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arXiv:cond-mat/9702241 · physica status solidi (b)

Two Interacting Particles in a Random Potential: Numerical Calculations of the Interaction Matrix Elements

Rudolf A. Römer, Michael Schreiber, Thomas Vojta

Published Feb 1, 1999• 11 citations
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Abstract

We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. This leads to a much weaker enhancement than proposed earlier. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by these data.

DOI
10.1002/(sici)1521-3951(199902)211:2<681::aid-pssb681>3.0.co;2-i

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We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. This leads to a much weaker enhancement than proposed earlier. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by these data.
We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we first study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. We further show that even for models where the dependency of the coupling matrix element on the single-particle localization length is correctly described by the corresponding random matrix model its predictions for the localization length can be qualitatively incorrect. These results indicate that the mapping of an interacting random system onto an effective random matrix model is potentially dangerous. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by the above arguments.
We study two interacting particles in a random potential chain by means of the transfer matrix method. The dependence of the two-particle localization length λ2λ_2λ2​ on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of λ2λ_2λ2​ as compared to the results for single particles is entirely due to the finite size of the systems considered. This is shown for a Hubbard-like onsite interaction and also a long-range interaction.
We study the scaling of the localization length of two interacting bosons in a one-dimensional random lattice with the single particle localization length. We consider the short-range interaction assuming that the particles interact when located both on the same site. We discuss several regimes, among them one interesting weak Fock space disorder regime. In this regime we obtain a weak logarithmic scaling law. Numerical benchmark data support the absence of any strong enhancement of the two particle localization length.
We investigate the localization of two interacting particles in a one-dimensional random potential. Our definition of the two-particle localization length, \ensuremath{\xi}, is the same as that of von Oppen et al. [Phys. Rev. Lett. 76, 491 (1996)]. \ensuremath{\xi}'s for chains of finite lengths are calculated numerically using the recursive Green's function method for several values of the strength of the disorder, W,W,W, and the strength of interaction, U.U.U. When U=0,U=0,U=0, \ensuremath{\xi} approaches a value larger than half the single-particle localization length as the system size tends to infinity and behaves as $\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}{\ensuremath{\nu}}_{0}}$ for small WWW with ${\ensuremath{\nu}}_{0}=2.1\ifmmode\pm\else\textpm\fi{}0.1.$ When $U\ensuremath{\ne}0,$ we use the finite size scaling ansatz and find the relation $\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=2.9\ifmmode\pm\else\textpm\fi{}0.2.$ Moreover, data show the scaling behavior $\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}{\ensuremath{\nu}}_{0}}g(b|U|{/W}^{\ensuremath{\Delta}})$ with $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$
In a recent letter [Phys. Rev. Lett. 78, 515 (1997); cond-mat/9612034] Roemer and Schreiber report on numerical calculations that led them to conclude that the previously observed enhancement of the localization length L2L_2L2​ of two interacting particles (TIP) vanishes in the thermodynamic limit. We point out that such a claim (i) is in conflict with the scaling theory of localization, (ii) ignores a consistent picture from a wealth of published numerical data and analytical investigations, and (iii) directly contradicts new numerical results obtained with a Green function method that is well adapted to the problem under study. The claim of Roemer and Schreiber is based on extrapolating from L<360 to infinity. Our data clearly shows the two particle localization length L2L_2L2​ to be significantly enhanced and independent of system size between L=100 and L=1000, as expected on physical grounds.
We study two interacting particles in a random potential chain by means of the transfer matrix method. The dependence of the two-particle localization length ${\ensuremath{\lambda}}_{2}$ on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of ${\ensuremath{\lambda}}_{2}$ as compared to the results for single particles is entirely due to the finite size of the systems considered. This is shown for a Hubbard-like on site interaction and also a long-range interaction.