Quasi-particle residue and charge of the one-dimensional Fermi polaron

Abstract

We consider a mobile impurity coupled to an ideal Fermi gas in one spatial dimension through an attractive contact interaction. We calculate the quasi-particle residue $Z$ exactly, based on Bethe Ansatz and diagrammatic Monte Carlo methods, and with varational Ansatz up to one particle-hole excitation of the Fermi sea. We find that the exact quasi-particle residue vanishes in the thermodynamic limit as a power law in the number of particles, consistent with the Luttinger-liquid paradigm and the breakdown of Fermi-liquid theory. The variational Ansatz, however, predicts a finite value of $Z$, even in the thermodynamic limit. We also study how the presence of the impurity affects the density of the spin-up sea by calculating the pair correlation function. Subtracting the homogeneous background and integrating over all distances gives the charge $Q$. This charge turns out to grow continuously from 0 at zero coupling to 1 in the strong-coupling limit. The varational Ansatz predicts $Q=0$ at all couplings. So, although the variational Ansatz has been shown to be remarkably accurate for the energy and the effective mass, it fails even qualitatively when predicting $Z$ and the pair correlation function in the thermodynamic limit.

Figures (5)

• Figure 1: The quasi-particle residue $Z$ as function of the number $N_{\uparrow}$ of spin-up fermions for two interaction strengths, $\gamma=-4$ and $\gamma=-10$. The open symbols are obtained via Bethe Ansatz. The red and black symbols with error bars are results of diagrammatic Monte Carlo simulation with the PDet algorithm. Both are in excellent agreement. The full lines are power law fits to $Z = a N_{\uparrow}^{-\theta}$ [ with $a = 0.982$ and $\theta=0.0644$ for $\gamma=-4$, and $a= 0.775$ and $\theta=0.2040$ for $\gamma=-10$ ]. Results obtained with the variational Ansatz are also shown. Here, the values of $Z$ quickly saturate to a constant upon increasing $N_{\uparrow}$.
• Figure 2: The exponent $\theta$, describing the Anderson orthogonality catastrophe, as a function of the interaction strength $\gamma$. The blue circles correspond to the numerical data obtained from the finite-size correction of the polaron energy calculated with $N_\uparrow=99$, see Eq. (\ref{['DeltaE']}). The dashed line corresponds to the analytical result, $\theta(\gamma)=2 \arctan^2(\gamma/(2\pi))/\pi^2$, obtained from Eqs. (\ref{['theta']}) and (\ref{['deltaN']}).
• Figure 3: The polaron energy $E_P$ (i.e. the ground state energy difference $E-E_{FS}$ of the interacting and the non-interacting system) in units of the Fermi energy $E_F$ as function of the number $N_{\uparrow}$ of spin-up fermions. Data is obtained via Bethe Ansatz and variational polaron Ansatz for $\gamma=-4$ and $\gamma=-10$. For the variational Ansatz we observe exponential convergence, while the Bethe Ansatz converges as $1/N_\uparrow$.
• Figure 4: The charge $Q$ defined in Eq. (\ref{['eq:charge_def']}) as function of the coupling $\gamma$, calculated with Bethe Ansatz (red solid line). The charge changes continuously from 0 to 1, marking the continuous crossover from weak to strong coupling. The blue dashed line corresponds to $\Delta N$, as given in Eq. (\ref{['eq:DeltaN_final']}).
• Figure 5: The pair correlation function $\tilde{g}_2(x) = g_2(x)/n_{\downarrow}$ shifted by the homogeneous spin-up density $n_{\uparrow} = k_F/\pi$ for two values of the coupling strength $\gamma$. Exact Bethe Ansatz results as well as results based on the variational Ansatz are shown.