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Quantum and wave systems of mathematical physics

Priority areas of development: mathematics
The project has been carried out as part of the HSE Program of Fundamental Studies.

Goal of research:

The first part of the research is aimed at studying the influence of Coulomb interaction on the Landau levels in graphene, to analyze the coherence conservation in a system of semiconducting quantum dots which are elements of quantum neural networks, to perform a group-theoretical analysis of the general form of pseudomagnetic and pseudoelectric fields arising in three-dimensional Dirac materials under deformations, to study of the vortex structure of the Bose condensate of excitonic polaritons with regard to the ``blue shift'' of the Rabi splitting, and to study the influence of the coordinate dependence of the effective photon mass and the force of photon-photon interaction on the properties of the Bose condensate in a trap.

The second part is aimed at studying the models of controlled formation of the random network architecture, which is preferable with respect to one of motifs (directed evolution). An increase in the number of small connected subgraphs can lead to a competition between the energy gain and the entropy loss and, as a result, the network can demonstrate a ``critical behavior''. The artificial construction of such networks allows one to determine which of the motifs is responsible for the formation of certain structure in the network.

The third part is aimed at studying the phenomenon of different scales arising in the problem of flow past a half-infinite plate with small imperfections on its surface and at studying the related problem of Rayleigh-type equation on large  times and the development and applications of the method of explicit description of interaction of nonlinear waves for investigating generalizes solutions of hyperbolic conservation laws.

In the fourth part, the goal is to study the resonance regimes arising for different values of the controlling parameters in planar traps and the corresponding algebras of integrals of motion, as well to reduce the anharmonic part of the operator by the quantum averaging method, and to investigate the possibility of origination the tunnel effect and bilocalization of states in such a system. Also we study the resonance regime in the model of a charge motion in a layer with quadratic confinement and magnetic field, namely, to reveal the hidden symmetry algebra (pseudospin) and to calculate the asymptotics of the spectrum and wave functions.


The following methods are used: the quantum multiparticle calculations in the static approximation of chaotic phases, the solving of the von Neumann equation numerically by the quasi-adiabatic integration by paths, the theory of spatial groups and the theory of small groups for three-dimensional crystals, the numerical solving the Gross-Pitaevskii equation for coupled exciton and photon condensates, the Thomas-Fermi approximation, the Kosterlitz-Thouless theory, probabilistic Monte Carlo methods, different algorithms on graphs, algorithms for searching communities in graphs (Newman's methds, spectral analysis), algorithms for storage and operation with large data massifs, and various algorithms for network visualization, the set of statistical physics methods including the generating function method, cluster analysis, methods developed for analyzing anomalous diffusion, theory of random matrices, and algorithms for propagating the excitation in networks, the quantum averaging, the algebraic and geometric methods: construction of resonance algebras of hyperbolic type, determination of the creation-annihilation structure in them, analysis of the topology of symplectic leaves of the corresponding Poisson algebras of integrals of motion, study of the phase portraits of reduced Hamiltonians on symplectic leaves, methods of quantum adiabatic approximation, the method of constructing the global semiclassical asymptotics by using coherent states of the algebra.

Empirical base of research:

The results of studies in the field of the project, which were published in the world scientific literature, were used as the empirical basis of the project.

Results of research

Numerical modeling was developed for scanning tunnel spectroscopy experiments and the key role of screening was revealed. The coherence conservation in a system of semiconductor quantum dots with dipole-dipole interaction. The general form of pseudomagnetic and pseudoelectric fields was found for three-dimensional systems, where the Dirac dots are formed due to the crystal symmetry. The influence of the ``blue shift'' of the Rabi splitting on the structure of vortexes in the Bose condensate of excitonic polaritons was stated. The effect of spatial dependence of the phonon system parameters on the structure of room-temperature Bose condensate of photons in a microcavity filled with a dye was analyzed.

A model of directed evolution of a random network resulting in the formation of a certain network structure was proposed. The topological and spectral properties of equilibrium states were investigated in a wide range of parameters. The analytic description was constructed for Erdoes-Renyi random networks.

The time scales arising in two- and three-deck structures were studied. The solutions of Rayleigh-type equation were estimated at large and superlarge times for the problem of flow past periodic imperfections on a plate. A qualitative difference between the flow past periodic imperfections and the flow past a localized (hump-type) imperfection was demonstrated. For a scalar hyperbolic conservation law the nonphysical solutions – the propagating unstable jumps - were studied. It was proved ``near'' them there are stable solutions.

The scale of physical parameters determining the planar Penning trap with rectangular ring electrode was analyzed. In the regime of basic hyperbolic resonance 3:(-1) between the modified cyclotron and magnetron frequencies, an explicit formula was obtained for the averages anharmonic part of the Hamiltonian in terms of the generators of the non-Lie algebra of symmetries. It was shown that there arise pairs of periodic trajectories on energy levels of the averaged Hamiltonian of the trap, and the tunnel splitting of the spectrum and bilocalized states arise for the corresponding quantum operator. The asymptotics of the splitting value was derived, and the geometric interpretation of the obtained formula was given via characteristics of the complexified Hamiltonian system.

It was discovered that, for the Schroedinger operator in a layer with a quadratic confinement and a homogeneous magnetic field, under a resonance between the Lorentz and the transverse frequencies, there arise a pseudospin su(2)-field, which remove the resonance degeneracy of the Landau levels. It was shown that, as the layer geometry deviated from the plane shape, there arises a longitudinal geometric current and a phase shift of the wave function which contains a hidden quantum number. A quantization rule was obtained for the quasiparticle motions along the layer, which contains an additional parity index related to the algebraic number of geometric poles and the Landau level number.

Level of implementation,  recommendations on implementation or outcomes of the implementation of the results

The predicted coherence conservation in dipole-coupled semiconducting quantum dots can be used in models of quantum neural networks on the basis of semiconductor nanostructures. The prediction about the form of pseudomagnetic fields in deformed Dirac materials  can be used to seek the materials suitable for valleytronics (valley electronics). The numerical analysis of quantum multiparticle effects in graphene in a strong magnetic field can be used for understanding the recent experiments on excitation of cyclotron transitions and magnetoplasmons in graphene and graphene disks. The results of analysis of the internal structure of vortexes in the Bose condensate of excitonic polaritons can be verified experimentally.

The model of directed evolution and the corresponding programs of the network clusterization and visualization can be used for studying complex structurized systems of various nature, taking into account their special features and giving an opportunity to improve the tuning of the model.

The revealed tunnel bistates in the nanotrap can be used in the design of perspective quantum nanodevices. The discovered tunnel bilocalized states in planar resonance nanotraps can find application in creation of qubits.

The discovered pseudospin of quasiparticles in a layer with magneto-dimensional resonance and the Aharonov-Bohm type effect for the corresponding phase are of interest in the design of "pseudospintronics'' models. The effect of magneto-dimensional resonance and pseudospin phase transfer in the geometric current can be used in the geometric nano-electronics, in the Hall electronics.



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Karasev M., Novikova E., Vybornyi E. Instantons via breaking geometric symmetry in hyperbolic traps // Mathematical notes. 2017. Vol. 102. No. 5-6. P. 776-786. doi
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Gaydukov R., Danilov V. Time scales in the multi-deck structure of the boundary layer in the case of periodic irregularities on the plate surface // Journal of Fluid Mechanics. 2017
Danilov V. Stepwise (Multiparticle) Approximations and Limiting Generalized Solution to Hyperbolic Conservation Law // Archive for Rational Mechanics and Analysis. 2017
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