### Article

## Anisotropic superfluidity of two-dimensional excitons in a periodic potential

We study anisotropies of the helicity modulus, excitation spectrum, sound velocity, and angle-resolved luminescence spectrum in a two-dimensional system of interacting excitons in a periodic potential. Analytical expressions for anisotropic corrections to the quantities characterizing superfluidity are obtained. We consider particularly the case of dipolar excitons in quantum wells. For GaAs/AlGaAs heterostructures as well as MoS2/hBN/MoS2 and MoSe2/hBN/WSe2 transition-metal dichalcogenide bilayers estimates of the magnitude of the predicted effects are given. We also present a method to control superfluid motion and to determine the helicity modulus in generic dipolar systems.

Recently suggested subwavelength lattices offer remarkable prospects for the observation of novel superfluids of fermionic polar molecules. It becomes realistic to obtain a topological p-wave superfluid of microwave-dressed polar molecules in 2D lattices at temperatures of the order of tens of nanokelvins, which is promising for topologically protected quantum information processing. Another foreseen novel phase is an interlayer p-wave superfluid of polar molecules in a bilayer geometry.

The formation of the roton-maxon excitation spectrum and the roton instability effect for a *weakly* correlated Bose gas of dipolar excitons in a semiconductor layer are predicted. The stability diagram is calculated. According to our numerical estimations, the threshold of the roton instability for Bose-Einstein condensed exciton gas with roton-maxon spectrum is achievable experimentally, e.g., in GaAs semiconductor layers.

These notes have appeared as a result of a one-term course in superfluidity and superconductivity given by the author to fourth-year undergraduate students and first-year graduate students of the Department of Physics, Moscow State University of Education. The goal was not to give a detailed picture of these two macroscopic quantum phenomena with an extensive coverage of the experimental background and all the modern developments, but rather to show how the knowledge of undergraduate quantum mechanics and statistical physics could be used to discuss the basic concepts and simple problems, and draw parallels between superconductivity and superfluidity.

Superconductivity and superfluidity are two phenomena where quantum mechanics, typically constrained to the microscopic realm, shows itself on the macroscopic level. Conceptually and mathematically, these phenomena are related very closely, and some results obtained for one can, with a few modifications, be immediately carried over to the other. However, the student of these notes should be aware of important differences between superconductivity and superfluidity that stem mainly from two facts: (1) electrons in a superconductor carry a charge, therefore one has to take into account interaction with electromagnetic radiation; (2) electrons move in a lattice, therefore phonons play a role not only a mediators of attractive interaction between pairs of electrons, but also as scatterers of charge carriers.

Although these are notes on superfluidity *and *superconductivity, and there are a few cross-references, the two subjects can be studied independently with, perhaps, a little extra work by the student to fill in the gaps resulting from such study. The material of Chapter 1 introduces the method of second quantisation that is commonly used to discuss systems with many interacting particles. It is then applied in Chaper 2 to treat the uniform weakly interacting Bose gas within the approach by N. Bogoliubov, and in Chapter 4 to formulate the theory of the uniform superconducting state put forth by J. Bardeen, L. Cooper and R. Schrieffer. Chapter 3 presents the theory proposed independently by E. Gross and L. Pitaevskii of a non-uniform weakly interacting Bose gas, with a discussion of vortices, rotation of the condensate, and the Bogoliubov equations. In Chapter 5 we discuss the Ginzburd-Landau theory of a non-uniform superconductor near the critical temperature and apply it to a few simple problems such as the surface energy of the boundary between a normal metal and a superconductor, critical current and critical magnetic field, and vortices.

We predict the effect of the roton instability for a two-dimensional *weakly* interacting gas of tilted dipoles in a single homogeneous quantum layer. Being typical for strongly correlated systems, the roton phenomena appear to occur in a weakly interacting gas. It is important that in contrast to a system of normal to the layer dipoles, breaking of the rotational symmetry for a system of tilted dipoles leads to the convergence of the condensate depletion even up to the threshold of the roton instability, with mean-field approach being valid. Predicted effects can be observed in a wide class of dipolar systems. We suggest observing predicted phenomena for systems of ultracold atoms and polar molecules in optical lattices, and estimate optimal experimental parameters.

The anisotropic superfluidity in a weakly interacting two‐dimensional Bose gas of photons in a dye‐filled optical microcavity is investigated, taking into account the dependence of the photon effective mass on the in‐plane coordinate. With the use of the generalized Gross–Pitaevskii equation and the Bogoliubov approach, it is shown that the modulation of the microcavity width leads to an effective periodic potential and the periodicity of the condensate wave function, and both the condensate energy and the spectrum of elementary excitations depend on the direction of motion. The anisotropic character of the dynamical and superfluid properties, such as helicity modulus, superfluid density, and sound velocity, as well as experimentally observable manifestations of their anisotropy are described.

Overview This book concisely presents the latest trends in the physics of superconductivity and superfluidity and magnetismin novel systems, as well as the problem of BCS-BEC crossover in ultracold quantum gases and high-Tc superconductors. It further illuminates the intensive exchange of ideas between these closely related fields of condensed matter physics over the last 30 years of their dynamic development. The content is based on the author’s original findings obtained at the Kapitza Institute, as well as advanced lecture courses he held at the Moscow Engineering Physical Institute, Amsterdam University, Loughborough University and LPTMS Orsay between 1994 and 2011. In addition to the findings of his group, the author discusses the most recent concepts in these fields, obtained both in Russia and in the West. The book consists of 16 chapters which are divided into four parts. The first part describes recent developments in superfluid hydrodynamics of quantum fluids and solids, including the fashionable subject of possible supersolidity in quantum crystals of 4He, while the second describes BCS-BEC crossover in quantum Fermi-Bose gases and mixtures, as well as in the underdoped states of cuprates. The third part is devoted to non-phonon mechanisms of superconductivity in unconventional (anomalous) superconductors, including some important aspects of the theory of high-Tc superconductivity. |The last part considers the anomalous normal state of novel superconductive materials and materials with colossal magnetoresistance (CMR). The book offers a valuable guide for senior-level undergraduate students and graduate students, postdoctoral and other researchers specializing in solid-state and low-temperature physics.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.