Critical Phenomena, the Landau-Ginsburg Theory, Mean-Field Theory

C. Di Castro, Roberto Raimondi

Hidetoshi Nishimori, Gerardo Ortíz

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.

Daniel J. Amit, Víctor Martín-Mayor

Nabil Iqbal, Hong Liu, Márk Mezei

We consider several types of quantum critical phenomena from finite-density gauge-gravity duality which to different degrees lie outside the Landau-Ginsburg-Wilson paradigm. These include: (i) a ``bifurcating'' critical point, for which the order parameter remains gapped at the critical point, and thus is not driven by soft order parameter fluctuations. Rather it appears to be driven by ``confinement'' which arises when two fixed points annihilate and lose conformality. On the condensed side, there is an infinite tower of condensed states and the nonlinear response of the tower exhibits an infinite spiral structure; (ii) a ``hybridized'' critical point which can be described by a standard Landau-Ginsburg sector of order parameter fluctuations hybridized with a strongly coupled sector; (iii) a ``marginal'' critical point which is obtained by tuning the above two critical points to occur together and whose bosonic fluctuation spectrum coincides with that postulated to underly the ``Marginal Fermi Liquid'' description of the optimally doped cuprates.

Kenneth Geddes Wilson, Hugh C. Wolfe, C. D. Graham +1 more

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.

C. Di Castro, Roberto Raimondi