D. B. Abraham

Equilibrium Stranski-Krastanow and Volmer-Weber models

An equilibrium random surface model in 3d is defined which includes versions of both the Stranski-Krastanow and Volmer-Weber models of crystal surface morphology. In a limiting case, the model reduces to one studied previously in a different context for which exact results are available in part of the phase diagram, including the critical temperature, the associated specific heat singularity and the geometrical character of the transition. Through a connection to the 2d Ising model, there is a natural association with the Schramm-Loewner evolution that has also been observed experimentally in a nonequilibrium deposition setting.

Microscopic model for thin film spreading.

A microscopic, driven lattice gas model is proposed for the dynamics and spatiotemporal fluctuations of the precursor film observed in spreading experiments. Matter is transported both by holes and particles, and the distribution of each can be described by driven diffusion with a moving boundary. This picture leads to a stochastic partial differential equation for the shape of the boundary. Explicit analytic results are obtained which agree with the simulations of the lattice gas.

Nonperturbative analysis of a model of random surfaces

Abstract The truncated pair function for a particular system of random surfaces on Z d is analyzed for all temperatures strictly below the critical temperature. Throughout this regime, it is shown that (i) the pair function has Ornstein-Zernike behavior, (ii) the mass, or inverse correlation length, is analytic, (iii) the number of random surfaces has the expected asymptotic scaling, and (iv) the surfaces do not undergo a breathing transition. The results are established by using a random-surface Schwinger-Dyson equation which should be applicable in the nonperturbative analysis of other models of random paths and surfaces.

Surface states and the Casimir interaction in the Ising model

Using exact calculations, we elucidate the significance of surface states for the Casimir interactions in an Ising strip with a finite width. The surface states are responsible for the strong asymmetry between the super- and sub-critical regimes of the Casimir forces. We introduce an enhanced version of Fisher-Privman theory and justify it using another exact calculation. This gives a rather accurate account of the Casimir scaling function in the sub-critical regime and its exact asymptotic behaviour. We apply analogous ideas in three dimensions and obtain the Casimir scaling function which is in striking agreement with Monte Carlo simulations results for large negative scaling variable.

Nonequilibrium dynamics of finite interfaces.

We present an exact solution to an interface model representing the dynamics of a domain wall in a two-phase Ising system. The model is microscopically motivated, yet we find that in the scaling regime our results are consistent with those obtained previously from a phenomenological, coarse-grained Langevin approach.

Susceptibility of the rectangular Ising ferromagnet

The critical index valuesγ= 7/4 for the susceptibility andδ=15 for the critical isotherm are derived rigorously for the rectangular Ising ferromagnet with nearest neighbor interactions. The critical indices associated with the Fisher moment definition of the correlation length are obtained asT→Tc+. The index of the fluctuation sum definition of critical correlations is obtained.

Computational Study of a Multistep Height Model

An equilibrium random surface multistep height model proposed in [Abraham and Newman, EPL 86, 16002 (2009)] is studied using a variant of the worm algorithm. In one limit, the model reduces to the two-dimensional Ising model in the height representation. When the Ising model constraint of single height steps is relaxed, the critical temperature and critical exponents are continuously varying functions of the parameter controlling height steps larger than one. Numerical estimates of the critical exponents can be mapped via a single parameter, the Coulomb gas coupling, to the exponents of the O(n) loop model on the honeycomb lattice with n ≤ 1.

Correlation functions in an exactly solvable tefface-ledge-kink model

Abstract We solve exactly a terrace-ledge-kink (TLK) model describing a vicinal section of a crystal surface at a microscopic level, with either repulsive or attractive interactions between the ledges. As expected there is a faceting, or reconstructive, phase transition, driven either by temperature or by the chemical potential, that controls the mean slope of the surface. In the rough phase we carry out a thorough investigation of microscopic thermal fluctuations of the interface. This is done by combining Bethe ansatz and Conformal Field Theory methods in order to calculate appropriately defined correlators.

A continuum of pure states in the Ising model on a halfplane

We study the homogeneous nearest–neighbor Ising ferromagnet on the right half plane with a Dobrushin type boundary condition—say plus on the top part of the boundary and minus on the bottom. For sufficiently low temperature T, we completely characterize the pure (i.e., extremal) Gibbs states, as follows. There is exactly one for each angle