Friedel oscillations in a gas of interacting one-dimensional fermionic atoms confined in a harmonic trap

TL;DR

This paper analytically investigates Friedel oscillations in interacting one-dimensional fermionic gases confined in a harmonic trap, revealing how soft boundaries alter their decay compared to open boundaries.

Contribution

It introduces an analytical method to compute Friedel oscillations in a harmonically confined Tomonaga-Luttinger liquid, highlighting the impact of soft boundaries on oscillation decay.

Findings

Friedel oscillations decay with boundary exponent (K+1)/2

Decay differs from open boundary conditions

Soft boundaries modify oscillation behavior

Abstract

Using an asymptotic phase representation of the particle density operator $\hat{\rho}(z)$ in the one-dimensional harmonic trap, the part $\delta \hat{\rho}_F(z)$ which describes the Friedel oscillations is extracted. The expectation value $<\delta \hat{\rho}_F(z)>$ with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga-Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent $\nu = (K+1)/2$ away from the classical boundaries. This value differs from that known for open boundary conditions or strong pinning impurities. The soft boundary in the present case thus modifies the decay of Friedel oscillations. The case of two components is also discussed.