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Progress of Theoretical Physics

Modified Landau Theory of the Second Order Phase Transition

Yoshiki Kuramoto

Published Mar 1, 1969• 5 citations
Publisher

Abstract

The difficulty near the critical point encountered by the Landau theory or the classical theory of the second order phase transition is removed with the least modification of the original framework. The essential point of our idea is to note that the temperature region in which the Landau theory valids depends sensitively on the way of defining the local order parameter. It is shown that the relations among the singularities of various thermodynamic quantities predicted by this modified form of the Landau theory are in good agreement with those obtained by nonclassical theories such as the static scaling theory.

DOI
10.1143/ptp.41.604

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The difficulty near the critical point encountered by the Landau theory or the classical theory of the second order phase transition is removed with the least modification of the original framework. The essential point of our idea is to note that the temperature region in which the Landau theory valids depends sensitively on the way of defining the local order parameter. It is shown that the relations among the singularities of various thermodynamic quantities predicted by this modified form of the Landau theory are in good agreement with those obtained by nonclassical theories such as the static scaling theory.
It has been seen in the previous chapters that Landau mean field approach relies on an incomplete partition function, the use of which is only justified if the fluctuations are Gaussian. This approach is manifestly invalid in the vicinity of a second order phase transition, since the second derivative of the Landau free energy with respect to the order parameter is equal to zero at the critical temperature https://www.w3.org/1998/Math/MathML" display="inline"> T C 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_26.tif "/> and the magnitude of the fluctuations near the critical point become prohibitive for the use of the theory. This is a real paradox, since Landau theory is designed to be valid exactly in the vicinity of second order phase transition points (see Section 9.2.2 ). The present chapter focuses first on the discussion of this problem and the precise evaluation of the domain of validity of mean field theories. In a second part, mean field predictions are compared with experimental data and the notion of scale invariance in the vicinity of the critical point is discussed.
A general relation between the critical specific heat above and below a second-order phase transition is derived. It is expressed in terms of the effective four-point coupling u(l)u(l)u(l) of the Ginzburg-Landau Hamiltonian and is valid in the entire region between asymptotic criticality and noncritical background behavior. Application to the $\ensuremath{\lambda}$ transition of 4He^{4}\mathrm{He}4He yields excellent agreement with the fixed-point value u∗=u(0){u}^{*}=u(0)u∗=u(0) predicted by high-order perturbation theory.
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.
The Landau Paradigm The usual paradigm to describe phase transitions is that of Landau. It is based on an expansion of a free energy functional or an action in terms of an order parameter. Close to the phase transition the order parameter is small, and only terms dictated by symmetry are allowed in this expansion. Associated with this paradigm are many important ideas, such as that of broken symmetry and Goldstone modes. The extension of this paradigm to include quantum phase transitions certainly brought new ideas and progress, as we have seen in this book: the inextricability between the dynamics and static properties, the modified hyperscaling relation with the notion of an effective dimensionality, the role of the uncertainty principle linking energy and temporal fluctuations and so on. The underlying principle here is always that of symmetry. However, as we will see in this chapter, this paradigm is not sufficient to describe certain classes of phenomena that do not involve symmetry changes but are nonetheless still associated with diverging lengths, singularities and critical exponents: features that are clearly related to critical phenomena and phase transitions. We have already met similar behaviour when studying density-driven or Lifshitz transitions. We will see in this chapter that these transitions are much more common and interesting that initially thought. The keyword here is topology, which contains principles and constraints that protect quasi-particle excitations and are still more powerful than those of symmetry. Topological Quantum Phase Transitions We will refer to topological quantum phase transitions (TQPT) as those that separate two phases where at least one of them is topologically non-trivial, as characterised by some topological parameter like a winding number or a Berry phase. In some cases these transitions either give rise to singularities in a thermodynamic quantity or are accompanied by a diverging length. However, differently from Landau-type phase transitions they are not necessarily associated with a symmetry change. Also there is no identifiable order parameter to distinguish between the different phases. Topological quantities are not useful to play the role of an order parameter. They are in general discontinuous at topological transitions and this does not necessarily reflect the nature of the transition. On the other hand, we will show that the renormalisation group is a useful tool to describe TQPT.
Classical and quantum criticality In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions. For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T) β . Here T c is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, g c is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < g c .