The interplay of Anderson localization and electron-electron interactions is known to lead to enhancement of superconductivity due to multifractality of electron wave functions. We develop the theory of multifractally-enhanced superconducting states in two-dimensional systems in the presence of spin-orbit coupling. Using the Finkel'stein nonlinear sigma model, we derive the modified Usadel and gap equations that take into account renormalizations caused by the interplay of disorder and interactions. Multifractal correlations induce energy dependence of the superconducting spectral gap. We determine the superconducting transition temperature and the superconducting spectral gap in the case of Ising and strong spin orbit couplings. In the latter case the energy dependence of superconducting spectral gap is convex whereas in the former case (as well as in the absence of spin-orbit coupling) it is concave. Multifractality enhances not only the transition temperature but, in the same way, the spectral gap at zero temperature. Also we study mesoscopic fluctuations of the local density of states in the superconducting state. Similarly to the case of normal metal, spin-orbit coupling reduce the amplitude of fluctuations.
Results are reported in E.S. Andriyakhina, I.S. Burmistrov, ZhETF 162, 522 (2022).

The report is devoted to the problem of classifying integrable matrix systems of the Painlevé equations P_2-P_6 type. Using the Kovalevskaya-Painlevé test, all integrable matrix systems of type P_4 are found. All systems of types 2-6 are found that have matrix generalizations of the Okamoto Hamiltonians. All integrable matrix Hamiltonian systems of types P_2-P_6 are described.

Previously it was reported that universal spectrum of the inverse cascade of gravity waves on the surface of 3D fluid in the presence of condensate is different from the KZ-spectrum predicted by the Wave Turbelence Theory. Anflytical explanation of the difference in the spectrum slope is proposed. Interaction of short waves with a long wave backgrown (condensate) was considered in the framework of diffusion approximation for the Hasselmann waves kinetic equation. "Diffusion" coefficient in the wavenumbers space was obtained as a function of k in the presence of condensate. An obtained spectrum has a power -3.0 and is close to the one observed in the numerical experiment (power ~-3.1).

We develop a theory of anomalous elasticity in disordered two-dimensional flexible materials with orthorhombic crystal symmetry. Similar to the clean case, we predict existence of infinitely many flat phases with anisotropic bending rigidity and Young's modulus showing power-law scaling with momentum controlled by a single universal exponent the very same as in the clean isotropic case. With increase of temperature or disorder these flat phases undergo crumpling transition. Remarkably, in contrast to the isotropic materials where crumpling occurs in all spatial directions simultaneously, the anisotropic materials crumple into tubular phase. In distinction to clean case in which crumpling transition happens at unphysically high temperatures, a disorder-induced tubular crumpled phase can exist even at room-temperature conditions. Our results are applied to anisotropic atomic single layers doped by adatoms or disordered by heavy ions bombarding.

The following mathematical construction appears in many problems of mathematical physics including soliton solutions of the KP 2 equation and on-shell amplitude calculations for N=4 supersymmetric Yang-Mills. Consider a planar oriented graph in the upper half-plane with positive weights on edges, satisfying some additional conditions. Let the vectors on the boundary vertices - sources are expressed as sums along all paths from the boundary to the boundary. If the graph has non-trivial cycles, these sums are infinite, but these infinite sums can be expressed as rational expressions with positive denominators using Talaska formula. We show that a generalization of Talaska formula can be written for vectors at interlal edges of the graph.

In this work we consider the same graphs as in the previous talk. The vectors at intrernal graphs can be represented either as sums of all paths to the boundary or as solutions of some linear system of equation. This system of eqiations shall inculde some system signs (signature), which is sufficiently non-trivial. Lem proved that such system exists but provided no explicit formula. We suggesed an explicit rule for a signature and proved that, if the graph satisfy some natural additional condition (no dead ends), then the signature is unuque up to a natural gauge transformation.

Численно исследуется критическое поведение четырехкомпонентной модели Поттса на гексагональной решетке. Использован модифицированный метод Ванга-Ландау с контролем точности оценки плотности состояний. Конечномерный анализ полученных результатов подтверждает наличие фазового перехода второго рода с критическими показателями, соответствующими классу универсальности двумерной четырехкомпонентной модели Поттса.
Будет опубликовано в ЖЭТФ, том 162, выпуск 6

I will discuss how compact tetra and pentaquarks could be seen in the QQqq and QQqqq-quark systems.

We theoretically investigate asymmetric two-junction SQUIDs with different current-phase relations in the two Josephson junctions, involving higher Josephson harmonics. Our main focus is on the «minimal model» with one junction in the SQUID loop possessing the sinusoidal current-phase relation and the other one featuring additional second harmonic. The current-voltage characteristic (CVC) turns out to be asymmetric, I(−V) ≠ −I(V). The asymmetry is due to the presence of the second harmonic and depends on the magnetic flux through the interferometer loop, vanishing only at special values of the flux such as integer or half-integer in the units of the flux quantum. The system thus demonstrates the flux-tunable Josephson diode effect (JDE), the simplest manifestations of which is the direction dependence of the critical current. We analyze asymmetry of the overall I(V) shape both in the absence and in the presence of external ac irradiation. In the voltage-source case of external signal, the CVC demonstrates the Shapiro spikes. The integer spikes are asymmetric (manifestation of the JDE) while the half-integer spikes remain symmetric. In the current-source case, the CVC demonstrates the Shapiro steps. The JDE manifests itself in asymmetry of the overall CVC shape, including integer and half-integer steps.
arXiv:2208.10856

Scaling of various local observables with a system size at Anderson transition criticality is characterized by a generalized multifractality. We study the generalized multifractality in the spin quantum Hall symmetry class (class C) in the presence of interaction. We employ Finkel'stein nonlinear sigma model and construct the pure scaling derivativeless operators for class C. Within two-loop renormalization group analysis we compute the anomalous dimensions of these pure scaling operators and demonstrate that they are affected by the interaction. We find that the interaction breaks exact symmetry relations between generalized multifractal exponents known for a noninteracting problem.

The quantum fluctuations of the Dirac field in external classical gravitational and electromagnetic fields are studied. A self-consistent equation for torsion is calculated, which is obtained using one-loop fermion diagrams.
Class. Quantum Grav. 39, 155009 (2022); arXiv:2203.03625