Mobile impurity interacting with a Hubbard chain and the role of Friedel oscillations

TL;DR

This work analyzes a single mobile impurity in a one-dimensional open Hubbard chain coupled to a spin-$\frac{1}{2}$ fermionic bath using exact diagonalization. It reveals a spectrum of impurity configurations driven by bath–impurity interactions and boundary-induced Friedel oscillations, including miscible regimes, impurity–particle phase separation, and impurity–hole phase separation under particle–hole symmetry. The authors introduce quantitative diagnostics—average impurity position, two-body occupancies, and impurity entanglement—to map a detailed phase diagram as a function of $U_{fI}$ and $U/t$, with clear signatures at regime boundaries. The findings show that Friedel oscillations can induce nontrivial impurity localizations and suggest using impurities as probes of Friedel physics in ultracold-atom experiments. The results also highlight symmetry-driven connections between quarter- and three-quarters fillings and point to future work on dynamics and non-lattice realizations.

Abstract

This work examines a mobile impurity interacting with a bath of a few spin-$\uparrow$ and spin-$\downarrow$ fermions in a small one-dimensional open lattice system. We study ground-state properties using the exact diagonalization method, where the system is modeled by a three-component Fermi Hubbard Hamiltonian. We find that in addition to the standard phase separation between a strongly repulsive impurity and the bath, a strongly-attractive impurity also phase separates with the fermionic holes. Furthermore, we find that the impurity can show an oscillatory pattern in its density for intermediate bath-impurity interactions, which are induced by Friedel oscillations in the fermionic bath. This rich behavior of the impurity could be probed with fermionic ultracold mixtures in optical lattices.

Figures (14)

• Figure 1: Density profile of the fermions at $\nu_f=1/4$ in the absence of the impurity ($U_{fI}=0$) as a function of lattice sites. (a) considers a lattice with $M=8$, (b) with $M=12$, and (c) with $M=16$. The blank circles show results for $U=0$, the filled squares for $U=t$, and the filled diamonds for $U=4t$. The dashed horizontal lines indicate the filling factor of the fermions. The dash-dotted curves show the non-interacting profile from Eq. (\ref{['sec:1o4;sub:Friedel;eq:nonint']}).
• Figure 2: Density profiles of the impurity (orange circles) and of the fermions (blue squares) at $\nu_f=1/4$ for $U=4t$ as a function of sites in a lattice with $M=12$. The filled large markers consider the finite interaction $U_{fI}$ indicated in each panel, while the blank small markers consider a non-interacting impurity ($U_{fI}=0$) for reference. The top panels and bottom panels consider repulsive and attractive interactions, respectively. The dashed horizontal lines indicate the filling factor.
• Figure 3: Density profiles of the impurity [(a) and (c)] and of the fermions [(b) and (d)] at $\nu_f=1/4$ as a function of $U_{fI}/U$ and of sites in a lattice with $M=12$. The top and bottom panels consider $U/t=4$ and $U/t=0.5$, respectively.
• Figure 4: (a) Average positions $\text{site}^{(\text{max})}_I$ (orange dashed line) and $\bar{x}_I$ (solid blue line) for $U/t=4$ as a function of $U_{fI}/U$. (b) Average position $\bar{x}_I$ as a function of $U_{fI}/U$ and of $U/t$. The vertical lines in (a) and dashed lines in (b) indicate changes in $\text{site}^{(\text{max})}_I$. The dotted line in (b) indicates a change of $\text{site}^{(\text{max})}_I$ within the same sR regime [see Eq. (\ref{['sec:1o4;sub:x;eq:phasesR']})]. Both (a) and (b) consider $\nu_f=1/4$ and a lattice with $M=12$. A 0.5 is added to the average positions so that the sites are integer numbers.
• Figure 5: Fermion-impurity (a) and hole-impurity (b) double occupation at $\nu_f=1/4$ for $U/t=4$ as a function of $U_{fI}/U$ in a lattice with $M=12$. The different lines represent the sum of double occupation across the sites indicated in the legends. The vertical lines indicate changes in $\text{site}^{(\text{max})}_I$, as reported in Fig. \ref{['sec:1o4;sub:x;fig:x']}(a).