General localization lengths for two interacting particles in a disordered chain

Abstract

The propagation of an interacting particle pair in a disordered chain is characterized by a set of localization lengths which we define. The localization lengths are computed by a new decimation algorithm and provide a more comprehensive picture of the two-particle propagation. We find that the interaction delocalizes predominantly the center-of-mass motion of the pair and use our approach to propose a consistent interpretation of the discrepancies between previous numerical results. PACS number(s): 72.15. Rn, 71.30.+h The problem of interacting electrons in a disordered potential is one of the important unsolved problems in condensed matter physics. This has been emphasized again by the recent observation [1] of a metal-insulator transition in two dimensional (2d) systems which was theoretically unanticipated. Some time ago, Shepelyansky [2] proposed that it would be worthwhile to consider the simple case of two interacting particles in a random potential. He predicted that unexpectedly, such a particle pair could propagate coherently over distances ξ 2 much larger than the single-particle localization length ξ 1 as long as the two particles are within ξ 1 from each other.Specifically, Shepelyansky obtained for the twoparticle localization lengthwhere U denotes the interaction strength and W the disorder strength. Since ξ 1 ∼ 1/W 2 , Eq. (1) implies an enhancement of the localization length for weak disorder. Shepelyansky's original argument involved several uncontrolled assumptions for the single-particle eigenstates. This led to a number of (mostly numerical) attempts [3][4][5][6][7] to study the problem of two interacting particles more rigorously. Recently, Römer and Schreiber [7] concluded from the TMM that the enhancement effect does not exist. In view of this claim and of the quantitatively different expressions for ξ 2 quoted above, it appears that there are few secured results in this field. Our purpose in this paper is to present a more comprehensive picture of the twoparticle propagation by defining and computing a set of localization lengths. We unambiguously show that the effect exists and propose a resolution of the controversy in the previous works [7,10].It is currently not clear whether these ideas have any relevance to the degenerate finite-density Fermi gas. It appears to be the most promising direction to consider the localization properties of quasiparticle pairs. There have been a number of studies [3,8,9] whether quasiparticle excitations delocalize relative to single-particle ones. While a numerical study for a one-dimensional (1d) system showed delocalization only for unrealistically high excitation energy of the pair (of order of the bandwidth) [8], both arguments [3] and numerical studies [9] in higher dimensions suggest the possibility of a new pair mobility edge close to the ground state.The two-particle problem in one-dimension is described by the Hamiltonianwhere m labels the N sites of the 1d lattice and H 1 is the usual single-particle Anderson Hamiltonianǫ m is a random si...

Two interacting electrons in a disorder potential: localization properties

“…Numerical results in one dimension [6], although in line with the sign-independence, do not fully support the U 2 dependence predicted analytically. Moreover, the effect is numerically found to be strongest when the two electrons move at short distance [9], a feature which is also not covered by the analytical expression. Finally, the result (1) has either been derived heuristically [2], or it is based on statistical assumptions [8] which, although plausible, are not obviously valid.…”

“…The numerical work of Ref. [9] in one dimension has clearly demonstrated that the change of the localization length caused by the two-body interaction is biggest when the two electrons which move through the one-dimensional wire, keep the shortest distance from each other. This is our motivation for the constraint artificially imposed in our model.…”

“…As a result, the effect is now well established. In addition, Song and von Oppen [9] have demonstrated that it depends strongly on the distance between the two electrons and is strongest when these remain close together.…”

“…This surprising result was rederived by Imry [2] who used Thouless' block scaling picture. Further numerical work [3,4,5,6,7,8,9] in one dimension aimed at a better understanding and quantification of these findings, and at settling a controversy which had arisen. As a result, the effect is now well established.…”

“…Motivated by the numerical results of Ref. [9], we reduce Hilbert space and keep only states where the two electrons stay reasonably close together. This leads to a non-linear sigma model.…”