Exact calculation of spectral properties of a particle interacting with a one-dimensional Fermi gas in optical lattices

TL;DR

This work provides an exact, determinant-based expression for the impurity form factor in a 1D Hubbard model with one spin-down impurity, enabling exact computation of the impurity spectral function in finite lattices. Leveraging the Bethe ansatz and a Slater-determinant reformulation, the authors separately treat regular Bethe states and irregular states (ζ-spin-flip and η-pairing) to satisfy sum rules and capture spectral weight. They reveal a three-branch spectral structure comprising low- and high-energy Hubbard bands and a mid-band polaron-like feature, with the relative weights governed by filling, interaction strength, and momentum, including nontrivial behavior at Q=0 and Q=π. The results bridge exact many-body theory and cold-atom experiments, offering a route to observe Fermi polarons, Hubbard-band physics, and polaron quasiparticles in 1D optical lattices, and point toward extensions to dynamics and multi-impurity settings.

Abstract

By using the exact Bethe wavefunctions of the one-dimensional Hubbard model with $N$ spin-up fermions and one spin-down impurity, we derive an analytic expression of the impurity form factor, in the form of a determinant of a $(N+1)$ by $(N+1)$ matrix. This analytic expression enables us to exactly calculate spectral properties of one-dimensional Fermi polarons in lattices, when the masses of the impurity particle and the Fermi bath are equal. We present the impurity spectral function as functions of the on-site interaction strength and the filling factor of the Fermi bath, and discuss the origin of Fermi singularities in the spectral function at small momentum and the emergence of polaron quasiparticles at large momentum near the boundary of Brillouin zone. Our analytic expression of the impurity form factors pave the way to exploring the intriguing dynamics of a particle interacting with a Fermi bath. Our exact predictions on the impurity spectral function could be directly examined in cold-atom laboratories by using the radio-frequency spectroscopy and Ramsey spectroscopy.

Figures (6)

• Figure 1: Graphical illustration of the solutions for $\sin(k_{j})=u\cot(k_{j}L/2)$ in the ground state, where $\Lambda=0$. The blue dots and red dots are the roots for the repulsive interaction $u>0$ and the attractive interaction $u<0$, respectively.
• Figure 2: Upper panel: Residues of different (i.e., $n$-th) Bethe wavefunctions at zero total momentum ($Q=0$) as a function of the energy $\omega=E_{N+1}^{(n)}\left(\{k_{j}\},\Lambda\right)-E_{\textrm{FS},N}$ at the quarter filling $\nu=(N+1)/L=0.5$ and at the attraction $U=-4t$. We choose a small lattice size, i.e., $L=12$ and $N=5$, so the different many-body states can be well separated, for a better illustration. The stars and circles correspond to the $k-\Lambda$ solutions and the all real-$k$ solutions, respectively. The color of the symbols shows the value of the quasi-momentum $\Lambda$, which increases as the color changes from blue to red. At zero momentum, all the states are doubly degenerate, so we only include the states with $\Lambda\geq0$. The arrow emphasizes the broken-pair state, discovered by McGuire with the Gaudin-Yang model McGuire1966. Lower panel: Configurations of the quantum numbers $\{I_{j}\}$ for different Bethe wavefunctions. The yellow area highlights the pseudo Fermi sea of the broken-pair state.
• Figure 3: Residues (a and b) and the impurity spectral functions (c and d, in units of $t^{-1}$) as a function of the energy $\omega=E_{N+1}^{(n)}\left(\{k_{j}\},\Lambda\right)-E_{\textrm{FS},N}$ at the quarter filling $\nu=(N+1)/L=0.5$ and at the attraction $U=-2t$. In the left and right columns, we show the results at zero momentum $Q=0$ and at the momentum $Q=\pi$, respectively. Here, we choose $L=40$ and $N=19$. As in Fig. \ref{['fig2_states']}, in the upper panels, the residues of the real-$k$ solutions and the $k-\Lambda$ solutions are shown by circles and stars, respectively, with the color of symbols indicating the value of the quasi-momentum $\Lambda$. The two black dots in (b) show the residues of the spin-flip state (i.e., the right dot) and of the $\eta$-pairing state (the left dot). The inset in (c) highlights the repulsive branch of the spectral function starting at $\omega\sim t$, contributed by all the real-$k$ solutions.
• Figure 4: The same plots as in Fig. \ref{['fig3_uu05m']}, except that the attractive interaction strength becomes $U=-4t$. Due to the enhanced attraction, the repulsive branch of Fermi polarons at zero momentum $Q=0$ becomes more significant, as highlighted in the inset of Fig. \ref{['fig4_uu10m']}(c).
• Figure 5: Three-dimensional plots of the spectral function $A(Q,\omega)$, in units of $t^{-1}$ as indicated by the color map, at the interaction strengths $U=-2t$ (a) and $U=-4t$ (b). The corresponding two-dimensional contour plots are also shown at the top of the figure. Here, we take the lattice size $L=40$ and the number of spin-up fermions $N=19$.