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arXiv:1311.1284 · Physical Review A

Existence of an energy gap in a one-dimensional Lesanovsky model

Hosho Katsura

Published Dec 30, 2013• 1 citation
arXivHTMLPDF

Abstract

We study the quantum lattice gas model in one dimension introduced by Lesanovsky [Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012)], who showed that the exact ground state and a couple of excited states can be obtained analytically. The Hamiltonian of the model depends solely on the parameter $z$, the meaning of which is a fugacity in the corresponding classical lattice gas model. For small $z$ ($0<z<1$), we prove that there is an energy gap between the ground state and the excited states by applying Knabe's method [Knabe, J. Stat. Phys. 52, 627 (1988)].

DOI
10.1103/physreva.88.065602

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We study the quantum lattice gas model in one dimension introduced by Lesanovsky [Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012)], who showed that the exact ground state and a couple of excited states can be obtained analytically. The Hamiltonian of the model depends solely on the parameter zzz, the meaning of which is a fugacity in the corresponding classical lattice gas model. For small zzz (0<z<10<z<10<z<1), we prove that there is an energy gap between the ground state and the excited states by applying Knabe's method [Knabe, J. Stat. Phys. 52, 627 (1988)].
We introduce and solve a lattice version of an easily soluble one-dimensional continuum quantum system with a periodic potential. The relationship of the one problem to the other is identical to the relationship of the one-dimensional quantum many-body lattice gas recently introduced by Shastry [Phys. Rev. Lett. 60, 639 (1988)] and by Haldane [Phys. Rev. Lett. 60, 635 (1988)] to the one-dimensional quantum many-body continuum gas earlier solved by Sutherland. Thus it is hoped that by understanding the "missing" states in this simple case, we might better understand the missing states in the more complicated many-body case, which in many instances comprise all the states. The results are not encouraging.
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).
Abstract We study an attractive Hubbard model on bipartite lattices. In the grand canonical formalism, we prove the existence of superconducting long-range order in the ground state on Lieb lattices with the chemical potential corresponding to half filling. We also study the low-energy excitations above several ground states for the translationally invariant Hamiltonian. We prove the following: (i) The pairing excitations are gapless above the ground states when the number of fermions deviates from that of the half filling by the order of the volume. (ii) A certain class of single-fermion excitations shows a non-vanishing spectral gap above the ground state with an even number of fermions in a strong coupling and low-density regime.
We study a model of Rydberg atoms in a nearest-neighbor Rydberg blockaded regime, introduced by Lesanovsky [Phys. Rev. Lett. 108, 105301 (2012)]. This many-body model (which has one parameter zzz) has an exactly known gapped liquid ground state, and two exactly known low-lying excitations. We discover two exact low-lying eigenstates. We also discuss behavior of the model at small parameter zzz and its proximity to an integrable model. Lastly, we discuss connections between the Lesanovsky model at intermediate zzz and the so-called PXP model. The PXP model describes a recent experiment that observed unusual revivals from a charge-density-wave initial state, which are attributed to a set of many-body ``scar states'' which do not obey the eigenstate thermalization hypothesis. We discuss the possibility of approximate scar states in the Lesanovsky model and present two approximations for them.
We study atoms in optical lattices whose electronic ground state is off-resonantly coupled to a highly excited state with strong binary interactions. We present a time-dependent treatment of the resulting quantum dynamics, which---contrary to recent predictions [Li, Ates, and Lesanovsky, Phys. Rev. Lett. 110, 213005 (2013)]---proves that the strong repulsion between the weakly admixed Rydberg states does not lead to atomic trap loss. This finding provides an important basis for creating and manipulating coherent long-range interactions in optical lattice experiments.