From quantum disorder to quantum chaos

We study the statistics of wave functions in a ballistic chaotic system. The statistical ensemble is generated by adding weak smooth random potential, which allows us to apply the ballistic $\sigma$-model approach. We analyze conditions of applicability of the $\sigma$-model, emphasizing the role played by the single-particle mean free path and the Lyapunov exponent due to the random potential. In particular, we present a resolution of the puzzle of repetitions of periodic orbits counted differently by the $\sigma$-model and by the trace formula.Comment: 15 pages, no figures. Contribution to a special issue of J. Low Temp. Phys. dedicated to Peter Woelfle's 60th birthda

Interaction-induced magnetoresistance in a two-dimensional electron gas

We study the interaction-induced quantum correction \delta\sigma_{\alpha\beta} to the conductivity tensor of electrons in two dimensions for arbitrary T\tau (where T is the temperature and \tau the transport scattering time), magnetic field, and type of disorder. A general theory is developed, allowing us to express \delta\sigma_{\alpha\beta} in terms of classical propagators (``ballistic diffusons''). The formalism is used to calculate the interaction contribution to the longitudinal and the Hall resistivities in a transverse magnetic field in the whole range of temperature from the diffusive (T\tau 1) regime, both in smooth disorder and in the presence of short-range scatterers. Further, we apply the formalism to anisotropic systems and demonstrate that the interaction induces novel quantum oscillations in the resistivity of lateral superlattices.Comment: 35 pages, 14 figure

Low-temperature spin Coulomb drag in a two-dimensional electron gas

The phenomenon of low-temperature spin Coulomb drag in a two-dimensional electron gas is investigated. The spin transresistivity coefficient is essentially enhanced in the diffusive regime, as compared to conventional predictions. The origin of this enhancement is the quantum coherence of spin-up and spin-down electrons propagating in the same random impurity potential and coupled via the Coulomb interaction. A comprehensive analysis of spin and interlayer Coulomb drag effects is presented.Comment: 5 pages, 4 figure

Anomalous Hooke's law in disordered graphene

The discovery of graphene, a single monolayer of graphite, has closed the discussion on stability of 2D crystals. Although thermal fluctuations of such crystals tend to destroy the long-range order in the system, the crystal can be stabilized by strong anharmonicity effects. This competition is the central issue of the crumpling transition, i.e., a transition between flat and crumpled phases. We show that anharmonicity-controlled fluctuations of a graphene membrane around equilibrium flat phase lead to unusual elastic properties. In particular, we demonstrate that stretching $\xi$ of a flake of graphene is a nonlinear function of the applied tension at small tension: ${\xi\propto\sigma^{\eta/(2-\eta)}}$ and ${\xi\propto\sigma^{\eta/(8-\eta)}}$ for clean and strongly disordered graphene, respectively. Conventional linear Hooke's law, ${\xi\propto\sigma}$ is realized at sufficiently large tensions: ${\sigma\gg\sigma_*},$ where $\sigma_*$ depends both on temperature and on the disorder strength.Comment: 10 pages, 3 figure

Mesoscopic fluctuations of the local density of states in interacting electron systems

We review our recent theoretical results for mesoscopic fluctuations of the local density of states in the presence of electron-electron interaction. We focus on the two specific cases: (i) a vicinity of interacting critical point corresponding to Anderson-Mott transition, and (ii) a vicinity of non-interacting critical point in the presence of a weak electron-electron attraction. In both cases strong mesoscopic fluctuations of the local density of states exist.Comment: A brief review based on arXiv:1305.2888, arXiv:1307.5811, arXiv:1412.3306, arXiv:1603.0301

Multifractality at Anderson transitions with Coulomb interaction

We explore mesoscopic fluctuations and correlations of the local density of states (LDOS) near localization transition in a disordered interacting electronic system. It is shown that the LDOS multifractality survives in the presence of Coulomb interaction. We calculate the spectrum of multifractal dimensions in $2+\epsilon$ spatial dimensions and show that it differs from that in the absence of interaction. The multifractal character of fluctuations and correlations of the LDOS can be studied experimentally by scanning tunneling microscopy of two-dimensional and three-dimensional disordered structures.Comment: 16 pages, 2 figure

Local density of states and its mesoscopic fluctuations near the transition to a superconducting state in disordered systems

We develop a theory of the local density of states (LDOS) of disordered superconductors, employing the non-linear sigma-model formalism and the renormalization-group framework. The theory takes into account the interplay of disorder and interaction couplings in all channels, treating the systems with short-range and Coulomb interactions on equal footing. We explore 2D systems that would be Anderson insulators in the absence of interaction and 2D or 3D systems that undergo Anderson transition in the absence of interaction. We evaluate both the average tunneling density of states and its mesoscopic fluctuations which are related to the LDOS multifractality in normal disordered systems. The obtained average LDOS shows a pronounced depletion around the Fermi energy, both in the metallic phase (i.e., above the superconducting critical temperature $T_c$) and in the insulating phase near the superconductor-insulator transition (SIT). The fluctuations of the LDOS are found to be particularly strong for the case of short-range interactions -- especially, in the regime when $T_c$ is enhanced by Anderson localization. On the other hand, the long-range Coulomb repulsion reduces the mesoscopic LDOS fluctuations. However, also in a model with Coulomb interaction, the fluctuations become strong when the systems approaches the SIT

Superconductor-insulator transitions: Phase diagram and magnetoresistance

Influence of disorder-induced Anderson localization and of electron-electron interaction on superconductivity in two-dimensional systems is explored. We determine the superconducting transition temperature $T_c$, the temperature dependence of the resistivity, the phase diagram, as well as the magnetoresistance. The analysis is based on the renormalization group (RG) for a nonlinear sigma model. Derived RG equations are valid to the lowest order in disorder but for arbitrary electron-electron interaction strength in particle-hole and Cooper channels. Systems with preserved and broken spin-rotational symmetry are considered, both with short-range and with long-range (Coulomb) interaction. In the cases of short-range interaction, we identify parameter regions where the superconductivity is enhanced by localization effects. Our RG analysis indicates that the superconductor-insulator transition is controlled by a fixed point with a resistivity $R_c$ of the order of the quantum resistance $R_q = h/ 4e^2$. When a transverse magnetic field is applied, we find a strong nonmonotonous magnetoresistance for temperatures below $T_c$.Comment: 34 pages, 20 figure

Weak antilocalization in two-dimensional systems with large Rashba splitting

We develop the theory of quantum transport and magnetoconductivity for two-dimensional electrons with an arbitrary large (even exceeding the Fermi energy), linear-in-momentum Rashba or Dresselhaus spin-orbit splitting. For short-range disorder potential, we derive the analytical expression for the quantum conductivity correction, which accounts for interference processes with an arbitrary number of scattering events and is valid beyond the diffusion approximation. We demonstrate that the zero-field conductivity correction is given by the sum of the universal logarithmic "diffusive" term and a "ballistic" term. The latter is temperature independent and encodes information about spectrum properties. This information can be extracted experimentally by measuring the conductivity correction at different temperatures and electron concentrations. We calculate the quantum correction in the whole range of classically weak magnetic fields and find that the magnetoconductivity is negative both in the diffusive and in the ballistic regimes, for an arbitrary relation between the Fermi energy and the spin-orbit splitting. We also demonstrate that the magnetoconductivity changes with the Fermi energy when the Fermi level is above the "Dirac point" and does not depend on the Fermi energy when it goes below this point.Comment: 12 pages, 5 figures and Online Supplemental Material (15 pages, 7 figures

Transversal magnetoresistance and Shubnikov-de Haas oscillations in Weyl semimetals

We explore theoretically the magnetoresistance of Weyl semimetals in transversal magnetic fields away from charge neutrality. The analysis within the self-consistent Born approximation is done for the two different models of disorder: (i) short-range impurties and (ii) charged (Coulomb) impurities. For these models of disorder, we calculate the conductivity away from charge neutrality point as well as the Hall conductivity, and analyze the transversal magnetoresistance (TMR) and Shubnikov-de Haas oscillations for both types of disorder. We further consider a model with Weyl nodes shifted in energy with respect to each other (as found in various materials) with the chemical potential corresponding to the total charge neutrality. In the experimentally most relevant case of Coulomb impurities, we find in this model a large TMR in a broad range of quantizing magnetic fields. More specifically, in the ultra-quantum limit, where only the zeroth Landau level is effective, the TMR is linear in magnetic field. In the regime of moderate (but still quantizing) magnetic fields, where the higher Landau levels are relevant, the rapidly growing TMR is supplemented by strong Shubnikov-de Haas oscillations, consistent with experimental observations