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Statistical field theory

Hidetoshi Nishimori, Gerardo Ortíz

Published Dec 2, 2010• 71 citations
Publisher

Abstract

Abstract Statistical-mechanical systems often involve discrete elementary degrees of freedom such as spins in the Ising model. Field theories, on the other hand, have continuous fields, defined over the whole space-time or part of it, as fundamental degrees of freedom. These two seemingly different descriptions of physical phenomena can be related close to the critical point. The present chapter summarizes how the description by continuous fields emerges from discrete degrees of freedom in a more systematic manner than in previous chapters. The phenomenological Landau-Ginzburg approach, based on the concept of order parameter, is expanded to generate effective field theories. The important roles of symmetry and topology are also elucidated in some detail. Also shown are some of the important consequences of having a broken-symmetry phase, such as long-range order, the emergence of Nambu-Goldstone modes when the symmetry involved is continuous, and topological defects.

DOI
10.1093/acprof:oso/9780199577224.003.0005

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Abstract The present chapter explains the mean-field approximation, the Landau theory, the infinite-range model, and the Bethe approximation, and shows that all these (mean-field) theories are essentially equivalent to each other. The Landau theory is a phenomenological approach that uses the concept of symmetry and the order parameter, a measure of the breaking of that symmetry, as fundamental collective degrees of freedom. Also described are the Landau theory of tricritical behaviour, correlation functions, the limit of applicability of the mean-field theory, known as the Ginzburg criterion, and dynamic critical phenomena. Mean-field theories yield the exact critical exponents for dimensions larger than the upper critical dimension, and their solutions provide a reasonable starting point for more advanced methods including the renormalization group.
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.
The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.
In this chapter, we introduce topological defects as excitations complementary to the elementary Nambu–Goldstone modes and relate their classification to the presence of broken symmetries. We show that the proliferation of topological defects can lead to the suppression of long-range correlations and the emergence of algebraic long-range order. The Berezinskii–Kosterlitz–Thouless phase transition is discussed both generally and in the specific example of the one-dimensional Ising model. In the same context, the Kramers–Wannier duality between topological defects and elementary modes is introduced.
By analogy with the Landau-Ginzburg theory of ordinary zero-form symmetries, we introduce and develop a Landau-Ginzburg theory of one-form global symmetries, which we call mean string field theory. The basic dynamical variable is a string field – defined on the space of closed loops – that can be used to describe the creation, annihilation, and condensation of effective strings. Like its zero-form cousin, the mean string field theory provides a useful picture of the phase diagram of broken and unbroken phases. We provide a transparent derivation of the area law for charged line operators in the unbroken phase and describe the dynamics of gapless Goldstone modes in the broken phase. The framework also provides a theory of topological defects of the broken phase and a description of the phase transition that should be valid above an upper critical dimension, which we discuss. We also discuss general consequences of emergent one-form symmetries at zero and finite temperature.
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.