arXiv:1105.2637 · Journal of Experimental and Theoretical Physics Letters

Two interacting particles in a random potential

Dmitry O. Krimer, Рамаз Хомерики, Sergej Flach A

Published Nov 1, 2011• 30 citations

Abstract

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.

DOI: 10.1134/s0021364011170097

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arXiv:1105.2637 · Journal of Experimental and Theoretical Physics Letters

Two interacting particles in a random potential

Dmitry O. Krimer, Рамаз Хомерики, Sergej Flach A

Published Nov 1, 2011• 30 citations

arXiv HTML PDF

Abstract

DOI: 10.1134/s0021364011170097

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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

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.

W,

and the strength of interaction,

U.

When

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

W

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 $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$

N=20\phantom{\rule{0.16em}{0ex}}000

) and disorder strengths (down to

W=0.5

). We vary the energy

E

and the on-site interaction strength

u

. At a given disorder strength, the largest enhancement of ${\ensuremath{\xi}}_{2}$ occurs for

u

of the order of the single-particle bandwidth and for two-particle states with energies at the center of the spectrum,

E=0

. We observe a crossover in the scaling of ${\ensuremath{\xi}}_{2}$ with the single-particle localization length ${\ensuremath{\xi}}_{1}$ into the asymptotic regime for ${\ensuremath{\xi}}_{1}>100$ (

W<1.0

). This happens due to the recovery of translational invariance and momentum conservation rules in the matrix elements of interconnected Fock eigenstates for

u=0

. The entrance into the asymptotic scaling is manifested through a nonlinear scaling function ${\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=F(u{\ensuremath{\xi}}_{1})$.