Transport through nanostructures: Finite time vs. finite size

TL;DR

This paper investigates how finite size and finite measurement time affect the calculation of full counting statistics in quantum impurity systems, revealing universal corrections and methods for extrapolation to long-time limits.

Contribution

It demonstrates that the leading finite-time correction to the cumulant generating function scales as 1n t_m and is universally related to the steady state CGF, even with interactions.

Findings

Finite measurement time correction scales as 1n t_m.

Universal relation between correction and steady state CGF.

Numerical extrapolation matches Bethe-ansatz results.

Abstract

Numerical simulations and experiments on nanostructures out of equilibrium usually exhibit strong finite size and finite measuring time $t_m$ effects. We discuss how these affect the determination of the full counting statistics for a general quantum impurity problem. We find that, while there are many methods available to improve upon finite-size effects, any real-time simulation or experiment will still be subject to finite time effects: in short size matters, but time is limiting. We show that the leading correction to the cumulant generating function (CGF) at zero temperature for single-channel quantum impurity problems goes as $\ln t_m$ and is universally related to the steady state CGF itself for non-interacting systems. We then give detailed numerical evidence for the case of the self-dual interacting resonant level model that this relation survives the addition of interactions. This allows the extrapolation of finite measuring time in our numerics to the long-time limit, to excellent agreement with Bethe-ansatz results.