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

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.

💡 Research Summary

This paper investigates a single mobile impurity interacting with a balanced spin‑½ fermionic bath confined to a one‑dimensional open lattice. The system is modeled by a three‑component Hubbard Hamiltonian in which the two spin components experience a repulsive on‑site interaction U>0, while the impurity hops with the same tunnelling amplitude t and interacts equally with both spin species via an on‑site coupling U_fI that can be either repulsive or attractive. Exact diagonalization (ED) is employed on lattices up to M=12 sites (with additional size checks in the appendix) to obtain ground‑state properties such as site‑resolved densities, correlation functions, and entanglement entropy.

A central theme is the role of Friedel oscillations that naturally arise in open boundary conditions. For fillings ν_f=N_f/M<½ (the paper focuses on ν_f=1/4 and, by particle‑hole symmetry, ν_f=3/4) the non‑interacting fermion density displays a 2k_F standing‑wave pattern whose amplitude is largest near the edges and whose number of peaks equals the number of particles per spin. The position of the dominant peaks shifts from the third to the second site from the boundary as the intra‑bath repulsion U increases, reflecting the formation of a spin‑wave state.

The impurity’s behaviour is classified according to the magnitude and sign of U_fI:

1. Weak impurity‑bath coupling (|U_fI|≪U) – The impurity density n_I(i) adopts a sinusoidal profile with a maximum at the centre, essentially mirroring the non‑interacting case. The fermionic bath retains its Friedel oscillations, and both species occupy the same sites. This regime is termed miscible (M).

2. Strong repulsive impurity (U_fI≫U) – The impurity becomes localized at the lattice edges, with vanishing occupation in the centre, while the fermionic bath is expelled from the edges and concentrates in the middle. This is the familiar particle‑phase separation (pPS) observed for repulsive polarons in lattice systems. The impurity’s edge localisation extends over a few sites (typically the first three), reflecting the low filling ν_f=1/4.

3. Strong attractive impurity (U_fI≪−U) – Counter‑intuitively, the impurity also localizes at the edges, but now the fermions are drawn to the same edge sites, dramatically increasing their occupation there. Because the majority species in this limit are the holes (absence of fermions), the system undergoes a hole‑phase separation (hPS): the impurity sits at the boundaries while the holes occupy the centre, forcing the fermions to the edges. The impurity’s localisation is sharper than in pPS, essentially confined to the two outermost sites.

The particle‑hole symmetry of the Hubbard model, n_σ(i;U_fI,ν_f)+n_σ(i;−U_fI,1−ν_f)=1, and n_I(i;U_fI,ν_f)=n_I(i;−U_fI,1−ν_f), explains why the results for ν_f=1/4 map onto those for ν_f=3/4.

In the intermediate regime (|U_fI|≈U) the impurity density inherits the Friedel oscillation pattern of the bath, leading to a multi‑peak structure that directly reflects the underlying 2k_F standing wave. Thus, the impurity acts as a probe of Friedel physics: measuring the impurity’s site‑resolved density provides a non‑destructive way to infer the bath’s oscillatory response.

Correlation functions and entanglement entropy further substantiate the phase‑separation picture. In both pPS and hPS the impurity‑bath correlation drops sharply and the bipartite entanglement entropy decreases, indicating reduced quantum entanglement. In the miscible regime the entanglement remains high, consistent with a strongly hybridised polaronic state.

The authors discuss experimental feasibility with ultracold fermionic mixtures, e.g., 6Li atoms in three hyperfine states loaded into a 1D optical lattice with open ends. Feshbach resonances allow independent tuning of U and U_fI, while site‑resolved imaging can capture the predicted density patterns. The required lattice sizes (≈10–20 sites) are well within current quantum‑gas microscope capabilities, making the observation of both phase‑separation regimes and Friedel‑induced impurity oscillations realistic.

Overall, the work provides a comprehensive exact‑diagonalization study of a mobile impurity in a Hubbard chain, revealing (i) the conventional particle‑phase separation for strong repulsion, (ii) a novel hole‑phase separation for strong attraction rooted in particle‑hole symmetry, and (iii) impurity density modulations directly tied to Friedel oscillations of the bath. These findings enrich the understanding of lattice polarons and suggest new routes to probe many‑body Friedel physics using mobile impurities in ultracold‑atom platforms.