Surface tension and interfacial fluctuations in d-dimensional Ising model

TL;DR

This paper investigates surface tension and interfacial fluctuations in 2D and 3D Ising models, proposing a formula for finite-size scaling corrections and discussing the implications of analytic continuation to noninteger dimensions.

Contribution

It introduces a formula interpolating finite-size corrections for surface tension between 2D and 3D Ising models and explores the physical meaning of analytic continuation to noninteger dimensions.

Findings

Derived a formula for finite-size scaling corrections in d-dimensional Ising models.

Discussed the physical implications of analytic continuation to noninteger dimensions.

Identified the marginal value of d=2 for interface connectivity.

Abstract

The surface tension of rough interfaces between coexisting phases in 2D and 3D Ising models are discussed in view of the known results and some original calculations presented in this paper. The results are summarised in a formula, which allows to interpolate the corrections to finite-size scaling between two and three dimensions. The physical meaning of an analytic continuation to noninteger values of the spatial dimensionality d is discussed. Lattices and interfaces with properly defined fractal dimensions should fulfil certain requirements to possibly have properties of an analytic continuation from d-dimensional hypercubes. Here 2 appears as the marginal value of d below which the (d-1)-dimensional interface splits in disconnected pieces. Some phenomenological arguments are proposed to describe such interfaces. They show that the character of the interfacial fluctuations at d<2 is not the same as provided by a formal analytic continuation from d-dimensional hypercubes with d >= 2. It, probably, is true also for the related critical exponents.