The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter
s is coupled to a nonordering additional variable
y. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For
s2y coupling, the correlation function of
y was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is
s2y2, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to
s2y coupling is calculated.
〈y〉 has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of
y in the constrained case is discussed.
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.
For a system with a first or second order phase transition at
T∗, we construct a type of renormalization group transformation that maps the reduced temperature t=(T-
T∗)/
T∗ and the external field h for a system of volume V to t\ensuremath{'} and h\ensuremath{'} for another system of volume V\ensuremath{'}. Our construction is purely thermodynamic without reference to effective Hamiltonians. Using the transformation, we derive the finite size scaling form for the singular part of the free energy, which leads to the scaling laws in the thermodynamic limit. We thus provide a unified thermodynamic framework for phase transitions. \textcopyright{} 1996 The American Physical Society.