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arXiv:1903.06214 · Physical review. B./Physical review. B

Mobility edge of two interacting particles in three-dimensional random potentials

Filippo Stellin, Giuliano Orso

Published Jun 28, 2019• 5 citations
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Abstract

We investigate Anderson transitions for a system of two particles moving in a three-dimensional disordered lattice and subject to on-site (Hubbard) interactions of strength $U$. The two-body problem is exactly mapped into an effective single-particle equation for the center-of-mass motion, whose localization properties are studied numerically. We show that, for zero total energy of the pair, the transition occurs in a regime where all single-particle states are localized. In particular, the critical disorder strength exhibits a nonmonotonic behavior as a function of $|U|$, increasing sharply for weak interactions and converging to a finite value in the strong-coupling limit. Within our numerical accuracy, short-range interactions do not affect the universality class of the transition.

DOI
10.1103/physrevb.99.224209

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We investigate Anderson transitions for a system of two particles moving in a three-dimensional disordered lattice and subject to on-site (Hubbard) interactions of strength UUU. The two-body problem is exactly mapped into an effective single-particle equation for the center-of-mass motion, whose localization properties are studied numerically. We show that, for zero total energy of the pair, the transition occurs in a regime where all single-particle states are localized. In particular, the critical disorder strength exhibits a nonmonotonic behavior as a function of ∣U∣|U|∣U∣, increasing sharply for weak interactions and converging to a finite value in the strong-coupling limit. Within our numerical accuracy, short-range interactions do not affect the universality class of the transition.
We investigate Anderson localization of two particles moving in a two-dimensional (2D) disordered lattice and coupled by contact interactions. Based on transmission-amplitude calculations for relatively large strip-shaped grids, we find that all pair states are localized in lattices of infinite size. In particular, we show that previous claims of an interaction-induced mobility edge are biased by severe finite-size effects. The localization length of a pair with zero total energy exhibits a nonmonotonic behavior as a function of the interaction strength, characterized by an exponential enhancement in the weakly interacting regime. Our findings also suggest that the many-body mobility edge of the 2D Anderson-Hubbard model disappears in the zero-density limit, irrespective of the (bosonic or fermionic) quantum statistics of the particles.
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.
Most of our quantitative understanding of disorder-induced metal-insulator transitions comes from numerical studies of simple noninteracting tight-binding models, like the Anderson model in three dimensions.An important outstanding problem is the fate of the Anderson transition in the presence of additional Hubbard interactions of strength U between particles.Based on large-scale numerics, we compute the position of the mobility edge for a system of two identical bosons or two fermions with opposite spin components.The resulting phase diagram in the interaction-energy-disorder space possesses a remarkably rich and counterintuitive structure, with multiple metallic and insulating phases.We show that this phenomenon originates from the molecular or scatteringlike nature of the pair states available at given energy E and disorder strength W .The disorder-averaged density of states of the effective model for the pair is also investigated.Finally, we discuss the implications of our results for ongoing research on many-body localization.
Most of our quantitative understanding of disorder-induced metal-insulator transitions comes from numerical studies of simple noninteracting tight-binding models, like the Anderson model in three dimensions. An important outstanding problem is the fate of the Anderson transition in the presence of additional Hubbard interactions of strength UUU between particles. Based on large-scale numerics, we compute the position of the mobility edge for a system of two identical bosons or two fermions with opposite spin component. The resulting phase diagram in the interaction-energy-disorder space possesses a remarkably rich and counterintuitive structure, with multiple metallic and insulating phases. We show that this phenomenon originates from the molecular or scattering-like nature of the pair states available at given energy and disorder strength. The behavior of the disorder-averaged density of states of the effective model for the pair is also discussed.
We study the interplay of disorder localization and strong local interactions within the Anderson-Hubbard model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. For fermions, we find rich nonmonotonic behavior of the localization length ξ, particularly in two-dimensional systems, including an interaction-induced exponential enhancement of ξ for small and intermediate disorders and a strong reduction of ξ due to hopping suppression by strong interactions. In three dimensions, we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbardgapped phases. For small U there is re-entrant behavior from the Anderson localized phase to the metallic phase. For bosons, the unrestricted particle occupation number per lattice site yields a monotonic enhancement of ξ as a function of decreasing interaction, which we assume to persist until the superfluid Bose- Einstein condensate phase is entered. Besides, we study cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.
Using the generalized DMFT+Σ approach, we study the influence of disorder on single-particle properties of the normal phase and the superconducting transition temperature in the attractive Hubbard model. A wide range of attractive potentials U is studied, from the weak coupling region, where both the instability of the normal phase and superconductivity are well described by the BCS model, to the strong-coupling region, where the superconducting transition is due to Bose-Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures much higher than the superconducting transition temperature. We study two typical models of the conduction band with semi-elliptic and flat densities of states, respectively appropriate for three-dimensional and two-dimensional systems. For the semi-elliptic density of states, the disorder influence on all single-particle properties (e.g., density of states) is universal for an arbitrary strength of electronic correlations and disorder and is due to only the general disorder widening of the conduction band. In the case of a flat density of states, universality is absent in the general case, but still the disorder influence is mainly due to band widening, and the universal behavior is restored for large enough disorder. Using the combination of DMFT+Σ and Nozieres-Schmitt-Rink approximations, we study the disorder influence on the superconducting transition temperature T c for a range of characteristic values of U and disorder, including the BCS-BEC crossover region and the limit of strong-coupling. Disorder can either suppress T c (in the weak-coupling region) or significantly increase T c (in the strong-coupling region). However, in all cases, the generalized Anderson theorem is valid and all changes of the superconducting critical temperature are essentially due to only the general disorder widening of the conduction band.