International Journal of Modern Physics B

A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, Z_N-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach.

Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed.

The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.

These lecture notes are intended to provide an introduction to Chern-Simons theories and “anyons” for readers with interests in condensed matter physics. I discuss applications of anyons in the fractional quantum Hall effect and to superconductivity.

We present a systematic review of the connection between the complete integrability of (finite-dimensional and infinite-dimensional) Hamiltonian and non-Hamiltonian dynamical systems and the singularity structure of their equations of motion. We make a special attempt to separate the many conjectures in the field from the few rigorous results. The singularity analysis of ordinary and partial differential equations is illustrated by means of four examples. In the Appendix, we present a brief review of the special properties of soliton equations and the relations between them. We summarize these relations in a tabular form.

We use maps that generate a hierarchy of partitionings that can be organized onto a complete binary tree to ask the question when does the Gibbs distribution describe the frequency of distribution of iterates of a map over its natural partitioning. We are led to a generalization of Borel’s normal numbers, to symbol sequences where the blocks of symbols of a given length are distributed statistically independently, but with unequal frequencies. This result leads to a new universality principle based upon classes of initial conditions that limits the extent within which one can use an observed f(α) spectrum to infer the underlying dynamical system. An application to a turbulence experiment is discussed as an example. Our method is based upon the generation of symbol sequences by an algorithm and the consequent recovery of the corresponding initial condition by backward iteration of the map. Our analysis illustrates that measure one theorems in mathematics cannot serve as a guide to what should be the outcome of either computation or experiment.

The effect of radiation disorder on the electrical resistivity of the metallic non-superconducting BaNb₄O₆ oxide has been investigated. It is shown that variation of electrical resistivity ρ of this compound under disorder is typical of metallic systems, i.e. residual resistivity increases linearly with defect concentration while the temperature dependence of ρ changes slightly. Such a behavior qualitatively differs from the previously observed unusual behavior of HTSC with similar crystal structure.

Local electronic densities of states of Tl₂Ca₂Ba₂Cu₃O₁₀ (2223) are calculated by tight-binding approximation. The nearest and the next-nearest neighbor transitions are considered, and the results are compared with each other. After the latter is considered, the densities of states are smoothened and the semiconductor-like characteristic of the Tl-O layers disappears. Through comparison of 2223 with the other two high T_c ceramics La_2−xSr_xCuO₄ and YBa₂Cu₃O_7−x, it is believed that they have the same leading superconductive mechanism besides inter-plane interactions.

There is a purely electronic analog to spinodal decomposition in alloys. A number of examples of such phase separation are presented, including one of potential interest to high-T_c superconductivity. It is speculated that, under some circumstances, the heterogeneous state may be described by a modification of the anyon theory.

X-ray Absorption Near Edge Structure (XANES) measurements above L_II edge of Sm are employed to study the intermediate valence in Sm_1−xGd_xS(0.05≤x≤0.30) alloys. The split white lines at the L_II edge of Sm are used to determine the average valence of Sm in these alloys, The internal chemical pressure generated by substitution of smaller trivalent cation, Gd, for Sm drives the latter to a higher valence. Sm valence increases smoothly first and then undergoes an abrupt increment at a critical concentration of 16 at.% of Gd. At concentration of Gd higher than 16 at.% the increase in Sm valence is smooth and small. The data for Sm average valence for Sm_1−xGd_xS alloys obtained by L_II absorption edges shows fair agreement with other techniques.

The squeezing of the square of the field amplitude in the two-atom one-mode model with multiphoton transitions is investigated. The time-dependent squeezing factors are calculated. The conditions for the optimum squeezing are shown.

We study the structure of the phase space for a system of N molecules of ellipsoidal symmetry, as a function of concentration and temperature. A classical lattice gas approximation is considered and a single molecule is described by a rigid ellipsoidal core with weak attractive tails along the long axis. The method adopted is a second-order mean-field approach – designed in such a way as to keep into account the fluctuations from equilibrium of the order parameters up to the fourth order – combined with a cumulant-cluster expansion, and improved by keeping track of the short-range correlations. Preliminary numerical calculations show the existence, in the case of zero attractive tail, of a second order phase transition.