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arXiv:1504.06636 · Physical Review B

Kohn's localization in disordered fermionic systems with and without interactions

Vipin Kerala Varma, Sebastiano Pilati

Published Oct 15, 2015• 20 citations
arXivHTMLPDF

Abstract

Understanding the metal-insulator transition in disordered many-fermion systems, both with and without interactions, is one of the most challenging and consequential problems in condensed matter physics. In this paper, we address this issue from the perspective of the modern theory of the insulating state (MTIS), which has already proven to be effective for band and Mott insulators in clean systems. First, we consider noninteracting systems with different types of aperiodic external potentials: uncorrelated disorder (one-dimensional Anderson model), deterministic disorder (Aubry-Andr\'e Hamiltonian and its modification including next-nearest-neighbor hopping), and disorder with long-range correlations (self-affine potential). We show how the many-body localization tensor defined within the MTIS may be used as a powerful probe to discriminate the insulating and the metallic phases, and to locate the transition point. Then, we investigate the effect of weak repulsive interactions in the Aubry-Andr\'e Hamiltonian, a model which describes a recent cold-atoms experiment. By treating the weak interactions within a mean-field approximation we obtain a linear shift of the transition point towards stronger disorder, providing evidence for delocalization induced by interactions.

DOI
10.1103/physrevb.92.134207

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Understanding the metal-insulator transition in disordered many-fermion systems, both with and without interactions, is one of the most challenging and consequential problems in condensed matter physics. In this paper, we address this issue from the perspective of the modern theory of the insulating state (MTIS), which has already proven to be effective for band and Mott insulators in clean systems. First, we consider noninteracting systems with different types of aperiodic external potentials: uncorrelated disorder (one-dimensional Anderson model), deterministic disorder (Aubry-Andr'e Hamiltonian and its modification including next-nearest-neighbor hopping), and disorder with long-range correlations (self-affine potential). We show how the many-body localization tensor defined within the MTIS may be used as a powerful probe to discriminate the insulating and the metallic phases, and to locate the transition point. Then, we investigate the effect of weak repulsive interactions in the Aubry-Andr'e Hamiltonian, a model which describes a recent cold-atoms experiment. By treating the weak interactions within a mean-field approximation we obtain a linear shift of the transition point towards stronger disorder, providing evidence for delocalization induced by interactions.
Interacting fermions in the presence of disorder pose one of the most challenging problems in condensed matter physics, primarily due to the absence of accurate numerical tools. Our investigation delves into the intricate interplay between interaction-induced Mott insulation and disorder-driven Anderson localization in the Hubbard model subjected to a random potential. On the Cayley tree, the application of statistical dynamical mean-field theory proves adept at discerning among a metal and the two distinct insulators, Anderson or Mott. Our comprehensive analysis, accounting for subtle yet potent finite-size effects and fluctuations, yields a noteworthy finding: in the presence of disorder, we consistently observe an intervening Anderson-localized regime between the metallic and Mott insulator states. This observation intriguingly mirrors scenarios witnessed in dirty Bosons, where an insulating Bose glass phase consistently emerges between the superfluid and Mott phases.
The interplay between Mott and Anderson routes to localization in disordered interacting systems gives rise to different transitions and transport regimes. Here, we investigate the phase diagram at finite temperatures using dynamical mean-field theory combined with typical medium theory, which is an effective theory of the Mott-Anderson metal-insulator transition. We mainly focus on the properties of the coexistence region associated with the Mott phase transition. For weak disorder, the coexistence region is found to be similar to that in the clean case. However, as we increase disorder, Anderson localization effects are responsible for shrinking the coexistence region, and at sufficiently strong disorder (approximately equal to twice the bare bandwidth) it drastically narrows, the critical temperature Tc{T}_{c}Tc​ abruptly goes to zero, and we observe a phase transition in the absence of a coexistence of the metallic and insulating phases. In this regime, the effects of interaction and disorder are found to be of comparable importance for charge localization.
Real materials always contain, to some extent, randomness in the form of defects or irregularities. It is known since the seminal work of Anderson that randomness can drive a metallic phase to an insulating one, and the mechanism responsible for this transition is intrinsically different from the one of the interaction-induced transitions discovered by Mott. Lattice Hamiltonians, with their conceptual and computational advantages, permitted to investigate broadly the interplay of both mechanisms. However, a clear understanding of the differences (or not) with their real-space counterparts is lacking, especially in the presence of long-range Coulomb interactions. This work aims at shedding light on this challenging question by investigating a real-space one-dimensional model of interacting electrons in the presence of a disordered potential. The transition between delocalized and localized phases are characterized using two different indicators, namely the single-particle occupation entropy and the position-space information entropy. In addition, the performance of density functional approximations to reproduce the exact ground-state densities of this many-body localization model are gauged.
A metal can be driven to an insulating phase through distinct mechanisms. A possible way is via the Coulomb interaction, which then defines the Mott metal-insulator transition (MIT). Another possibility is the MIT driven by disorder, the so-called Anderson MIT. Here we analyze interacting particles in disordered Hubbard chains---thus comprising the Mott-Anderson physics---by investigating the ground-state entanglement with density functional theory. The localization signature on entanglement is found to be a decreasing on the entanglement degree in comparison to the clean (without interaction and disorder) case, with local minima at certain critical densities. Individually, the Mott (Anderson) MIT has a single critical density whose minimum entanglement decreases as the interaction (disorder) enhances. While in the Mott MIT entanglement saturates at finite values, characterizing partial localization, in the Anderson MIT the system reaches full localization, with zero entanglement, for sufficiently strong disorder. In the combined Mott-Anderson MIT, we find three critical densities referring to local minima on entanglement. One of them is the same as for the Anderson MIT, but now the presence of interaction requires a stronger disorder potential to induce full localization. A second critical density is related to the Mott MIT, but due to disorder it is displaced by a factor proportional to the concentration of impurities. The third local minimum on entanglement is unique to the concomitant presence of disorder and interaction, found to be related to an effective density phenomenon, thus referred to as a Mott-like MIT. Since entanglement has been intrinsically connected to the magnetic susceptibility---a quantity promptly available in cold-atom experiments---our detailed numerical description might be useful for the experimental investigation of the Mott-Anderson MIT.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.
We investigate the zero-temperature metal-insulator transition in a one-dimensional two-component Fermi gas in the presence of a quasiperiodic potential resulting from the superposition of two optical lattices of equal intensity but incommensurate periods. A mobility edge separating (low-energy) Anderson localized and (high-energy) extended single-particle states appears in this continuous-space model beyond a critical intensity of the quasiperiodic potential. To discern the metallic phase from the insulating phase in the interacting many-fermion system, we employ unbiased quantum Monte Carlo (QMC) simulations combined with the many-particle localization length familiar from the modern theory of the insulating state. In the noninteracting limit, the critical optical-lattice intensity for the metal-insulator transition predicted by the QMC simulations coincides with the Anderson localization transition of the single-particle eigenstates. We show that weak repulsive interactions induce a shift of this critical point towards larger intensities, meaning that repulsion favors metallic behavior. This shift appears to be linear in the interaction parameter, suggesting that even infinitesimal interactions can affect the position of the critical point.