Statistical properties of two-particle transmission at Anderson transition

Abstract

The ensemble of $L\times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/|i-j|{)}^a$, is known to present an Anderson localization transition at $a=1$, where one-particle eigenfunctions are multifractal. Here we study numerically, at this critical point, the statistical properties of the transmission $T_2$ for two distinguishable particles, two bosons or two fermions. We find that the statistics of $T_2$ is multifractal, i.e. the probability to have $T_2(L)\sim 1/L^{\kappa }$ behaves as $L^{{\Phi }_2(\kappa )}$, where the multifractal spectrum ${\Phi }_2(\kappa )$ for fermions is different from the common multifractal spectrum concerning distinguishable particles and bosons. However in the three cases, the typical transmission $T_2^{typ}(L)$ is governed by the same exponent ${\kappa }_2^{typ}$, which is much smaller than the naive expectation $2{\kappa }_1^{typ}$, where ${\kappa }_1^{typ}$ is the typical exponent of the one-particle transmission $T_1(L)$.

9 pages, 4 figures

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