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. Under "fractional" we understand the substitution of the ordinary second derivative (kinetic energy term) in the Gross-Pitaevskii equation by a so-called fractional Laplacian, characterized by the Lévy index µ. We show that the joint action of interparticle interaction, SOC, and Zeeman splitting in a synthetic magnetic field makes the dynamics of corresponding qubit highly nontrivial with evident chaotic features at both strong interactions and Lévy indices µâ†’1 when the Lévy trajectories of bosons with long jumps dominated over those derived from ordinary Gaussian distribution, corresponding to µ=2. Using analytical and numerical arguments, we discuss the possibilities to control the above qubit using the synergy of SOC, interaction strength, and "fractionality," characterized by the Lévy index µ.

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. Our main finding is that joint action of screening and disorder either destroys the exciton (strong screening) or enhances the bounding of electron and hole in an exciton, leading to its collapse in the extreme case. Latter effects may also be related to the quantum manifestations of chaotic exciton behavior in the above semiconductor structures. Hence, they should be considered in device applications, where the interplay between dielectric screening and disorder is important. Our theoretical results permit one to predict the various excitonic properties in semiconductor samples with different degrees of disorder and Coulomb interaction screenings.

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. For that we use famous Vakhitov-Kolokolov (VK) criterion. The variational results are confirmed by numerical solution of a one-dimensional cubic-quintic nonlinear Schrödinger equation. Direct numerical simulations of the linear stability problem of soliton texture gives the same soliton stability boundary as within variational method. Thus we confirm that simple variational approach combined with VK criterion gives reliable information about soliton structure and stability in our model. Our results may be relevant to both optical solitons and Bose-Einstein condensates in cold atomic gases.

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. To solve the obtained 3D spectral problem, we pass to the momentum space, where the problem simplifies greatly as fractional Laplacian becomes simply [Formula: see text], k is a modulus of the momentum vector and [Formula: see text] is Lévy index, characterizing the degree of disorder. In this case, [Formula: see text] corresponds to the strongest disorder, while [Formula: see text] to the weakest so that the case [Formula: see text] corresponds to "ordinary" (i.e. that without fractional derivatives) 3D quantum harmonic oscillator. Combining analytical (variational) and numerical methods, we have shown that in the fractional (disordered) 3D oscillator problem, the famous orbital momentum degeneracy is lifted so that its energy starts to depend on orbital quantum number l. These features can have a strong impact on the physical properties of many solids, ranging from multiferroics to oxide heterostructures, which, in turn, are usable in modern microelectronic devices.

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. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index [Formula: see text]), the soliton is unstable, even infinitesimal difference [Formula: see text] from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of [Formula: see text] dependence ([Formula: see text] is soliton frequency and N its mass) show (within the famous Vakhitov-Kolokolov criterion) the stability of our soliton texture in the fractional [Formula: see text] case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at [Formula: see text], which is in accord with existing literature data. These results may be relevant to both Bose-Einstein condensates in cold atomic gases and optical solitons in the disordered media.

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. Our main finding is that above synergy between screening and disorder either destroys the exciton (strong screening) or promote the creation of a bound state, leading to its collapse in the extreme case. Our second finding is energy levels crossing, i.e. the degeneracy (with respect to index [Formula: see text]) of the exciton eigenenergies at certain discrete value of screening radius. Latter effects may also be related to the quantum manifestations of chaotic exciton behavior in above 2D semiconductor structures. Hence, they should be considered in device applications, where the interplay between dielectric screening and disorder is important.

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. We model the disorder phenomenologically substituting the ordinary Laplacian in the corresponding Schrödinger equation by the fractional one with Lévy index µ, characterizing the degree of disorder. Our variational treatment (corroborated by numerical results) shows that an exciton exists for 1 < µ≤ 2. The case µ = 2 corresponds to the "ordered" hydrogenic problem, while in the opposite case µ = 1 the exciton collapses. The exciton spectrum is dominated by the sample areas with moderate disorder. Our theory permits controlled predictions to be made of the excitonic properties in semiconductor samples with different degrees of disorder.

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. Combining analytical and numerical methods, we observe that the exciton exists only for µ > 1, while the point µ = 1 (strongest disorder) corresponds to the exciton collapse. We show also that in the fractional (disordered, corresponding to 1 < µ < 2; µ = 2 corresponds to the ordered case) 2D hydrogenic problem, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number n but also on orbital m. These features can have a profound influence on the lifetime of optically generated excitons in the above 2D semiconductor structures. They should be taken into account while designing the photovoltaic cells, nanolasers and optical spintronics devices, where 2D excitons play a significant role.

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. Dynamically, the dependence of the exciton energy on the spin orientation in the presence of SOC produces an anomalous spin-dependent velocity resulting in chaotic motion. We observe numerically the critical dependence of the dynamics on the initial conditions, where the system can return to and exit a stability domain through very small changes in the initial spin orientation. This chaos can have a strong influence on the lifetime of optically injected carriers in semiconductors and organometallic perovskites. Hence, this effect should be taken into account while designing structures for photovoltaic and optical spintronics applications, where excitons play a significant role.

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 La2-xSrxCuO4, 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-Tc materials. We demonstrate that (1) at temperature T = 0, the superfluid density ns turns out to be considerably smaller than the total electron density; (2) the critical temperature Tc is controlled by ns rather than by doping, and is a linear function of the ns; (3) at T > Tc the resistivity ρ(T) varies linearly with temperature, ρ(T) ∝ αT, where α diminishes with Tc → 0, whereas in the normal (non superconducting) region induced by overdoping, Tc = 0, and ρ(T) ∝ T2. Our results are in good agreement with recent experimental observations.

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. At the same time, the xy components are coupled by much weaker exchange interaction of ferromagnetic sign. In the system with spatial disorder (half of the corresponding lattice sites are occupied by spinless Nb(5+) ions) such frustrating interaction results in the fact that the antiferromagnetic order is formed by the z projection of the spins, while their xy components contribute to spin glass behaviour. Our theoretical findings are supported by the experimental evidence of such a coexistence of antiferromagnetic and spin glass phases in chemically disordered Fe-containing complex perovskite multiferroics.

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. Physically, the interfacial potential can result from either a mechanical stress at the interface or other structural imperfections. We show that depending on the potential strength, charge carriers (electrons or holes) may form an energy band which is localized at the interface and is within the band gaps of the constitutent materials (LAO and STO). Moreover, our description predicts a valve effect at a certain critical potential strength, U0cr, when the interface potential works as a valve suppressing the interfacial conductivity. In other words, the interfacial electrons become dispersionless at U0 = U0cr, and thus cannot propagate. This critical value separates the quasielectron (U0 < U0cr) and quasihole (U0 > U0cr) regimes of the interfacial conductivity.

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

We demonstrate that recent experimental data (Castel et al 2009 J. Phys.: Condens. Matter 21 452201) on the tungsten bronze compound (TBC) Ba(2)Pr(x)Nd(1-x)FeNb(4)O(15) can be well explained in our model predicting a crossover from ferroelectric (x = 0) to orientational (dipole) glass (x = 1), rather then relaxor, behavior. We show that, since a 'classical' perovskite relaxor like Pb(Mn(1/3)Nb(2/3))O(3) is never a ferroelectric, the presence of ferroelectric hysteresis loops in the TBC shows that this substance actually transits from ferroelectric to orientational glass phase with x growth. To describe the above crossover theoretically, we use the simple replica-symmetric solution for the disordered Ising model.

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. Our comprehensive theoretical analysis of the experimental situation in 2D 3He allows us to propose a simple expression for the effective mass M*(T,x), describing all the diverse experimental facts in 2D 3He in a unified manner and demonstrating that the universal behavior of M*(T,x) coincides with that observed in 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.