Phase transitions and tricritical points: an exactly soluble model for magnetic or distortive systems

Craig E. Zaspel

The free energy of the elastic Ising chain with a dimerizing distortion in an external magnetic field is obtained exactly. With the lattice distortion as an order parameter it is shown that the phase diagram in the external field, temperature plane contains a tricritical point. This is probably the simplest known system that exhibits a tricritical point.

Victor J. Emery

Michael Ellis Fisher

This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!

W. Speth

St. Sarbach, T. Schneider

An exactly soluble $n$-component continuous spin model with tricritical points is shown to be equivalent, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, to a one-component model with anharmonic long-range interactions.