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arXiv:2410.13302 · Physical Review A

Lieb excitations and topological flat mode of the spectral function of a Tonks-Girardeau gas in a Kronig-Penney potential

Wen-Bin He, Giedrius Žlabys, Hoshu Hiyane, Sarika Sasidharan Nair, Thomas Busch

Published Jan 21, 2025• 1 citation
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Abstract

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.

DOI
10.1103/physreva.111.013312

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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.
In this thesis I study the topological properties exhibited in cold atom systems trapped in periodic and finite-size one-dimensional potentials. Ultracold atoms provide an excellent platform for exploring various many-body quantum models that are challenging to identify in traditional condensed matter systems. In particular, recent advancements in experimental cold atom physics have enabled the creation of subwavelength nanoscale potentials, making it possible to realize the Kronig-Penney model and its variations. One of these considers shifting a regular lattice of point-like barriers within a finite-sized box, which allows topologically protected edge states to emerge as a function of the shift parameter. The continuous shift parameter then provides a virtual dimension and the system becomes effectively two-dimensional, allowing topological states to be explored over a large and flexible parameter space. In my work I go beyond the standard singleparticle treatment of topological system and analyze the equilibrium and non-equilibrium physics of fermionic and bosonic many-body systems. The first project I present focuses on the non-equilibrium dynamics of a ground-state fermionic many-body gas following a quench between different lattice shift parameters that allow for edge states with different chirality. I examine the role of the single-particle chiral edge states in the non-equilibrium dynamics of the system and demonstrate that the usual monotonic decay associated with the orthogonality catastrophe, typically observed with increasing system size, is significantly altered. The second study explores the topological characteristics of a system of two interacting particles in the Kronig-Penney-type nanoscale potential and examines the appearance of two-particle topological states within the spectrum. This investigation is extended to the Rice-Mele model, where similar topological features are observed. Finally, the third study considers the optical realization of a height-modulated nanoscale potential, which generates a complex potential landscape. To probe the many-body properties of such a system, I examined the spectral function of a Tonks–Girardeau gas in a quasiperiodically modulated KP potential. By varying the modulation amplitude and frequency, I identified a transition between delocalized and localized single-particle eigenstates. In the localized regime, the spectral function shows flat, non-dispersive modes that reflect localized states, whereas in the delocalized regime it becomes continuous and dispersive.
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 study the exact nonequilibrium spectral function of a gas of strongly correlated Tonks-Girardeau bosons subjected to a strong periodic drive. Utilizing the theory of Floquet spectral function in conjunction with the Bose-Fermi mapping theorem, we show that nonequilibrium Lieb modes emerge if the underlying mapped fermions form a Floquet-Fermi sea. In the low-frequency regime, the exact analysis reveals the emergence of characteristic linear Lieb excitations for the bosonic system, while the underlying mapped fermions displays the wide Dirac-like linear dispersion.
We determine the exact many-body properties of a bosonic Tonks-Girardeau gas confined in a harmonic potential with a tunable $\ensuremath{\delta}$-function barrier at the trap center. This is done by calculating the reduced single-particle density matrix, the pair-distribution function, and the momentum distribution of the gas as a function of barrier strength and particle number. With increasing barrier height we find that the ground-state occupation in a diagonal basis diverges from the N\sqrt{N}N​ behavior that is expected for the case of a simple harmonic trap. In fact, the scaling of the occupation number depends on whether one has an even or odd number of particles. Since this quantity is a measure of the coherence of our sample we show how the odd-even effect manifests itself in both the momentum distribution of the Bose gas and interference fringe visibility during free temporal evolution.
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.
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.