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arXiv:0808.1586 · Physical Review B

Glassy states in fermionic systems with strong disorder and interactions

David J. Schwab, Sudip Chakravarty

Published Mar 4, 2009• 8 citations
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Abstract

We study the competition between interactions and disorder in two dimensions. Whereas a noninteracting system is always Anderson localized by disorder in two dimensions, a pure system can develop a Mott gap for sufficiently strong interactions. Within a simple model, with short-ranged repulsive interactions, we show that, even in the limit of strong interaction, the Mott gap is completely washed out by disorder for an infinite system for dimensions $D\ensuremath{\le}2$, leading to a glassy state. Moreover, the Mott insulator cannot maintain a broken symmetry in the presence of disorder. We then show that the probability of a nonzero gap as a function of system size falls onto a universal curve, reflecting the glassy dynamics. An analytic calculation is also presented in one dimension that provides further insight into the nature of slow dynamics.

DOI
10.1103/physrevb.79.125102

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We study the competition between interactions and disorder in two dimensions. Whereas a noninteracting system is always Anderson localized by disorder in two dimensions, a pure system can develop a Mott gap for sufficiently strong interactions. Within a simple model, with short-ranged repulsive interactions, we show that, even in the limit of strong interaction, the Mott gap is completely washed out by disorder for an infinite system for dimensions D\ensuremath≤2D\ensuremath{\le}2D\ensuremath≤2, leading to a glassy state. Moreover, the Mott insulator cannot maintain a broken symmetry in the presence of disorder. We then show that the probability of a nonzero gap as a function of system size falls onto a universal curve, reflecting the glassy dynamics. An analytic calculation is also presented in one dimension that provides further insight into the nature of slow dynamics.
The competition between the Mott transition and the Anderson localization in one-dimensional electron systems is studied with the bosonization and renormalization-group methods. The $\ensuremath{\beta}$ function is calculated up to second order in the strength of diagonal disorder by using a replica trick. It is found that sufficiently strong forward scattering by random impurities destroys the Mott-Hubbard gap, and backward scattering gives rise to Anderson localization for the resulting gapless state. On the other hand, if the umklapp interaction is strong enough, the Mott insulating state still overwhelms the Anderson localization.
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The interplay between Anderson localization and Mott localization is considered in the vicinity of one dimension for a collection of interacting spinless fermions or repulsive bosons. We demonstrate that in the absence of commensurability effects an Anderson localization transition does take place for d\ensuremath{\le}1. It is characterized by a critical dynamical exponent z having value z=2-d for d\ensuremath{\rightarrow}1-0, in disagreement with the previous claims that z always equals d. We find that there is only a localized phase for d>1 and expect that this similarity between free fermions and interacting quantum particles will be valid up to two dimensions. We also consider the competition between the effects of commensurability, disorder, and interaction in exactly one dimension. The outcome strongly depends on the order of umklapp scattering responsible for the formation of the Mott insulator in the pure case. If it is \ensuremath{\Delta}k=2kF{\mathit{k}}_{\mathit{F}}kF​ scattering and the disorder is sufficiently weak, the Mott phase survives and the Anderson phase intervenes almost everywhere between the Mott insulator and the conductor, except at one special symmetry-enhanced point corresponding to the fixed commensurate filling: here, as in the pure case, the Mott transition belongs to the Kosterlitz-Thouless universality class. For sufficiently strong disorder, or for any disorder and higher-order scattering, the path from the Mott insulator phase (if it survives) to the conductor always crosses the Anderson phase.
We study the zero-temperature superfluid-insulator transition for a two-dimensional model of interacting, lattice bosons in the presence of quenched disorder and particle-hole symmetry. We follow the approach of a recent series of papers by Altman, Kafri, Polkovnikov, and Refael, in which the strong disorder renormalization group is used to study disordered bosons in one dimension. Adapting this method to two dimensions, we study several different species of disorder and uncover universal features of the superfluid-insulator transition. In particular, we locate an unstable finite disorder fixed point that governs the transition between the superfluid and a gapless, glassy insulator. We present numerical evidence that this glassy phase is the incompressible Mott glass and that the transition from this phase to the superfluid is driven by a percolation-type process. Finally, we provide estimates of the critical exponents governing this transition.
We study a system of fermions in one spatial dimension with linearly confining interactions and short-range disorder. We focus on the zero-temperature properties of this system, which we characterize using bosonization and the Gaussian variational method. We compute the static compressibility and ac conductivity, and thereby demonstrate that the system is incompressible, but exhibits gapless optical conductivity. This corresponds to a ``Mott glass'' state, distinct from an Anderson and a fully gapped Mott insulator, arising due to the interplay of disorder and charge confinement. We argue that this Mott glass phenomenology should persist to nonzero temperatures.
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.