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arXiv:2005.13646 · Physical Review Letters

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

Jacopo Settino, Nicola Lo Gullo, Francesco Plastina, Anna Minguzzi

Published Feb 9, 2021• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.126.065301

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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.
We report on the derivation of determinant representations for the Green's functions and spectral function of the trapped Tonks-Girardeau gas on the lattice and in the continuum. Our results are valid for any type of statistics of the constituent particles, at zero and finite temperature and arbitrary confining potentials, including nonequilibrium scenarios induced by sudden changes of the external potential. In addition, they are also extremely efficient and easy to implement numerically with the main computational effort being represented by the calculation of partial overlaps of the dynamically evolved single-particle wave functions. In the lattice case we show that the spectral function of a system with a strong harmonic potential presents only two singular lines compared with three singular lines in the case of a homogeneous system.
Lieb excitations are fundamental to the understanding of the low-energy behavior of many-body quantum gases. Here we study the spectral function of a Tonks-Girardeau gas in a finite-sized Kronig-Penney potential and show that the Lieb-I and Lieb-II excitations can become gapped as a function of the barrier height. Moreover, we reveal the existence of a topological flat mode near the Fermi energy and at zero momentum and show that this is robust to perturbations in the system. Through a scaling analysis, we determine the divergent behavior of the spectral function. Our results provide a significant reference for the observation and understanding of the gapped Lieb excitations and the topological flat mode of quantum gases in experimentally realistic subwavelength optical lattice potentials.
The dynamical correlations of a strongly correlated system are an essential ingredient to describe its nonequilibrium properties. We present a general method to calculate exactly the dynamical correlations of hard-core anyons in one-dimensional lattices, valid for any type of confining potential and any temperature. We obtain exact explicit expressions of the Green's function, the spectral function, and the out-of-time-ordered correlators (OTOCs). We find that the anyonic spectral function displays three main singularity lines which can be explained as a double spectrum in analogy to the Lieb-Liniger gas. The dispersion relations of these lines can be given explicitly and they cross at a hot point $({q}_{m},{\ensuremath{\omega}}_{m})$, which induces a peak in the momentum distribution function at qm{q}_{m}qm​ and a power-law singularity in the local spectral function at ${\ensuremath{\omega}}_{m}$. We also find that the anyonic statistics can induce spatial asymmetry in the Green's function, its spectrum, and the OTOC. Moreover, the information spreading characterized by the OTOCs shows light-cone dynamics, asymmetric for general statistics and low temperatures, but symmetric at infinite temperature. Our results pave the way toward studying the nonequilibrium dynamics of hard-core anyons and experimentally probing anyonic statistics through spectral functions.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.