# Keldysh Derivation of Oguri's Linear Conductance Formula for Interacting Fermions

## TL;DR

This paper derives a Keldysh-based formula for linear conductance in interacting fermionic systems, providing a new approach that simplifies calculations and is applicable within approximation schemes, demonstrated on a quantum point contact model.

## Contribution

The paper presents a Keldysh formalism derivation of Oguri's linear conductance formula, enabling easier application in approximation schemes and extending the theoretical toolkit for interacting fermion transport.

## Key findings

- Derived a Keldysh-based conductance formula from Meir-Wingreen
- Applied the formula to a quantum point contact model with perturbation theory
- Reduced numerical costs by using non-uniform lattice spacing

## Abstract

We present a Keldysh-based derivation of a formula, previously obtained by Oguri using the Matsubara formalisum, for the linear conductance through a central, interacting region coupled to non-interacting fermionic leads. Our starting point is the well-known Meir-Wingreen formula for the current, whose derivative w.r.t.\ to the source-drain voltage yields the conductance. We perform this derivative analytically, by exploiting an exact flow equation from the functional renormalization group, which expresses the flow w.r.t.\ voltage of the self-energy in terms of the two-particle vertex. This yields a Keldysh-based formulation of Oguri's formula for the linear conductance, which facilitates applying it in the context of approximation schemes formulated in the Keldysh formalism. (Generalizing our approach to the non-linear conductance is straightforward, but not pursued here.) -- We illustrate our linear conductance formula within the context of a model that has previously been shown to capture the essential physics of a quantum point contact in the regime of the 0.7 anomaly. The model involves a tight-binding chain with a one-dimensional potential barrier and onsite interactions, which we treat using second order perturbation theory. We show that numerical costs can be reduced significantly by using a non-uniform lattice spacing, chosen such that the occurence of artificial bound states close to the upper band edge is avoided.