Average Density Of States Research Articles

Disorder-induced topological quantum phase transitions in multigap Euler semimetals

We study the effect of disorder in systems having a nontrivial Euler class. As these recently proposed multigap topological phases come about by braiding non-Abelian charged band nodes residing between different bands to induce stable pairs within isolated band subspaces, novel properties may be expected. Namely, a modified stability and critical phases under the unbraiding to metals can arise when the disorder preserves the underlying C2T or PT symmetry on average. Employing elaborate numerical computations, we verify the robustness of associated topology by evaluating the changes in the average densities of states and conductivities for different types of disorders. Upon performing a scaling analysis around the corresponding quantum critical points, we retrieve a universality for the localization length exponent of ν=1.4±0.1 for Euler-protected phases, relating to two-dimensional percolation models. We generically find that quenched disorder drives Euler semimetals into critical metallic phases. Finally, we show that magnetic disorder can also induce topological transitions to quantum anomalous Hall plaquettes with local Chern numbers determined by the initial value of the Euler invariant. Published by the American Physical Society 2024

Treating disorder in introductory solid state physics

Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.

Magnetic‐Flux‐Induced Tunable Completely Flat Band and Topological Nearly Flat Band in Square Kagome Lattice

AbstractThe flat bands in a square kagome lattice containing square and triangle plaquettes, are respectively investigated, and the origin of the doubly degenerated completely flat bands and the corresponding compact localized states are elucidated. It is found that the introduction of external magnetic flux enriches the modulation parameters, making the system present many interesting results. In the case that the magnetic flux penetrates through each square plaquette, the tunable completely flat band has one more tunable parameter. And when the external magnetic flux penetrates through each triangle plaquette, excepting a completely flat band, the band dispersions of the system present the topological nearly flat band, which is very useful to realize the interesting fractional quantum Hall physics. The average density of states is also calculated to corroborate the completely FB generated by the highly localized eigenstates. Furthermore, the implementation of the square kagome lattice system based on the current photonic waveguide network technology is demonstrated. The scheme opens up a way to generate the tunable completely flat band and topological nearly flat band in square kagome lattice under multi‐parameter variable conditions.

Localization phenomena and electronic transport in irradiated Aubry–André–Harper systems

The role of light irradiation on electronic localization is critically investigated for the first time in a tight-binding lattice where site energies are modulated in the cosine form following the Aubry–André–Harper (AAH) model. The critical point of transition from delocalized-to-localized phase can be monitored selectively by regulating the light parameters that is extremely useful to have controlled electron transmission across the system. Starting with a strictly one-dimensional (1D) AAH chain, we extend our analysis considering a two-stranded ladder model which brings peculiar signatures in presence of irradiation. Unlike 1D system, AAH ladder exhibits a mixed phase (MP) zone where both extended and localized energy eigenstates co-exist. This is the fundamental requirement to have mobility edge in energy band spectrum. A mathematical description is given for decoupling the irradiated ladder into two effective 1D AAH chains. The underlying mechanism of getting a MP zone relies on the availability of two distinct critical points (CPs) of the decoupled chains, in presence of second-neighbor hopping between the two strands. Using a minimal coupling scheme the effect of light irradiation is incorporated following the Floquet–Bloch ansatz. The localization behaviors of different energy eigenstates are studied by calculating inverse participation ratio, and, are further explained in a more compact way by calculating two-terminal transmission probabilities together with average density of states. Finally, the decoupling procedure is extended for a more general multi-stranded AAH ladders where multiple CPs and thus multiple mobility edges are found. Our analysis may provide a new route of engineering localization properties in similar kind of other fascinating quasiperiodic systems.

Finite-size scaling of the density of states inside band gaps of ideal and disordered photonic crystals

We study the density of states (DOS) in band gaps of ideal and disordered three-dimensional photonic crystals of finite size. The ideal crystal is a diamond lattice of resonant point scatterers (atoms) whereas the disordered one is obtained from it by displacing the scatterers by random distances in random directions. We find that DOS inside a band gap of the ideal crystal decreases as the inverse of the crystal size. Disorder narrows the band gap and DOS exhibits enhanced fluctuations near the new band edges. However, the average DOS still exhibits the same scaling with the crystal size within the remaining band gap. A phenomenological explanation of this scaling suggests that it should hold for one- and two-dimensional photonic crystals as well.

Optical Properties of px + ipy Superconductors with Strong Impurities

We study effects of strong impurities on the optical properties of a chiral px + ipy superconductor with broken time-reversal symmetry. Potential impurities create subgap states in the superconductor’s spectrum similar to Yu–Shiba–Rusinov states on magnetic impurities in a conventional superconductor. We calculate the average density of states and find both the profile of the subgap impurity band due to localized states and modifications to the continuous spectrum above the gap. We also consider the anomalous Hall response of the superconductor at arbitrary frequencies and temperatures arising due to skew scattering on impurities. Hall conductivity is responsible for the polar Kerr effect and can be directly measured in an optical experiment. The frequency dependence of the Hall conductivity exhibits a rich structure corresponding to various possible transitions of quasiparticles between superconducting condensate, localized impurity states, and continuous spectrum above the gap.

Multifractality at the Weyl-semimetal–diffusive-metal transition for generic disorder

A Weyl semimetal is a three dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal-diffusive metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potentials. We also discuss the relation between the geometric fluctuations of critical wave functions and the broad distribution of the local density of states (DOS) at the transition. We derive a new scaling relation for the typical local DOS and argue that it holds for other disorder-driven transitions in which both the average and typical local DOS vanish on one side of the transition. As an illustration we apply it to the recently discussed unconventional quantum transition in disordered semiconductors with power-law dispersion relation near the band edge.

Short Note on the Density of States in 3D Weyl Semimetals.

The average density of states in a disordered three-dimensional Weyl system is discussed in the case of a continuous distribution of random scattering. Our results clearly indicate that the average density of states does not vanish, reflecting the absence of a critical point for a metal-insulator transition. This calculation supports recent suggestions of an avoided quantum critical point in the disordered three-dimensional Weyl semimetal. However, the effective density of states can be very small such that the saddle-approximation with a vanishing density of states might be valid for practical cases.

Effective tunnel conductance and effective ac conductivity of randomly strained graphene

We consider a single-layer graphene with high ripples, so that the pseudo-magnetic fields due to these ripples are strong. If the magnetic length corresponding to a typical pseudo-magnetic field is smaller than the ripple size, the resulting Landau levels are local. Then the effective properties of the macroscopic sample can be calculated by averaging the local properties over the distribution of ripples. We find that this averaging does not wash out the Landau quantization completely. Average density of states (DOS) contains a feature (inflection point) at energy corresponding to the first Landau level in a {\em typical} field. Moreover, the frequency dependence of the ac conductivity %while the average ac conductivity contains a maximum at a frequency corresponding to the first Landau level in a typical field. This nontrivial behavior of the effective characteristics of randomly strained graphene is a consequence of non-equidistance of the Landau levels in the Dirac spectrum.

Global Phase Diagram of a Dirty Weyl Liquid and Emergent Superuniversality

Pursuing complementary field-theoretic and numerical methods, we here paint the global phase diagram of a three-dimensional dirty Weyl system. The generalized Harris criterion, augmented by a perturbative renormalization-group (RG) analysis shows that weak disorder is an irrelevant perturbation at the Weyl semimetal(WSM)-insulator quantum critical point (QCP). But, a metallic phase sets in through a quantum phase transition (QPT) at strong disorder across a multicritical point (MCP). The field theoretic predictions for the correlation length exponent $\nu=2$ and dynamic scaling exponent $z=5/4$ at this MCP are in good agreement with the ones extracted numerically, yielding $\nu=1.98 \pm 0.10$ and $z=1.26 \pm 0.05$, from the scaling of the average density of states (DOS). Deep inside the WSM phase, generic disorder is also an irrelevant perturbation, while a metallic phase appears at strong disorder through a QPT. We here demonstrate that in the presence of generic, but strong disorder the WSM-metal QPT is ultimately always characterized by the exponents $\nu=1$ and $z=3/2$ (to one-loop order), originating from intra-node or chiral symmetric (e.g., regular and axial potential) disorder. We here anchor such emergent \emph{chiral superuniversality} through complementary RG calculations, controlled via $\epsilon$-expansions, and numerical analysis of average DOS across WSM-metal QPT. In addition, we also discuss a subsequent QPT (at even stronger disorder) of a Weyl metal into an Anderson insulator by numerically computing the typical DOS at zero energy. The scaling behavior of various physical observables, such as residue of quasiparticle pole, dynamic conductivity, specific heat, Gr$\ddot{\mbox{u}}$neisen ratio, inside various phases as well as across various QPTs in the global phase diagram of a dirty Weyl liquid are discussed.

Anomalous 125Te Nuclear Spin Relaxation Coincident with Charge Kondo Behavior in Superconducting Pb1−xTlxTe

We report a ^{125}Te-NMR study of single-crystalline Pb_{1-x}Tl_{x}Te for x= 0 (undoped non-superconducting "parent"), 0.35 at% (doped but on the borderline of superconductivity), and 1.0 at% (superconducting, with a critical temperature Tc ~ 1.0 K). The Knight shift in the normal state is enhanced as x increases, corresponding to an increase in the average density of states (DOS). The NMR line-width also increases significantly with increasing x, indicative of a strong spatial variation in the local DOS surrounding each Tl dopant. Remarkably, for the superconducting composition, the ^{125}Te nuclear spin relaxation rate (1/T1T) for Te ions that are close to the Tl dopants is unexpectedly enhanced in the normal state below a characteristic temperature of ~10 K. This temperature coincides with the temperature below which the normal state resistivity experiences an upturn. Such a simultaneous upturn in both the resistivity and (1/T1T) is highly anomalous, and puts Pb_{1-x}Tl_{x}Te in a distinct new class of doped semiconductors. These observations provide microscopic evidence for dynamical charge fluctuations in the absence of paramagnetism, and are consistent with expectations for charge Kondo behavior associated with the Tl dopant ions. In contrast, such anomalies were not detected in the non-superconducting samples (x=0 and 0.35 at%), suggesting a connection between dynamical valence fluctuations and the occurrence of superconductivity in Pb_{1-x}Tl_{x}Te.

Quantitative analytical theory for disordered nodal points

Disorder effects are especially pronounced around nodal points in linearly dispersing bandstructures as present in graphene or Weyl semimetals. Despite the enormous experimental and numerical progress, even a simple quantity like the average density of states cannot be assessed quantitatively by analytical means. We demonstrate how this important problem can be solved employing the functional renormalization group method and, for the two dimensional case, demonstrate excellent agreement with reference data from numerical simulations based on tight-binding models. In three dimensions our analytic results also improve drastically on existing approaches.

Subgap states in disordered superconductors with strong magnetic impurities

We study the density of states (DOS) in diffusive superconductors with pointlike magnetic impurities of arbitrary strength described by the Poissonian statistics. The mean-field theory predicts a nontrivial structure of the DOS with the continuum of quasiparticle states and (possibly) the impurity band. In this approximation, all the spectral edges are hard, marking distinct boundaries between spectral regions of finite and zero DOS. Considering instantons in the replica sigma-model technique, we calculate the average DOS beyond the mean-field level and determine the smearing of the spectral edges due interplay of fluctuations of potential and nonpotential disorder. The latter, represented by inhomogeneity in the concentration of magnetic impurities, affects the subgap DOS in two ways: via fluctuations of the pair-breaking strength and via induced fluctuations of the order parameter. In limiting cases, we reproduce previously reported results for the subgap DOS in disordered superconductors with strong magnetic impurities.

Multiple mobility edges in a 1D Aubry chain with Hubbard interaction in presence of electric field: Controlled electron transport

Electronic behavior of a 1D Aubry chain with Hubbard interaction is critically analyzed in presence of electric field. Multiple energy bands are generated as a result of Hubbard correlation and Aubry potential, and, within these bands localized states are developed under the application of electric field. Within a tight-binding framework we compute electronic transmission probability and average density of states using Green's function approach where the interaction parameter is treated under Hartree–Fock mean field scheme. From our analysis we find that selective transmission can be obtained by tuning injecting electron energy, and thus, the present model can be utilized as a controlled switching device.

Кінетика елементарних актів окисно-відновних реакцій на межі фаз «тверде тіло–електроліт»

Within the framework of the quantum-mechanical theory of elementary acts of non-adiabatic electrochemical reactions and using the model of isotropic spherically symmetric band, the calculation of the discharge current of ions at the “solid–electrolyte” interface has been carried out. It is shown that our results generalize the analogical formulae, which are available in the literature. The average densities of states in the valence band and in the conduction band of the solid electrode under the heterogeneous charge transfer are found.

Kinetics of the Elementary Act of Electrochemical Reactions at the Semiconductor–Electrolyte Solution Interface

Abstract In the framework of the quantum-mechanical theory of elementary act of non-adiabatic electrochemical reactions, it is carried out the calculation of the discharge current of ions at the semiconductor-electrolyte solution interface using the model of isotropic spherically symmetric band. It is shown that our results generalize the well-known formulae for the current density obtained by Dogonadze, Kuznetsov, and Chizmadzhev [R. R. Dogonadze, A. M. Kuznetsov, and Yu. A. Chizmadzhev, The kinetics of some heterogeneous reactions at semiconductor-electrolyte interface, Zhur. Fiz. Khim. 38, 1195 (1964)]. The average densities of states in the valence band and the conduction band of the semiconductor electrode in the heterogeneous charge transfer are found.

T c of disordered superconductors near the Anderson transition

According to the Anderson theorem, the critical temperature T c of a disordered superconductor is determined by the average density of states and does not change at the localization threshold. This statement is valid under assumption of a self-averaging order parameter, which can be violated in the strong localization region. Stimulating by statements on the essential increase of T c near the Anderson transition, we carried out the systematic investigation of possible violations of self-averaging. Strong deviations from the Anderson theorem are possible due to resonances at the quasi-discrete levels, resulting in localization of the order parameter at the atomic scale. This effect is determined by the properties of individual impurities and has no direct relation to the Anderson transition. In particular, we do not see any reasons to say on “fractal superconductivity” near the localization threshold.

Wegner Estimate and Anderson Localization for Random Magnetic Fields

We consider a two dimensional magnetic Schrödinger operator with a spatially stationary random magnetic field. We assume that the magnetic field has a positive lower bound and that it has Fourier modes on arbitrarily short scales. We prove the Wegner estimate at arbitrary energy, i.e. we show that the averaged density of states is finite throughout the whole spectrum. We also prove Anderson localization at the bottom of the spectrum.

Bogoliubov excitations of disordered Bose-Einstein condensates

We describe repulsively interacting Bose-Einstein condensates in spatially correlated disorder potentials of arbitrary dimension. The first effect of disorder is to deform the mean-field condensate. Secondly, the quantum excitation spectrum and condensate population are affected. By a saddle-point expansion of the many-body Hamiltonian around the deformed mean-field ground state, we derive the fundamental quadratic Hamiltonian of quantum fluctuations. Importantly, a basis is used such that excitations are orthogonal to the deformed condensate. Via Bogoliubov-Nambu perturbation theory, we compute the effective excitation dispersion, including mean free paths and localization lengths. Corrections to the speed of sound and average density of states are calculated, due to correlated disorder in arbitrary dimensions, extending to the case of weak lattice potentials.

The effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian

The average density of states (DoS) of the one-dimensional Dirac Hamiltonian with a random mass on a finite interval [0, L] is derived. Our method relies on the eigenvalue distributions (extreme value statistics problem) which are obtained explicitly. The well-known Dyson singularity is recovered above the crossover energy . Below ϵc we find a log-normal suppression of the average DoS .