Exciton spectra in disordered quantum wells.

We study the effects of disorder on the exciton spectra in quantum well (QW) semiconductor structures. We model the disorder by introducing the fractional Laplacian into the Schrödinger equations, which describe the exciton spectra of the above QW structures. We calculate the exciton binding energies in its ground state and a few low-lying excited states as a function of the GaAs QW size.

Quantum chaotic features of the spin-orbit coupled excitons in disordered two-dimensional insulators.

We study the synergy between disorder (phenomenologically modeled by the introduction of Riesz fractional derivative in the corresponding Schrödinger equation) and spin-orbit coupling (SOC) on the exciton spectra in two-dimensional (2D) semiconductor structures. We demonstrate that the joint impact of "fractionality" and SOC considerably modifies the spectrum of corresponding "ordinary" (i.e.

Spin-orbit-coupled fractional oscillators and trapped Bose-Einstein condensates.

We study the ensemble of pseudo-spin 1/2 ultracold bosons, performing Lévy flights, confined in a parabolic potential. The (pseudo-) spin-orbit coupling (SOC) is additionally imposed on these particles. We consider the structure and dynamics of macroscopic pseudospin qubits based on Bose-Einstein condensates, obtained from the above "fractional" bosons.

Screened Coulomb interaction in insulators with strong disorder.

We study the effect of disorder on the excitons in a semiconductor with screened Coulomb interaction. Examples are polymeric semiconductors and/or van der Waals structures. In the screened hydrogenic problem, we consider the disorder phenomenologically using the so-called fractional Scrödinger equation.

1D solitons in cubic-quintic fractional nonlinear Schrödinger model.

We examine the properties of a soliton solution of the fractional Schrö dinger equation with cubic-quintic nonlinearity. Using analytical (variational) and numerical arguments, we have shown that the substitution of the ordinary Laplacian in the Schrödinger equation by its fractional counterpart with Lévy index [Formula: see text] permits to stabilize the soliton texture in the wide range of its parameters (nonlinearity coefficients and [Formula: see text]) values. Our studies of [Formula: see text] dependence ([Formula: see text] is soliton frequency and N its norm) permit to acquire the regions of existence and stability of the fractional soliton solution.

Fractional quantum oscillator and disorder in the vibrational spectra.

We study the role of disorder in the vibration spectra of molecules and atoms in solids. This disorder may be described phenomenologically by a fractional generalization of ordinary quantum-mechanical oscillator problem. To be specific, this is accomplished by the introduction of a so-called fractional Laplacian (Riesz fractional derivative) to the Scrödinger equation with three-dimensional (3D) quadratic potential.

Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity.

We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under "fractional" we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index [Formula: see text]. We speculate that the latter substitution corresponds to phenomenological account for disorder in a system.

The influence of Coulomb interaction screening on the excitons in disordered two-dimensional insulators.

We study the joint effect of disorder and Coulomb interaction screening on the exciton spectra in two-dimensional (2D) structures. These can be van der Waals structures or heterostructures of organic (polymeric) semiconductors as well as inorganic substances like transition metal dichalcogenides. We consider 2D screened hydrogenic problem with Rytova-Keldysh interaction by means of so-called fractional Scrödinger equation.

Lévy distributions and disorder in excitonic spectra.

We study analytically the spectrum of excitons in disordered semiconductors like transition metal dichalcogenides, which are important for photovoltaic and spintronic applications. We show that ambient disorder exerts a strong influence on the exciton spectra. For example, in such a case, the well-known degeneracy of the hydrogenic problem (related to Runge-Lenz vector conservation) is lifted so that the exciton energy starts to depend on both the principal quantum number n and orbital l.

The influence of disorder on the exciton spectra in two-dimensional structures.

We study the role of disorder in the exciton spectra in two-dimensional (2D) semiconductors. These can be heterostructures, thin films and multilayers (so-called van der Waals structures) of organometallic perovskites, transition metal dichalcogenides and other semiconductors for optoelectronic applications. We model the disorder by introduction of a fractional Laplacian (with Lévy index μ, defining the degree of disorder) to the Scrödinger equation with 2D Coulomb potential.

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights.

The fractional Laplacian (-Δ)^{α/2}, α∈(0,2), has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α-stable stochastic processes in R^{n}. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data-respecting fractional Laplacian should actually be. This ambiguity not only holds true for each specific choice of the process behavior at the boundary (e.

Chaotization of internal motion of excitons in ultrathin layers by spin-orbit coupling.

We show that Rashba spin-orbit coupling (SOC) can generate chaotic behavior of excitons in two-dimensional semiconductor structures. To model this chaos, we study a Kepler system with spin-orbit coupling and numerically obtain a transition to chaos at a sufficiently strong coupling. The chaos emerges since the SOC reduces the number of integrals of motion as compared to the number of degrees of freedom.

The influence of topological phase transition on the superfluid density of overdoped copper oxides.

We show that a quantum phase transition, generating flat bands and altering Fermi surface topology, is a primary reason for the exotic behavior of the overdoped high-temperature superconductors represented by LaSrCuO, whose superconductivity features differ from what is predicted by the classical Bardeen-Cooper-Schrieffer theory. This observation can open avenues for chemical preparation of high-T materials. We demonstrate that (1) at temperature T = 0, the superfluid density n turns out to be considerably smaller than the total electron density; (2) the critical temperature T is controlled by n rather than by doping, and is a linear function of the n; (3) at T > T the resistivity ρ(T) varies linearly with temperature, ρ(T) ∝ αT, where α diminishes with T → 0, whereas in the normal (non superconducting) region induced by overdoping, T = 0, and ρ(T) ∝ T.

Lévy flights in an infinite potential well as a hypersingular Fredholm problem.

We study Lévy flights with arbitrary index 0<μ≤2 inside a potential well of infinite depth. Such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrödinger equation into that for Fredholm integral equation with hypersingular kernel.

Transversal spin freezing and re-entrant spin glass phases in chemically disordered Fe-containing perovskite multiferroics.

We propose experimental verification and theoretical explanation of magnetic anomalies in the complex Fe-containing perovskite multiferroics like PbFe1/2Nb1/2O3 and PbFe1/2Ta1/2O3. The theoretical part is based on our model of coexistence of the long-range magnetic order and spin glass in the above compounds. In our model, the exchange interaction is anisotropic, coupling antiferromagnetically z spin components of Fe(3+) ions.

Two-dimensional electron gas at the LaAlO3/SrTiO3 inteface with a potential barrier.

We present a tight binding description of electronic properties of the interface between LaAlO3 (LAO) and SrTiO3 (STO). The description assumes LAO and STO perovskites as sets of atomic layers in the x-y plane, which are weakly coupled by an interlayer hopping term along the z axis. The interface is described by an additional potential, U0, which simulates a planar defect.

Lévy targeting and the principle of detailed balance.

We investigate confining mechanisms for Lévy flights under premises of the principle of detailed balance. In this case, the master equation of the jump-type process admits a transformation to the Lévy-Schrödinger semigroup dynamics akin to a mapping of the Fokker-Planck equation into the generalized diffusion equation. This sets a correspondence between above two stochastic dynamical systems, within which we address a (stochastic) targeting problem for an arbitrary stability index μ ε (0,2) of symmetric Lévy drivers.

Tetragonal tungsten bronze compounds: relaxor versus mixed ferroelectric-dipole glass behavior.

We demonstrate that recent experimental data (Castel et al 2009 J. Phys.: Condens.

Lévy flights in confining potentials.

We analyze confining mechanisms for Lévy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump-type processes are considered: those driven by Langevin equation with Lévy noise and those, named topological Lévy processes (occurring in systems with topological complexity such as folded polymers or complex networks), whose Langevin representation is unknown and possibly nonexistent.

Universal behavior of two-dimensional 3He at low temperatures.

On the example of two-dimensional (2D) 3He we demonstrate that the main universal features of its experimental temperature T-density x phase diagram [see Neumann, Nyéki, and Saunders, Science 317, 1356 (2007)10.1126/science.1143607] look like those in the heavy-fermion metals.

Domain-enhanced interlayer coupling in ferroelectric/paraelectric superlattices.

We investigate the ferroelectric phase transition and domain formation in a periodic superlattice consisting of alternate ferroelectric (FE) and paraelectric (PE) layers of nanometric thickness. We find that the polarization domains formed in the different FE layers can interact with each other via the PE layers. By coupling the electrostatic equations with those obtained by minimizing the Ginzburg-Landau functional, we calculate the critical temperature of transition Tc as a function of the FE/PE superlattice wavelength Lambda and quantitatively explain the recent experimental observation of a thickness dependence of the ferroelectric transition temperature in KTaO3/KNbO3 strained-layer superlattices.