Depletion in fermionic chains with inhomogeneous hoppings

TL;DR

This paper investigates how inhomogeneous hopping patterns in fermionic chains cause depletion and density modulations, providing a semiclassical framework and methods to compensate for these effects through tailored chemical potentials.

Contribution

It introduces a semiclassical approach to describe density profiles in inhomogeneous fermionic chains and shows how to counteract depletion effects with specific chemical potentials.

Findings

Density modulations occur away from half-filling due to inhomogeneous hoppings.

A semiclassical expression for single-particle modes and densities is derived.

Depletion effects can be compensated by adding a chemical potential proportional to hoppings.

Abstract

The ground state of a free-fermionic chain with inhomogeneous hoppings at half-filling can be mapped into the Dirac vacuum on a static curved space-time, which presents exactly homogeneous occupations due to particle-hole symmetry. Yet, far from half-filling we observe density modulations and depletion effects. The system can be described by a 1D Schr\"odinger equation on a different static space-time, with an effective potential which accounts for the depleted regions. We provide a semiclassical expression for the single-particle modes and the density profiles associated to different hopping patterns and filling fractions. Moreover, we show that the depletion effects can be compensated for all filling fractions by adding a chemical potential proportional to the hoppings. Interestingly, we can obtain exactly the same density profiles on a homogeneous chain if we introduce a chemical potential which is inverse to the hopping intensities, even though the ground state is different from the original one.