Impurity immersed in a two-component few-fermion mixture in a one-dimensional harmonic trap

TL;DR

This work analyzes a 1D three-component fermionic system in a harmonic trap, consisting of a balanced two-component mixture and a single impurity, using exact diagonalization to study the ground state under zero-range interactions. By tracking densities, two-particle correlations, and fidelity susceptibility across inter-component coupling $g_0$ and impurity coupling $g$, the authors identify a structural transition where the impurity localizes at the system edges and the external structure reorganizes while the mixture remains largely intact. Remarkably, the fidelity susceptibility exhibits a universal scaling with the particle number $N$, collapsing onto a single curve after rescaling by $N^2$ and shifting by a transition point $G_N(g_0)$, well described by a Lorentzian form. These findings reveal collective multi-component fermionic behavior and universal transition properties, with implications for extensions to more components or impurities and to different interaction regimes.

Abstract

We investigate a one-dimensional three-component few-fermion mixture confined in a parabolic external trap, where one component contains a single particle acting as an impurity. Focusing on the many-body ground state, we analyze how the interactions between the impurity and the other components influence the system's structure. For fixed interaction strengths within the mixture, we identify a critical interaction strength with the impurity for which the system undergoes a structural transition characterized by a substantial change in its spatial features. We explore this transition from the point of view of correlations and ground-state susceptibility. We remarkably find that this transition exhibits unique universality features not previously observed in other systems, highlighting novel many-body properties existing in multi-component fermionic mixtures.

Figures (6)

• Figure 1: Low-energy spectrum of the many-body Hamiltonian \ref{['Hamiltonian']} obtained for a system containing an impurity interacting with the two-component mixture of four particles ($N=2$). (a) Eigenenergies as functions of interactions with the impurity $g$ for a fixed internal interaction $g_0$. (b) Eigenenergies as functions of internal interactions $g_0$ for a fixed interactions with the impurity $g$. In all plots, energies are expressed in natural units of energy $\hbar\Omega$, and for clarity, we shift them by the ground-state energy at $g=0$ for (a) and $g_0=0$ for (b), respectively.
• Figure 2: Impact of interactions with the third-component particle on the two-component mixture containing four particles ($N=2$). Different columns correspond to different values of interaction strength with the impurity $g$. In all plots, inter-component interaction strength is fixed, $g_0=1$. (a) Single-particle density profiles of the mixture's components $n_\sigma(x)$ (thick blue) and the third-component particle $m(x)$ (thin red). For a better comparison, the non-interacting distribution of the mixture's components is additionally displayed with a light blue shadow. (b) Single-particle reduced density matrix of the impurity $\rho(x,x')$. (c) Two-particle density profile $\mu_{\sigma}(x,z)$. (d) Distribution of correlations ${\cal G}_\sigma(x,z)$ between impurity and particle from the component $\sigma$. Note that in the left plot, we consider the interaction strength $g$ slightly positive ($g=0.01$) since for vanishing interactions the distribution vanishes. In this case, the distribution intensity is also scaled by a factor $10^2$ to make it visible. The upper and lower values on the color bar refer to plots (c) and (d), respectively. In all plots, positions are given in units of $a_0$. In panels (a) and (b), densities are expressed in units of $a_0^{-1}$, while in panels (c)-(d), in units of $a_0^{-2}$.
• Figure 3: Distribution of internal two-particle correlations ${\cal K}(x,y)$ in the system containing impurity interacting with the two-component mixture containing four particles ($N=2$) for different internal interaction strengths $g_0$ and interactions with the impurity $g$. For any fixed interaction $g_0$, along with increasing repulsion $g$, the transition between two different regimes is visible. In all plots, positions are expressed in units of $a_0$ while densities are in units of $a_0^{-2}$.
• Figure 4: Impact of interactions with the third-component particle on the two-component mixture containing four particles ($N=2$) when the system is confined in a rectangular potential well \ref{['HamiltonianWell']}. Different columns correspond to different values of interaction strength with the impurity $g$. In all plots, inter-component interaction strength is fixed, $g_0=1$. (a) Single-particle density profiles of the mixture's components $n_\sigma(x)$ (thick blue) and the third-component particle $m(x)$ (thin red). (b) Single-particle reduced density matrix of the impurity $\rho(x,x')$. Note that for sufficiently large interaction $g$, the impurity is separated similarly as in the case of a harmonic confinement. In all plots, positions are expressed in the width of the well $L$, while densities are in $L^{-1}$.
• Figure 5: Properties of the fidelity susceptibility for different numbers of particles in the system. (a) The fidelity susceptibility $\chi(g_0;g)$ as a function of interaction $g$ for different internal interactions $g_0$ (legend in panel (c)). (b) Position of the transition $G_N(g_0)$ as a function of internal interaction $g_0$. (c) Susceptibilities from the left plot after rescaling in magnitude by $\chi_0(g_0)$ and centering around the transition interaction $G_N(g_0)$. All plots collapse to the same universal curve, signalling the specific universality of the transition. (d) Numerical collapse to the universal curve $\Phi$ (dashed black) of all the fidelity susceptibility curves (different points and colors) after applying appropriate scaling \ref{['FidelFinal']}. Note that deviations from the universality are diminished when the number of particles in the mixture increases.