# Statistical properties of two-particle transmission at Anderson transition

## TL;DR

This study investigates the multifractal statistical properties of two-particle transmission at the Anderson transition in power-law random banded matrices, revealing differences based on particle statistics and a common decay exponent for typical transmission.

## Contribution

It provides the first numerical analysis of two-particle transmission statistics at the Anderson transition, highlighting multifractality and differences between particle types.

## Key findings

- Transmission $T_2$ exhibits multifractal statistics.
- Fermions have a different multifractal spectrum than distinguishable particles and bosons.
- Typical two-particle transmission decays with an exponent smaller than twice the one-particle exponent.

## 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 $T_1(L)$.