Exact Spectral Function of a Tonks-Girardeau Gas in a Lattice

The single-particle spectral function of a strongly correlated system is an essential ingredient to describe its dynamics and transport properties. We develop a method to evaluate exactly the spectral function for a gas of one-dimensional bosons with infinitely strong repulsions valid for any type of external confinement. Focusing on the case of a lattice confinement, we find that the spectral function displays three main singularity lines. One of them is due uniquely to lattice effects, while the two others correspond to the Lieb-I and Lieb-II modes occurring in a uniform fluid. Differently from the dynamical structure factor, in the spectral function the Lieb-II mode shows a divergence, thus providing a route to probe such mode in experiments with ultracold atoms.

Figure 1

Sketch of the expected spectral function of a Bose fluid on a lattice in the $(k,\omega )$ plane for (a) zero, (b) weak, and (c) strong interactions. In this Letter, we provide an exact solution for the regime of infinitely strong interaction, which describes generically the case (c) (with the chemical potential $\mu$ corresponding to the Fermi energy $e_F$ of the mapped Fermi gas). The shaded areas indicate the regions where the spectral function is nonzero. (d)–(f) show the processes that give rise to the singularity lines ${\epsilon }_{1,2,3}(k)$.

Figure 2

Spectral function of the TG gas on a lattice in the $(k,\omega )$ plane. Violet, red, and yellow lines mark the excitation singularity lines ${\epsilon }_1(k)$, ${\epsilon }_2(k)$, and ${\epsilon }_3(k)$, respectively, which correspond to the elementary processes depicted at the bottom panels of Fig. 1, from left to right, respectively. Wave numbers $k$ and $k^{*}$ are in units of $a^{-1}$. Parameters used in the calculation: $N_s=256$, $N=64$.

Figure 3

Details of the spectral function for the TG gas on a lattice. (a) Cut of the spectral function for the TG gas in a lattice (Fig. 2) for $k^{*}a=1.0$. It shows power-law behavior in the vicinity of each depicted line [$\pm {\epsilon }_1(k^{*}),\pm {\epsilon }_2(k^{*})$]. (b),(c) Power-law behavior of the spectral function fitted with $c[\omega -e_F-{\epsilon }_{1,2}(k^{*}){]}^{-{\mu }_j}$, together with the fitted values of the exponents. The black circles indicate the points used to extract the power-law fits, whose result is displayed as a solid line with exponents specified inside each plot. The dashed lines correspond to the predicted power laws for the homogeneous system (see [79]). (d) Cut of the spectral function at $k^{**}a=2.71$. Vertical lines correspond to the peaks at $\omega -e_F={\epsilon }_1(k^{**})$ and $\omega -e_F={\epsilon }_3(k^{**})$.