Droplet theory in low dimensions: Ising systems in an ordering field

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Abstract
The authors extend the microscopic droplet theory of Ising systems, developed recently, to incorporate the effects of an ordering field. The theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting, and which has a full scaling form. The behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models, but with parameters renormalised in a physically intelligible way. The droplet population function implied by the theory is parametrised in a simple way, in space dimension d=2, to yield predictions for various thermodynamic properties, including the equation of state, which are found to be in fair accord with series-based results. The structure of the field-dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories. The domain of validity of the theory is assessed. It is concluded that the theory fails to describe (sufficiently) large droplets in the presence of a finite field, and is thus in principle trustworthy only in a region close to the coexistence curve.

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