These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
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.
A renormalization-group scheme is developed for the three-dimensional
O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a
1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at
z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at
z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of
lny.The critical behavior of semi-infinite systems in fixed dimensions
d<4 is investigated theoretically. The appropriate extension of Parisi's massive field theory approach is presented. Two-loop calculations and subsequent Pad\'e-Borel analyses of surface critical exponents of the special and ordinary phase transitions yield estimates in reasonable agreement with recent Monte Carlo results. This includes the crossover exponent $\ensuremath{\Phi}(d=3)$, for which we obtain the values $\ensuremath{\Phi}(n=1)\ensuremath{\simeq}0.54$ and $\ensuremath{\Phi}(n=0)\ensuremath{\simeq}0.52$, considerably lower than the previous $\ensuremath{\epsilon}$-expansion estimates.
The theoretical framework and the details of calculations of nonasymptotic critical behavior, above
Tc, previously published for
n=1, are presented and extended to the cases
n=2and3. The complete description of the real preasymptotic critical domain
Dpreas needs only three adjustable parameters. We show that the ${\ensuremath{\varphi}}^{4}$ model at infinite cutoff is sufficient to obtain this description. The higher-transient and finite-cutoff effects in
Dpreas affect only the adjustable parameters. A precise nonperturbative treatment of the ${\ensuremath{\varphi}}^{4}$ field theory exactly for
d=3, in the spirit of the Parisi work, yields nonasymptotic functions of temperature which, including all the quantitative universal characteristics of the critical behavior in
Dpreas, are adapted to a suitable comparison with experiments. We emphasize that this work allows a determination, within the experimental accuracy, of the size of
Dpreas and an appreciation of the effects of the higher corrections to scaling. We also give the estimates for
n=1,2,and3 of a new universal combination of amplitudes (
RBcr) which concerns the specific heat alone.
The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics