is a matter of thermodynamics. According to the Landau theory, the Gibbs free energy can be expanded to a Taylor series with respect to an order parameter η. For a continuous phase transition, only even powers of η may appear in the Taylor series. For a temperature-driven phase transition, η is proportional to [(Tc − T)/Tc]β, T being the temperature, Tc the critical temperature, and β the critical exponent. Landau theory shows discrepancies with experimental findings, especially when there are short-range atomic interactions. An improvement has been achieved with the renormalization-group theory, where fluctuations are considered.
Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
Critical exponents are functions of ddd, the dimension, and nnn, the number of components of the order parameter involved in the second-order phase transition. They can be expanded in power series of 1n\frac{1}{n}n1​ when nnn is large. We present details of calculating by perturbation theory the critical exponents above Tc{T}_{c}Tc​ for arbitrary ddd and to O(1n)O(\frac{1}{n})O(n1​) for short-range interacting systems and also for long-range interacting systems.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
In this chapter, the author shows the theory of phase transition using some systems of interacting spins. Note that various systems of different nature can be mapped into spin systems and solved using the spin language. However, the mean-field theory gives some artifacts at low dimensions and cannot determine with precision the critical exponents which characterize the nature of a phase transition. A transition from one phase to another may take place when an external parameter varies. Such a parameter can be the temperature or an external applied magnetic field. Renormalization group analysis shows that the universality class depends only on a few very general parameters such as the space dimension, the symmetry of the order parameter and the nature of the interaction. The Landau–Ginzburg theory is an extension of the mean-field theory which includes a great part of fluctuations so far neglected near the transition.
The theory of static critical phenomena in systems at equlibrium is discussed. The nature of the classical phenomenological theory of Landau is explained. Critical point exponents β, γ, γ', δ, . . . obtained from the Landau theory differ from the results of the theory for Ising model and experimental results. Statistical evaluation of the order parameter is obtained by taking into account the fluctuation. Then the correlation effect is considered. These approaches make implicit use of the assumption that the free energy if expandable in power series in the order parameter. From the calculation of spatial fluctuation and long range correlation effect the validity of the Landau theory is discussed.