Table of contents

The key problems of modern cosmology, such as the cosmological singularity, initial conditions, and the quantum tunneling hypothesis, are discussed. The relationship between the latest cosmological trends and L D Landau's old ideas is analyzed. Particular attention is given to the oscillatory approach to singularity; quantum tunneling processes determining wave function of the Universe in the presence of a compex scalar field; and the role of quantum corrections in these processes. The classical dynamics of closed models with a real scalar field is investigated from the standpoint of chaotic, fractal, and singularity-avoiding properties.

This paper considers tunneling traversal time; tunneling through an alternating potential, its enhancement and its fractal (Hofstadter butterfly) and chaotic resonances; and Hartree liquid resonance tunneling and its (specifically quantum) instability and turbulence.

A brief review is given of the state-of-the-art in elementary particle physics based on the talk of the same title given on January 22, 1998, at the seminar marking the 90th anniversary of the birth of L D Landau. (The seminar was hosted by the P L Kapitza Institute for Physical Problems in cooperation with the L D Landau Institute of Theoretical Physics).

Prospects for the observation of CP violation in B-meson decays are briefly reviewed based on the talk given at the Russian Academy of Sciences Presidium session of January 20, 1998.

The recent realization of Bose–Einstein condensation in atomic gases opens new possibilities for the observation of macroscopic quantum phenomena. There are two important features of these systems — weak interaction and significant spatial inhomogeneity. Because of this a non-trivial 'zeroth-order' theory exists, compared to the 'first-order' Bogolubov theory. The zeroth-order theory is based on the mean-field Gross–Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the constant explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. Elementary excitations define the thermodynamic behavior of the system and result in a Landau-type damping of collective modes. Fluctuations of the phase of the condensate wave function restrict the monochromaticity of the Josephson current. Fluctuations of the numbers of quanta result in quantum collapse-revival of the collective oscillations.

The nature of superfluid, superconducting, and magnetic ordering is elucidated for mesoscopic systems in which the single-particle level spacing is much larger than both the temperature and the critical temperature of ordering. Ordering is defined as a spontaneous violation of symmetry, the gauge invariance and time reversal being by definition symmetries violated in superfluidity (superconductivity) and magnetism contexts, respectively. Superfluidity and superconductivity are realized in thermodynamic equilibrium states with a non-integral average number of particles and are accompanied by the spontaneous violation of time homogeneity. In Fermi systems two types of superfluidity and superconductivity are possible which are characterized by the presence of pair or single-particle 'condensates'. The latter is remarkable in that spontaneous violation of fundamental symmetries such as spatial 2π rotation and double time reversal takes place. Possible experiments on metallic nanoparticles and ultracold atomic gases in magnetic traps are discussed.

The 'double exchange' mechanism and Jahn—Teller instabilities are shown to account for the low-temperature properties of slightly doped LaMnO₃ in the framework of the band insulator model. Analysis of the doping of La_1-xA_xMnO₃ with divalent A atoms suggests that Coulomb forces cause holes to be localized near dopants, which makes the formation of conducting clusters along these charged centers a major factor in the physics of such compounds. A percolation theory analysis of experimental data is given. The two-phase coexistence regime and the large-volume Fermi surface at high concentrations are discussed. The relevance of some of the results to doping physics in cuprates is suggested.

The experimental discovery of a number of 'strange metals' has reopened the question of the low temperature behavior of interacting Fermi systems. Here we provide a subjective overview of some aspects of the resulting theoretical work. It seems to us that from a theoretical standpoint Landau's Fermi-liquid theory has proven to be a remarkably robust description of clean Fermi systems. The only well documented theoretical examples of non-Fermi-liquid behavior are metals subject to gauge interactions or at quantum critical points. The experimental anomalies which prompted the reexamination of Fermi-liquid theory remain in many cases mysterious.

A review is presented of the author's work on resonance tunneling as an electron transport mechanism along the c axis in high-temperature layered cuprates. A formulation of the problem is given, qualitative aspects of the mechanism are described, and physical properties calculated. Comparisons are made with experimental data for the temperature dependence of normal conductivity, the frequency dependence of optical conductivity, and the stationary supercurrent along the c axis. For the latter, the resonance tunneling coherence of different centers is shown to be of crucial importance. Weakened interplane coupling and vortex fluctuations are invoked to explain the sharp drop in T_c and the rise in the 2Δ(0)/T_c ratio with decreasing oxygen content. Simple example models are given to demonstrate major aspects of resonance tunneling.