Modern theories of melting of two-dimensional systems are discussed that are mainly based on the concepts of the Berezinskii–Kosterlitz–Thouless (BKT) theory of phase transitions in two-dimensional systems with continuous symmetry. Today there exist three basic scenarios of melting of two-dimensional crystals. First of all, this is the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young (BKTHNY) theory, in which two-dimensional crystals are melted through two BKT-type continuous transitions with an intermediate hexatic phase. In this case a first-order phase transition can also occur. The third scenario has recently been proposed by Bernard and Krauth (BK), in which melting can occur through a BKT-type transition; in this case the hexatic phase–isotropic fluid transition is a first-order transition. The review presents a critical analysis of the approaches used to determine the parameters and the type of transition by computer simulation methods.

К 50-летию Института физики высоких давлений им. Л.Ф. Верещагина. Выездная научная сессия Отделения физических наук Российской академии наук, 23 апреля 2008 г., Стишов С.М., Хвостанцев Л.Г., Слесарев В.Н., Попова С.В., Бражкин В.В., Дюжева Т.И., Джавадов Л.Н., Громницкая Е.Л., Степанов Г.Н., Тимофеев Ю.А., Дижур Е.М., Вентцель В.А., Вороновский А.Н., Рыжов В.Н., Барабанов А.Ф., Магницкая М.В., Тареева Е.Е.

A simple model for the angle-dependent interaction betweenC 60 molecules in the face-centered cubic lattice is proposed. The bifurcations of the solutions of the nonlinear integral equations for orientational distribution functions in the mean-field approximation are analyzed, and the orientational phase transition in solidC 60 is described. The quantitative results for the orientational phase transition agree with the experimental data.

The statistical mechanics of a vortex system in a two-dimensional superconductor is constructed using the cyclic approximation, accounting for the finiteness of the vortex core. This leads to a crossover of the vortex antivortex pair unbinding transition from the usual continuous Kosterlitz-Thouless-like behavior for large core energies to the first order transition for low energies.

The statistical mechanics and thermodynamics of a two-dimensional system of vortices in a thin superconducting film are constructed. The long-range interaction of the vortices is taken into account in the ring approximation well known from plasma theory.

In a model of an isotropic quadrupole glass with two order parameters, a macroscopically large number of metastable states is found at zero temperature. The replica symmetry is broken in the immediate vicinity of the transition to the glass state, and it is shown that the obtained solution is stable.

Microscopic expressions are obtained for the shear and bulk moduli in the vicinity of the melting line and for the Frank constant of the hexatic phase in terms of the potential, the radial distribution function, and the direct correlation function.

A spin glass that differs from the Sherrington-Kirkpatrick model only in the spin's taking three values: +1, 0, -1, is investigated. The free energy of the system is obtained in the replica-symmetric approximation, in which the equations for the order parameters are also solved; the entropy at zero temperature is negative. At zero temperature, all spins take values +1, -1. This makes it possible to calculate the number of metastable states by the method of Edwards and Tanaka. The number is equal to the well-known result for the Sherrington-Kirkpatrick model.

A replica-symmetric solution of a model of quadrupole glass in which the coupling constants of the axial and nonaxial interactions are distributed in accordance with a Gaussian law is considered. When the parameters of the distributions are identical, a transition to the quadrupole-glass phase is possible in the system. The temperature dependence of the order parameter of the quadrupole glass and the stability of the obtained solution are investigated.