We study the tunneling density of states (TDOS) of a disordered electronic system with Coulomb interaction on the insulating side of the Anderson localization transition. The average TDOS shows a critical behavior at high energies, with a crossover to a soft Coulomb gap $\ensuremath{\Delta}$ at low energies. We demonstrate that the single-particle excitations experience a localization transition (which belongs to the noninteracting universality class) at an energy $E=\ifmmode\pm\else\textpm\fi{}{E}_{c}$. The mobility edge ${E}_{c}$ scales with the distance ${\ensuremath{\mu}}_{c}\ensuremath{-}\ensuremath{\mu}$ from the interacting critical point according to ${E}_{c}\ensuremath{\propto}{({\ensuremath{\mu}}_{c}\ensuremath{-}\ensuremath{\mu})}^{\ensuremath{\nu}z}$, where $\ensuremath{\nu}$ and $z$ are the localization-length and the dynamical critical exponents. Local TDOS shows strong fluctuations and long-range correlations which reflect the multifractality associated with interacting and noninteracting fixed points as well the localization of low-energy excitations.
We study the tunneling density of states (TDOS) of a disordered electronic system with Coulomb interaction on the insulating side of the Anderson localization transition. The average TDOS shows a critical behavior at high energies, with a crossover to a soft Coulomb gap \ensuremathΔ at low energies. We demonstrate that the single-particle excitations experience a localization transition (which belongs to the noninteracting universality class) at an energy E=\ifmmode±\else\textpm\fiEc. The mobility edge Ec scales with the distance \ensuremathμc\ensuremath−\ensuremathμ from the interacting critical point according to Ec\ensuremath∝(\ensuremathμc\ensuremath−\ensuremathμ)\ensuremathνz, where \ensuremathν and z are the localization-length and the dynamical critical exponents. Local TDOS shows strong fluctuations and long-range correlations which reflect the multifractality associated with interacting and noninteracting fixed points as well the localization of low-energy excitations.
Interplay of electron correlation and randomness is studied by using the Anderson-Hubbard model within the Hartree-Fock approximation. Under the coexistence of short-range interaction and diagonal disorder, we obtain the ground-state phase diagram in three dimensions, which includes an antiferromagnetic insulator, an antiferromagnetic metal, a paramagnetic insulator (Anderson-localized insulator) and a paramagnetic metal. Although only the short-range interaction is present in this model, we find unconventional soft gaps in the insulating phases irrespective of electron filling, spatial dimensions and long-range order, where the single-particle density of states (DOS) vanishes with a power-law scaling in one dimension (1D) or even faster in two dimensions (2D) and three dimensions (3D) toward the Fermi energy. We call it soft Hubbard gap. Moreover, exact-diagonalization results in 1D support the formation of the soft Hubbard gap beyond the mean-field level. The formation of the soft Hubbard gap cannot be attributed to a conventional theory by Efros and Shklovskii (ES) owing the emergence of soft gaps to the long-range Coulomb interaction. Indeed, based on a picture of multivalley energy landscape, we propose a phenomenological scaling theory, which predicts a scaling of the DOS in perfect agreement with the numerical results. We further discuss a correction of the scaling of the DOS by the long-range part of the Coulomb interaction, which modifies the scaling of Efros and Shklovskii. Furthermore, explicit formulae for the temperature dependence of the DC resistivity via variable-range hopping under the influence of the soft gaps are derived. Finally, we compare the present theory with experimental results of SrRu_{1-x}Ti_xO_3.
We study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.
The interplay of interactions and disorder is studied using the Anderson-Hubbard model within the typical medium dynamical cluster approximation. Treating the interacting, nonlocal cluster self-energy $[{\mathrm{\ensuremath{\Sigma}}}_{c}[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}](i,j\ensuremath{\ne}i)]$ up to second order in the perturbation expansion of interactions, U2, with a systematic incorporation of nonlocal spatial correlations and diagonal disorder, we explore the initial effects of electron interactions (U) in three dimensions. We find that the critical disorder strength (WcU), 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 WcU and demonstrate that the mobility edge (${\ensuremath{\omega}}_{\ensuremath{\epsilon}}$) is preserved as long as the chemical potential, $\ensuremath{\mu}$, is at or beyond the mobility edge energy.
Abstract We review our recent results on Anderson localization in systems of two interacting particles coupled by contact interactions. Based on an exact mapping to an effective single-particle problem, we numerically investigate the occurrence of metal-insulator phase transitions for the pair in two-(2D) and three-dimensional (3D) disordered lattices. In two dimensions, we find that interactions cause an exponential enhancement of the pair localization length with respect to its single-particle counterpart, but do not induce a delocalization transition. In particular we show that previous claims of 2D interaction-induced Anderson transitions are the results of strong finite-size effects. In three dimensions we find that the pair undergoes a metal-insulator transition belonging to the same (orthogonal) universality class of the noninteracting model. We then explore the phase diagram in the space of energy E, disorder W and interaction strength U, which reveals a rich and counterintuitive structure, endowed with multiple metallic and insulating phases. We point out that this phenomenon originates from the molecular and scattering-like nature of the pair states available at given energy and disorder strength.
We study noninteracting systems with a power-law quasiparticle dispersion ${\ensuremath{\xi}}_{\mathbf{k}}\ensuremath{\propto}{k}^{\ensuremath{\alpha}}$ and a random short-range-correlated potential. We show that, unlike the case of lower dimensions, for $d>2\ensuremath{\alpha}$, there exists a critical disorder strength (set by the bandwidth), at which the system exhibits a disorder-driven quantum phase transition at the bottom of the band that lies in a universality class distinct from the Anderson transition. In contrast to the conventional wisdom, it manifests itself in, e.g., the disorder-averaged density of states. For systems in symmetry classes that permit localization, the striking signature of this transition is a nonanalytic behavior of the mobility edge, which is pinned to the bottom of the band for subcritical disorder and grows for disorder exceeding a critical strength. Focusing on the density of states, we calculate the critical behavior (exponents and scaling functions) at this novel transition, using a renormalization group, controlled by an $\ensuremath{\varepsilon}=d\ensuremath{-}2\ensuremath{\alpha}$ expansion. We also apply our analysis to Dirac materials, e.g., Weyl semimetals, where this transition takes place in physically interesting three dimensions.
Effects of electron correlations on Anderson insulators have been one of the central themes for recent two decades, suggesting that the Anderson insulating phase turns into a novel insulating state referred to as many body localization (MBL). However, the role of spin degrees of freedom in this dynamical phase transition still remains unclarified as a function of the interaction strength. In this study, we perform real-space spin-resolved Hartree-Fock-Anderson simulations to investigate metal-insulator transitions above a critical disorder strength in three spatial dimensions, where all single-particle states are Anderson-localized without interactions. Here, relatively weak correlations below the Mott regime are taken into account in the mean-field fashion but disorder effects are introduced essentially exactly. We find two types of single-particle mobility edges, where the multifractal spectrum of the interaction-driven low-energy mobility edge deviates from that of the high-energy one smoothly connected with the multifractal spectrum of the metal-insulator transition without interactions. We show that the weakly interacting insulating phase remains to be a paramagnetic Anderson insulating state up to the temperature of the order of the band width. On the other hand, we uncover that the relatively strongly interacting insulating phase still below the Mott regime is ferromagnetic, which turns into a ferromagnetic metallic state at a critical temperature much lower than the order of the bandwidth. Based on all these results, we propose a quantum phase transition from a paramagnetic Anderson insulating state to a ferromagnetic MBL insulating phase via an intermediate ferromagnetic metallic state, which intervenes between these two insulators at the Fermi energy.