Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach

TL;DR

This paper presents a new analytical discrete WKB approach to compute local fermion densities in inhomogeneous free-fermion chains, enabling better understanding of entanglement and density effects beyond traditional methods.

Contribution

The authors develop a novel discrete WKB method for analyzing inhomogeneous free-fermion chains, providing a closed-form density profile applicable to various configurations.

Findings

Accurate density profiles for different inhomogeneous chains

Reproduction of depletion and saturation effects

Framework for entanglement entropy analysis

Abstract

We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques.