Phase theory and critical exponents for the Tomonaga-Luttinger model with harmonic confinement

Abstract

A phase operator formulation for a recent model of interacting one-dimensional fermions in a harmonic trap is developed. The resulting theory is similar to the corresponding approach for the Luttinger model with open boundary conditions (OBC). However, in place of the spatial coordinate $z$, a dimensionless variable $u$ defined on the unit circle appears as argument of the phase fields and $u$ is non-linearly related to $z$. Furthermore, form factors appear which reflect the harmonic trap geometry. The theory is applied to calculate one-particle correlation functions. In a properly defined thermodynamic limit, bulk and boundary critical exponents are calculated for the static two-point correlation function and the dynamic local correlation function. The local spectral density is also considered. The critical exponents found are in agreement with those known for OBC with the exception of the boundary scaling exponent ${\Delta }_{\perp }$.

arXiv comment20 pages, minor changes, typos and journal-ref added

References in corpus (22)

• Theory of Bose-Einstein condensation in trapped gases
• One-Dimensional Fermi liquids
• Bosonizing one-dimensional cold atomic gases
• Bose-Einstein Condensation in a Surface Micro Trap
• Tuning p-wave interactions in an ultracold Fermi gas of atoms
• Interacting one dimensional electron gas with open boundaries
• Stability and phase coherence of trapped 1D Bose gases
• Guiding Neutral Atoms with a Wire
• Spin-charge separation in ultra-cold quantum gases
• Commensurate-incommensurate transition of cold atoms in an optical lattice
• Properties of a Luttinger Liquid with Boundaries at Finite Temperature and Size
• One-dimensional optical lattices and impenetrable bosons
• Is a Trapped One-Dimensional Bose Gas a Luttinger Liquid?
• Luttinger liquids with boundaries: Power-laws and energy scales
• Low-Energy Properties of a One-dimensional System of Interacting bosons with Boundaries
• Bounded Luttinger liquids as a universality class of quantum critical behavior
• Luttinger model approach to interacting one-dimensional fermions in a harmonic trap
• Two-component Fermi gas in a one-dimensional harmonic trap
• Treatment of backscattering in a gas of interacting fermions confined to a one-dimensional harmonic atom trap
• Interacting fermions in one dimension: The Tomonaga-Luttinger model
• Friedel oscillations in a gas of interacting one-dimensional fermionic atoms confined in a harmonic trap
• Newton's law for Bloch electrons, Klein factors and deviations from canonical commutation relations

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• A many-fermion generalization of the Caldeira-Leggett model
• Dynamic response of interacting one-dimensional fermions in the harmonic atom trap: Phase response and the inhomogeneous mobility