Crossover scaling function for the free energy

David R. Nelson

We express the free energy of a system in its critical region as a line integral along a renormalization group trajectory. The kernal associated with the trajectory integral is just the spin independent part of the Hamiltonian generated by each renormalization group iteration. If the trajectory integral is parametrized in terms of nonlinear scaling fields, a closed expression for the crossover scaling function is obtained. Crossover scaling functions associated with free energy are calculated explicitly for recursion relation models treated by Riedel and Wegner, and the relevance to ε‐expansions near four dimensions is discussed. A breakdown of hyperscaling above the borderline dimensions (d=4 for critical points, and d=3 for tricritical points) arises naturally in the formalism.

James W. Dufty, Valentin V. Karasiev, Samuel B. Trickey

Christian Borgs, John Z Imbrie

David R. Nelson

By expressing the free energy as a line integral along a renormalization-group trajectory, a technique for calculating the crossover scaling functions which describe tricritical systems, anisotropic spin systems, etc., is developed. This formalism is applied to the model recursion relations of Riedel and Wegner, which simulate crossover behavior. A simple mechanism for the breakdown of dimensionality dependent hyperscaling relationships emerges from the analysis. The specific-heat crossover scaling function describing crossovers from Gaussian to Heisenberg critical behavior is constructed to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$.

Shin‐itiro Goto, Yoshiyuki Y. Yamaguchi

Ian D. Lawrie

The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.