Scaling in Interaction-Assisted Coherent Transport

Abstract

The pair localization length L2 of two interacting electrons in one-dimensional disordered systems is studied numerically. Using two direct approaches, we find L2 ∝ L α 1 , where L1 is the one-electron localization length and α ≈ 1.65. This demonstrates the enhancement effect proposed by Shepelyansky, but the value of α differs from previous estimates (α = 2) in the disorder range considered. We explain this discrepancy using a scaling picture recently introduced by Imry and taking into account a more accurate distribution than previously assumed for the overlap of one-electron wavefunctions.PACS numbers: 71.30, 72.15.R Very recently, Shepelyansky [1] considered the problem of two interacting electrons in a random potential, defined by the Schrödinger equation, U characterizes the on-site interaction and V i is a random potential distributed uniformly in the interval [−W, W ]. The indices n 1 and n 2 denote the positions of the first and the second electron, respectively. Shepelyansky proposed that, as a consequence of the interaction U , certain eigenstates extend over a range L 2 much larger than the one-particle localization length L 1 ∝ W −2 . A key quantity in the derivation of this spectacular result in one dimension is the matrix representation U of the Hubbard interaction in the disorder diagonal basis of localized oneelectron eigenstates. With R n,i the amplitude at site n of the one-particle eigenstate with energy E i we have U ij,lm = U Q ij,lm with Q ij,lm = n R n,i R n,j R n,l R n,m .(The Q ij,lm vanish unless all four eigenstates are roughly localized within the same box of size L 1 . Assuming that R n,i ∝ a n / √ L 1 inside the box, with a n a random number of order unity, and neglecting correlations among the a n at different sites n, one finds Q ≈ 1/L 3/2 1 for a typical nonvanishing matrix element. In [1], this estimate was adopted and used to reduce the original problem to a certain band matrix model, eventually giving L 2 ∝ L 2 1 . Later, Imry [2] employed the Thouless scaling block picture to reinforce, interpret and generalize this result. The key step in this approach involves the pair conductance g 2 = (U Q/δ 2 ) 2 , where δ 2 is the two-particle level spacing in a block of size L 1 and Q is evaluated between adjacent blocks. Using Q ≈ 1/L 3/2 1 as before, Imry finds that g 2 ≈ 1 on the scale L 2 ∝ L 2 1 , in agreement with Shepelyansky. As a second important result both approaches predict that the effect does not depend on the sign of U .In this letter, we confirm the enhancement effect by studying both the original model (1) for finite size samples and an infinite "bag model" with medium-range interaction. However, we find L 2 ∝ L α 1 with α ≈ 1.65 instead of α = 2 in both cases. Moreover, the sign of U is not entirely irrelevant. We suggest that the small value for α is due to a very peculiar distribution of the Q ij,lm .

Sparse-random-matrix configurations for two or three interacting electrons in a random potential

“…This conclusion was confirmed and extended to higher dimensions via Thouless block scaling picture by Imry [2]. The subsequent numerical studies [3,4,5,6,7,8,9,10] verified the main qualitative results concerning the presence of Shepelyansky states, mostly by supressing single particle transport via efficient Green function or bag model methods which examine pair propagation [8].The deviations from the predicted behavior of the two-particle localization length ξ found were usually attributed to the oversimplified statistical assumptions concerning the band random matrix model of the original Shepelyansky construction.However, there is an ongoing debate whether coherent pair propagation actually exists for two interacting electrons in infinite disordered systems [11,12], which began by a recent transfer matrix study where no propagation enhancement is found at E = 0 for an infinite chain [11]. Moreover, it was pointed out that the reduction to a SBRM relies on questionable assumptions regarding chaoticity of the non-interacting electron localized states within ξ 1 , so that the relevant matrix model could be prob-2 ably different [13,14].…”

Two interacting particles at a metal-insulator transition

“…is the wave vector of slice n, H ⊥ is the SP hopping term for the second (m) particle (corresponding to the transverse direction) and (χ n ) i,m = [µ n + µ m + U n,m ]δ i,m codes the QP potential and the interaction [8]. Note that in this approach the symmetry of the wave function remains unspecified and we cannot distinguish between boson and fermion statistics.…”

“…(6) only N = M times will not give convergence. Frahm et al [8] have solved this problem in their TMM study by exploiting the Hermiticity of the product matrix…”

“…This raises serious questions about the validity of the TMM approach to TIP and in fact it has been argued very recently [18], that the TMM approach of Ref. [8,11] may systematically underestimate the localization length of a pair state, since it automatically measures a mixture of localization lengths originating also from unpaired states. Thus in this work, we will use the TIP-TMM not as a tool to extract information about the pair states only, but rather aim at describing the general influence of the presence of one particle onto the transport properties of the other.…”

Interacting particles at a metal-insulator transition

“…This behavior is contrary to the localization of non-interacting particles in a onedimensional random potential, where the ground state wave function becomes localized for infinite small disorder strength. An ingeneous theoretical approach to the interplay of interactions and disorder is based on the twointeracting-particles (TIP) problem in one-dimensional random potential [8][9][10][11]. Furthermore, numerical results for spinless fermions in a random potential at finite particle density have given additional insight [12,13].…”