Landau's mean field theory of critical phenomena is corrected to take fluctuations into account. It is shown that fluctuations of intermediate wavelengths are the crucial effect producing non‐classical exponents (intermediate wavelengths are much larger than the range of forces, much smaller than the correlation length). A simple‐minded analysis of these fluctuations produces improved exponents for spatial dimension d less than 4. The exponents have an expansion in ε = d−4. The Landau theory is the 0th order term; the terms of order ε give better agreement with experiment than Landau theory. Terms of order ε2 have been calculated using a more sophisticated theory.
Landau's mean field theory of critical phenomena is corrected to take fluctuations into account. It is shown that fluctuations of intermediate wavelengths are the crucial effect producing non‐classical exponents (intermediate wavelengths are much larger than the range of forces, much smaller than the correlation length). A simple‐minded analysis of these fluctuations produces improved exponents for spatial dimension d less than 4. The exponents have an expansion in ε = d−4. The Landau theory is the 0th order term; the terms of order ε give better agreement with experiment than Landau theory. Terms of order ε2 have been calculated using a more sophisticated theory.
Mean-field theory, presented in the preceding chapter, correctly describes the qualitative features of most phase transitions and, in some cases, the quantitative features. Since mean-field theory replaces the actual configurations of the local variables (e.g. spins) by their average value, it neglects the effects of fluctuations about this mean. These fluctuations may or may not be important. The more spins that interact with a particular test spin, the more the test spin sees an effective average or mean field. If the test spin interacts with two neighbors, the averaging is minimal and the fluctuations are large and important. The number of spins producing the effective field increases with the range of the interaction and with the dimension. Thus we will find that mean-field theory is a good approximation in high dimensions but fails to provide a quantitatively correct description of second-order critical points in low dimensions. This chapter will be devoted to the study of second-order phase transitions when mean-field theory is not a good approximation. We will begin by considering fluctuations of the order parameter about its spatially uniform mean-field value. We will find that for spatial dimensions below an upper critical dimension d c (typically d c = 4), fluctuations always become important and invalidate mean-field theory for temperatures sufficiently close to T c . This will motivate a generalization of Landau theory that incorporates spatial fluctuations about a mean-field free energy minimum.
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.
Abstract We systematically develop theories of critical phenomena with a prior formed memory of a power-law decaying long-range temporal interaction parameterized by a constant θ > 0 for a space dimension d both below and above an upper critical dimension d c = 6 − 2/θ. We first provide more evidences to confirm the previous theory that a dimensional constant d t is demanded to rectify a hyperscaling law, to produce correct unique mean-field critical exponents via an effective spatial dimension originating from temporal dimension, and to transform the time and change the dynamic critical exponent. Next, for d < d c , we develop a renormalization-group theory by employing the momentum-shell technique to the leading nontrivial order explicitly and to higher orders formally in ϵ = d c − d but to zero order in ε = 1 − θ and find that more scaling laws besides the hyperscaling law are broken due to the breaking of the fluctuation-dissipation theorem. Moreover, because dynamics and statics are intimately interwoven, even the static critical exponents involve contributions from the dynamics and hence do not restore the short-range exponents even for θ = 1 and the crossover between the short-range and long-range fixed points is discontinuous contrary to the case of long-range spatial interaction. In addition, a new scaling law relating the dynamic critical exponent with the static ones emerges, indicating that the dynamic critical exponent is not independent. However, once d t is displaced by a series of ϵ and ε such that most values of the critical exponents are changed, all scaling laws are saved again, even though the fluctuation-dissipation theorem keeps violating. Then, for d ≥ d c , we develop an effective-dimension theory by carefully discriminating the corrections of both temporal and spatial dimensions and find three different regions. For d ≥ d c0 = 4, the upper critical dimension of the usual short-range theory, the usual Landau mean-field theory with fluctuations confined to the effective-dimension equal to d c0 correctly describe the critical phenomena with memory, while for d c < d ≤ 4, there exist new universality classes whose critical exponents depend only on the space dimension but not at all on θ. Yet another region consists of d = d c only and the previous theory is retrieved. All these results show that the dimensional constant d t is the fundamental ingredient of the theories for critical phenomena with memory. However, its value continuously varies with the space dimension and vanishes exactly at d = 4, reflecting the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Moreover, special finite-size scaling appears ubiquitously except for d = d c0.
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.
This chapter shows that mean field theory in its simplest form can be improved by inclusion of Gaussian fluctuations. The transition temperature is shifted, and the heat capacity does indeed exhibit a divergence rather than the discontinuity predicted for a uniform system. The expansion, in which the Gaussian and mean field theories are the lowest terms, has an expansion parameter which diverges as the critical point is approached. The problem of anomalous dimensions in critical phenomena was discovered and solved apparently without knowledge or recognition of these other phenomena exhibiting case asymptotics. The chapter briefly discusses the notion of renormalisation in field theory. It is an interesting historical point that Landau had first-hand knowledge of anomalous dimensions in both critical phenomena and the problem of the converging shock wave, but apparently made no reference to a connection between them.
This chapter discusses a very popular method of studying critical phenomena analytically: the so-called ϵ-expansion. It describes the renormalisation group calculation of the critical exponents in the complete Landau theory, near d = 4. The chapter outlines the strategy of the calculation of 0, to prepare the reader for the algebraic horrors to follow. R. P. Feynman developed useful shorthand for the terms, which is to write down diagrams indicating how different fields are paired up. There is one-to-one correspondence between each Feynman diagram or Feynman graph and a term in the perturbation series. This correspondence is not unique: there are many ways to write down diagrams for the terms, and people have their own preferences. Once one has figured out this correspondence, then one only has to be able to draw all the appropriate diagrams to see what one should compute.