Disordered Electron Systems Research Articles

Multifractal correlations of the local density of states in dirty superconducting films

Mesoscopic fluctuations of the local density of states encode multifractal correlations in disorderedelectron systems. We study fluctuations of the local density of states in a superconducting state of weakly disordered films. We perform numerical computations in the framework of the disordered attractive Hubbard model on two-dimensional square lattices. Our numerical results are explained by an analytical theory. The numerical data and the theory together form a coherent picture of multifractal correlations of local density of states in weakly disordered superconducting films.

Nonhydrodynamic initial conditions are not soon forgotten.

Solutions to hydrodynamic equations, which are used for a vast variety of physical problems, are assumed to be specified by boundary conditions and initial conditions on the hydrodynamic variables only. Initial values of other variables are assumed to be irrelevant for a hydrodynamic description. We show that this assumption is not correct because of the existence of long-time-tail effects that are ubiquitous in systems governed by hydrodynamic equations. We illustrate this breakdown of a hydrodynamic description by means of the simple example of diffusion in a disordered electron system.

Inhomogeneous Kondo destruction by RKKY correlations

The competition between the indirect exchange interaction (IEC) of magnetic impurities in metals and the Kondo effect gives rise to a rich quantum phase diagram, the Doniach Diagram (Doniach, 1977). A Kondo screened phase is separated from a spin ordered phase when the local exchange coupling J and the concentration of magnetic moments nM are varied. In disordered metals, both the Kondo temperature and the IEC are widely distributed due to the scattering of the conduction electrons from the impurity potential. Therefore, it is a question of fundamental importance, how this Doniach diagram is modified by the disorder, and if one can still identify separate phases. Recently, Nejati et al. (2017) investigated the effect of Ruderman–Kittel–Kasuya–Yosida (RKKY) correlations on the Kondo effect of two magnetic impurities, renormalizing the Kondo interaction based on the Bethe–Salpeter equation and performing the poor men’s renormalization group (RG) analysis with the RKKY-renormalized Kondo coupling. In the present study, we extend this theoretical framework, allowing for different Kondo temperatures of two RKKY-coupled magnetic impurities due to different local exchange couplings and density of states. As a result, we find that the smaller one of the two Kondo temperatures is suppressed more strongly by the RKKY interaction, thereby enhancing their initial inequality. In order to find out if this relevance of inequalities between Kondo temperatures modifies the distribution of the Kondo temperature in a system of a finite density of randomly distributed magnetic impurities, we present an extension of the RKKY coupled Kondo RG equations. We discuss the implication of these results for the interplay between Kondo coupling and RKKY interaction in disordered electron systems and the Doniach diagram in disordered electron systems.

Observation of a marginal Fermi glass

A long-standing open problem in condensed-matter physics is whether or not a strongly disordered interacting insulator can be mapped to a system of effectively non-interacting localized excitations. Using terahertz two-dimensional coherent spectroscopy, we investigate this issue in phosphorus-doped silicon, a classic example of a correlated disordered electron system in three dimensions. Despite the intrinsically disordered nature of these materials, we observe coherent excitations and strong photon echoes that provide us with a powerful method for the study of their decay processes. We extract the energy relaxation and decoherence rates close to the metal–insulator transition. We observe that both rates are linear in excitation frequency with a slope close to unity. The energy relaxation timescale counterintuitively increases with increasing temperature, and the coherence relaxation timescale has little temperature dependence below 25 K, but increases as the material is doped towards the metal–insulator transition. Here we argue that these features imply that the system behaves as a well-isolated electronic system on the timescales of interest, and relaxation is controlled by electron–electron interactions. Our observations constitute a distinct phenomenology, driven by the interplay of strong disorder and strong electron–electron interactions, which we dub the marginal Fermi glass.

Supersymmetry method for interacting chaotic and disordered systems: The Sachdev-Ye-Kitaev model

The nonlinear supermatrix $\sigma $-model is widely used to understand the physics of Anderson localization and the level statistics in noninteracting disordered electron systems. In contrast to the general belief that the supersymmetry method applies only to systems of noninteracting particles, we adopt this approach to the disorder averaging in the interacting models. In particular, we apply supersymmetry to study the Sachdev-Ye-Kitaev (SYK) model, where the disorder averaging has so far been performed only within the replica approach. We use a slightly modified, time-reversal invariant version of the SYK model and perform calculations in real-time. As a demonstration of how the supersymmetry method works, we derive saddle point equations. In the semiclassical limit, we show that the results are in agreement with those found using the replica technique. We also develop the formally exact superbosonized representation of the SYK model. In the latter, the supersymmetric theory of original fermions and their superpartner bosons is reformulated as a model of unconstrained collective excitations. We argue that the supersymmetry description of the model paves the way for precise calculations in SYK-like models used in condensed matter, gravity, and high energy physics.

The effect of superconducting fluctuations on the ac conductivity of a 2D electron system in the diffusive regime

We report a complete analytical expression for the one-loop correction to the ac conductivity σ(ω) of a disordered two-dimensional electron system in the diffusive regime. The obtained expression includes the weak localization and Altshuler–Aronov corrections as well as the corrections dueto superconducting fluctuations above superconducting transition temperature. The derived expression has no 1∕(iω) divergency in the static limit, ω→0, in agreement with general expectations for the normal state conductivity of a disordered electron system.

Finkel’stein Nonlinear Sigma Model: Interplay of Disorder and Interaction in 2D Electron Systems

In this paper, I briefly review recent theoretical results derived within the Finkel’stein nonlinear sigma model approach for description of two-dimensional interacting disordered electron systems. The examples include an electron system with two valleys, electrons in a double quantum well, electrons on the surface of a topological insulator, an electron system with superconducting correlations, and the integer quantum Hall effect.

Locally self-consistent embedding approach for disordered electronic systems

We present a new embedding scheme for the locally self-consistent method to study disordered electron systems. We test this method in a tight-binding basis and apply it to the single band Anderson model. The local interaction zone is used to efficiently compute the local Green's function of a supercell embeded into a local typical medium. We find a quick convergence as the size of the local interaction zone which reduces the computational costs as expected. This method captures the Anderson localization transition and accurately predicts the critical disorder strength. The present work opens the path towards the development of a typical medium embedding scheme for the $O(N)$ multiple scattering methods.

Quantum corrections to conductivity of disordered electrons due to inelastic scattering off magnetic impurities

We study the quantum corrections to the conductivity of the two-dimensional disordered interacting electron system in the diffusive regime due to inelastic scattering off rare magnetic impurities. We focus on the case of very different g-factors for electrons and magnetic impurities. Within the Born approximation for the inelastic scattering off magnetic impurities we find additional temperature-dependent corrections to the conductivity of the Altshuler-Aronov type. Our results demonstrate that the low temperature transport in interacting disordered electron systems with rare magnetic impurities is more interesting than it was commonly believed on the basis of treatment of magnetic impurity spins as classical ones.

Phase Diagrams of Three-Dimensional Anderson and Quantum Percolation Models Using Deep Three-Dimensional Convolutional Neural Network

The three-dimensional Anderson model is a well-studied model of disordered electron systems that shows the delocalization--localization transition. As in our previous papers on two- and three-dimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. {\bf 85}, 123706 (2016), {\bf 86}, 044708 (2017)], we used an image recognition algorithm based on a multilayered convolutional neural network. However, in contrast to previous papers in which 2D image recognition was used, we applied 3D image recognition to analyze entire 3D wave functions. We show that a full phase diagram of the disorder-energy plane is obtained once the 3D convolutional neural network has been trained at the band center. We further demonstrate that the full phase diagram for 3D quantum bond and site percolations can be drawn by training the 3D Anderson model at the band center.

Calculation of two-particle quantities in the typical medium dynamical cluster approximation

The mean-field theory for disordered electron systems without interaction is widely and successfully used to describe equilibrium properties of materials over the whole range of disorder strengths. However, it fails to take into account the effects of quantum coherence and information of localization. Vertex corrections due to multiple back-scatterings may drive the electrical conductivity to zero and make expansions around the mean field in strong disorder problematic. Here, we present a method for the calculation of two-particle quantities which enable us to characterize the metal-insulator transitions (MIT) in disordered electron systems by using the Typical Medium Dynamical Cluster Approximation (TMDCA). We show how to include vertex corrections and the information about the mobility edge to the typical mean-field theory. We successfully demonstrate the application of the developed method by showing that the conductivity formulated in this way properly characterizes the MIT in disordered systems.

Glassy Dynamics in Disordered Electronic Systems Reveal Striking Thermal Memory Effects.

Memory is one of the unique qualities of a glassy system. The relaxation of a glass to equilibrium contains information on the sample's excitation history, an effect often refer to as "aging." We demonstrate that under the right conditions a glass can also possess a different type of memory. We study the conductance relaxation of electron glasses that are fabricated at low temperatures. Remarkably, the dynamics are found to depend not only on the ambient measurement temperature but also on the maximum temperature to which the system was exposed. Hence the system "remembers" its highest temperature. This effect may be qualitatively understood in terms of energy barriers and local minima in configuration space and therefore may be a general property of the glass state.

Transmission resonances anomaly in one-dimensional disordered quantum systems

Connections between the electron eigenstates and conductivity of one-dimensional disordered electron systems is studied in the framework of the tight-binding model. We show that for weak disorder only part of the states exhibit resonant transmission and contribute to the conductivity. The rest of the eigenvalues are not associated with peaks in transmission and the amplitudes of their wave functions do not exhibit a significant maxima within the sample. Moreover, unlike ordinary states, the lifetimes of these `hidden' modes either remain constant or even decrease (depending on the coupling with the leads) as the disorder becomes stronger. In a wide range of the disorder strengths, the averaged ratio of the number of transmission peaks to the total number of the eigenstates is independent of the degree of disorder and is close to the value $\sqrt{2/5}$, which was derived analytically in the weak-scattering approximation. These results are in perfect analogy to the spectral and transport properties of light in one-dimensional randomly inhomogeneous media, which provides strong grounds to believe that the existence of hidden, non-conducting modes is a general phenomenon inherent to 1D open random systems, and their fraction of the total density of states is the same for quantum particles and classical waves.

Ground-state energy of spin polarized two-dimensional electron gas in the presence of impurity scattering

In the present work we study the spin polarization effects on the ground-state energy of an interacting two-dimensional electron gas in the presence of disorder scattering (impurity scattering) at arbitrarily spin polarization for various electron densities. For this purpose we employ the modified density-density correlation function at small values of wave-vector and frequency in the framework of linear response theory to calculate the exchange, correlation and ground-state energy for a spin polarized two-dimensional disordered electron gas. Our results show that the effect of spin polarization on the exchange, correlation and ground-state energy becomes considerable at high densities. We also conclude that the two-dimensional disordered electron system becomes unstable by increasing the spin polarization degree.

Metal-insulator transition in a weakly interacting disordered electron system

The interplay of interactions and disorder is studied using the Anderson-Hubbard model within the typical medium dynamical cluster approximation. Treating the interacting, non-local cluster self-energy ($\Sigma_c[{\cal \tilde{G}}](i,j\neq i)$) up to second order in the perturbation expansion of interactions, $U^2$, with a systematic incorporation of non-local spatial correlations and diagonal disorder, we explore the initial effects of electron interactions ($U$) in three dimensions. We find that the critical disorder strength ($W_c^U$), required to localize all states, increases with increasing $U$; implying that the metallic phase is stabilized by interactions. Using our results, we predict a soft pseudogap at the intermediate $W$ close to $W_c^U$ and demonstrate that the mobility edge ($\omega_\epsilon$) is preserved as long as the chemical potential, $\mu$, is at or beyond the mobility edge energy.

Two-loop renormalization of the Finkel’stein theory: The specific heat

We explore the two-loop renormalization of the specific heat for an interacting disordered electron system in the case of broken time reversal symmetry. Within the nonlinear sigma model approach we derive the two-loop result for the anomalous dimension which controls scaling of the specific heat with temperature. As an example, we elaborate the metal–insulator transition in d=2+ϵ dimensions for the case of broken time reversal and spin rotational symmetries and in the presence of Coulomb interaction. In this situation scaling of the specific heat is determined by the anomalous dimension of the Finkel’stein operator which is the eigenoperator of the renormalization group complementary to the eigenoperator corresponding to the second moment of the local density of states. We find that the absolute values of the anomalous dimensions of these operators differ beyond one-loop approximation contrary to the noninteracting case.

Magnetic quantum phase diagram of magnetic impurities in two-dimensional disordered electron systems

The quantum phase diagram of disordered electron systems as function of the concentration of magnetic impurities nm and the local exchange coupling J is studied in the dilute limit. We take into account the Anderson localisation of the electrons by a nonperturbative numerical treatment of the disorder potential. The competition between RKKY interaction JRKKY and the Kondo effect, as governed by the temperature scale TK, is known to gives rise to a rich magnetic quantum phase diagram, the Doniach diagram. Our numerical calculations show that in a disordered system both the Kondo temperature TK and JRKKY are widely distributed. Accordingly, also their ratio, JRKKY /TK is widely distributed as shown in Fig. 1 (a). However, we find a sharp cutoff of that distribution, which allows us to define a critical density of magnetic impurities nc below which Kondo screening wins at all sites of the system above a critical coupling Jc, forming the Kondo phase[see Fig. 1 (b)]. As disorder is increased, Jc increases and a spin coupled phase is found to grow at the expense of the Kondo phase. From these distribution functions we derive the magnetic susceptibility which show anomalous power law behavior. In the Kondo phase that power is determined by the wide distribution of the Kondo temperature, while in the spin coupled phase it is governed by the distribution of JRKKY. At low densities and small J < Jc we identify a paramagnetic phase. We also report results on a honeycomb lattice, graphene, where we find that the spin coupled phase is more stable against Kondo screening, but is more easily destroyed by disorder into a PM phase.

The beginnings of theoretical condensed matter physics in Rome: a personal remembrance

This oral history interview provides a personal view on how theoretical condensed matter physics developed in Rome starting in the sixties of the last century. It then follows along the lines of research pursued by the interviewee up to the date of the interview, in March 2006. The topics considered range from the phenomenology of superfluid helium and superconductors, critical phenomena and renormalisation group approach, quantum fluids to strongly correlated electron systems and high temperature superconductors. Within these topics, fundamental problems of condensed matter physics are touched upon, such as the microscopic derivation of scaling, the metal-insulator transition and the interaction effects on disordered electron systems beyond the Anderson localisation, and the existence of heterogeneous states in cuprates.

Vertex corrections to the mean-field electrical conductivity in disordered electron systems

The mean-field theory for noninteracting disordered electron systems is widely and successfully used to describe equilibrium properties of alloys over the whole range of disorder strengths. However, it fails to take into account the effects of quantum coherence and localizing backscattering effects when applied to transport phenomena. Vertex corrections due to multiple backscatterings may turn the electrical conductivity negative and make expansions around the mean field in strong disorder problematic. We show how to stabilize such an expansion with the inverse of the number of nearest neighbors on hypercubic lattices as a small parameter and how to include vertex corrections to the mean-field approximation in such a way that the conductivity remains non-negative in all disorder regimes.

Anderson localization versus charge-density-wave formation in disordered electron systems

We study the interplay of disorder and interaction effects including bosonic degrees of freedom in the framework of a generic one-dimensional transport model: the Anderson-Edwards model. Using the density-matrix-renormalization group technique, we extract the localization length and the renormalization of the Tomonaga-Luttinger-liquid parameter from the charge-structure factor by a elaborate sample-average finite-size scaling procedure. The properties of the Anderson localized state can be described in terms of scaling relations of the metallic phase without disorder. We analyze how disorder competes with the charge-density-wave correlations triggered by the bosons and give evidence that disorder will destroy the long-range charge-ordered state.