N. M. Švrakić

Finite-Size Corrections for Inclined Interfaces in Two Dimensions: Exact Results for Ising and Solid-on-Solid Models

Analysis of finite-size corrections for the surface tension and surface stiffness coefficients in two-dimensional models with inclined interfaces is presented. We obtain a universal leading contribution proportional to (lnL)/L for the 2D system of sizeL. By explicit calculations for restricted and unrestricted solid-on-solid models and the square lattice Ising model, we demonstrate the Gaussian nature of rough interfaces with fixed ends, and derive the leading 1/L-type corrections for appropriate surface quantities.

Asymptotic degeneracy of the transfer matrix spectrum for systems with interfaces: Relation to surface stiffness and step free energy

Two- and three-dimensional Ising-type systems are considered in the finite-cross-section cylindrical geometry. An interface is forced along the cylinder (strip in 2d) by the antiperiodic or +− boundary conditions. Detailed predictions are presented for the largest asymptotically degenerate set of the transfer matrix eigenvalues. For rough interfaces, i.e., for 0<T<Tc in 2d,TR<T<Tc in 3d, the eigenvalues are split algebraically, and the spectral gaps are governed by thesurface stiffness coefficient. For “rigid” interfaces, i.e., 0<T<TR in 3d, the eigenvalues are split exponentially, with the gaps determined by thestep free energy.

Difference equations in statistical mechanics. I. Cluster statistics models

A review and some new results are presented for several cluster statistics models, solutions of which can be reduced to difference equations. Mathematical techniques suitable for solving these equations are surveyed.

Transfer matrix spectrum for the finite-width Ising model with adjustable boundary conditions: Exact solution

Using the spinor approach, we calculate exactly the complete spectrum of the transfer matrix for the finite-width, planar Ising model with adjustable boundary conditions. Specifically, in order to control the boundary conditions, we consider an Ising model wrapped around the cylinder, and introduce along the axis a “seam” of defect bonds of variable strength. Depending on the boundary conditions used, the mass gap is found to vanish algebraically or exponentially with the size of the system. These results are compared with recent numerical simulations, and with random-walk and capillary-wave arguments.

Difference equations in statistical mechanics. II. Solid-on-solid models in two dimensions

A review and some new results are presented for the solid-on-solid models of wetting in two dimensions (i.e., line interfaces) with an emphasis on the difference equations arising in the transfer matrix calculations for these models. Methods for solving the appropriate difference equations exactly or approximately are surveyed, including specific results for short-range, long-range power-law, and applied field-like (binding) external potentials.

Temperley's triangular lattice compact cluster model: exact solution in terms of the q series

Temperleys model (1952) of self-supporting stackings of circles in a triangular lattice array against a line wall is solved exactly in terms of q hypergeometric functions. For N circles, the number of different configurations is described by the large-N asymptotic law A lambda N, with A=0.312 36. . . and lambda =1.735 66. . . .

Restoration of universality for the rod-to-coil transition scaling in the infinite-dimensionality limit: exact results for directed walks

We derive scaling forms for the thermodynamic and correlation quantities for the turn-weighted fully and partially directed self-avoiding walks on the hypercubic lattices ind⩾2. In the grand canonical (fixed fugacity per step) ensemble, the conformational rod-to-coil transition sets up in the regimew¯N=O(1), wherew is the weight of each 90° turn and¯N is the (fugacity-dependent) average number of steps. Contrary to the conventional critical phenomena wisdom, the scaling functions for the two different walk models, directed and partially directed, become universal only in the limitd→∞.

Roughening at wetting: step free energy

Build-up of roughening fluctuations in an interface at the wetting (unbinding) transitions can be measured by the free energy of a step-like formation due to a stepped substrate. For the 2D solid-on-solid model of line interface unbinding, exact calculations are presented. It is found that the step free energy vanishes linearly as the wetting transition is approached, as opposed to the well known quadratic vanishing of the interfacial binding energy (per unit length). Step-step interaction free energy is also calculated, for large step separation.