Exact equivalence between one-dimensional Bose gases interacting via hard-sphere and zero-range potentials

Abstract

We prove the equivalence between the hard-sphere Bose gas and a system with momentum-dependent zero-range interactions in one spatial dimension, which we call extended hard-sphere Bose gas. The two-body interaction in the latter model has the advantage of being a regular pseudopotential. The most immediate consequence is the existence of its Fourier transform, permitting the formulation of the problem in momentum space, not possible with the original hard-core interaction. In addition, in the extended system, interactions are defined in terms of the scattering length, positive or negative, identified with the hard-sphere diameter only when it is positive. We are then able to obtain, directly in the thermodynamic limit, the ground state energy of the strongly repulsive Lieb-Liniger gas and, more importantly, the energy of the lowest-lying super Tonks-Girardeau gas state with finite, strongly attractive interactions, in perturbation theory from the novel extended hard-sphere Bose gas. Tan relations involving the large-momentum behavior of the Lieb-Liniger gas are also derived, and then applied to the super Tonks-Girardeau gas within our perturbative approach.

5 pages, 2 figures. Final published version

References in corpus (8)

• Generalized Virial Theorem and Pressure Relation for a strongly correlated Fermi gas
• Exact Relations for a Strongly-interacting Fermi Gas from the Operator Product Expansion
• Verification of universal relations in a strongly interacting Fermi gas
• Universal relations for the two-dimensional spin-1/2 Fermi gas with contact interactions
• Local Correlations in the Super Tonks-Girardeau Gas
• Dynamics of pair correlations in the attractive Lieb-Liniger gas
• Effective super Tonks-Girardeau gases as ground states of strongly attractive multi-component fermions
• Tan's distributions and Fermi-Huang pseudopotential in momentum space

Cited by in corpus (21)

• Fermionization of two-component few-fermion systems in a one-dimensional harmonic trap
• Universal properties of Fermi gases in arbitrary dimensions
• Quantum Impurity in a One-dimensional Trapped Bose Gas
• Universal One-dimensional Atomic Gases Near Odd-wave Resonance
• Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low energy regime
• High-momentum distribution with subleading $k^{-3}$ tail in the odd-wave interacting one-dimensional Fermi gases
• Fractional energy states of strongly-interacting bosons in one dimension
• Field-theoretical aspects of one-dimensional Bose and Fermi gases with contact interactions
• Universal Tan relations for quantum gases in one dimension
• Comparative study of one-dimensional Bose and Fermi gases with contact interactions from the viewpoint of universal relations for correlation functions
• Hydrodynamic VS collisionless dynamics of a 1D harmonically trapped Bose gas
• Dissipative Effects on the Superfluid to Insulator Transition in Mixed-dimensional Optical Lattices
• Scaling properties of the Tan's contact: embedding pairs and correlation effect in the Tonks limit
• High-momentum tail and universal relations of a Fermi gas near a Raman-dressed Feshbach resonance
• Few-Body Route to One-Dimensional Quantum Liquids
• Ground-state properties of dilute spinless fermions in fractional dimensions
• Universal relations and normal-state properties of a Fermi gas with laser-dressed mixed-partial-wave interactions
• Mobile Spin Impurity in an Optical Lattice
• Phase Equilibrium of Binary Mixtures in Mixed Dimensions
• Fermi super-Tonks-Girardeau state for attractive Fermi gases in an optical lattice
• From few to many body degrees of freedom