Cold Fermi-gas with long range interaction in a harmonic trap

Abstract

We study equilibrium density and spin density profiles for a model of cold one-dimensional spin 1/2 fermions interacting via inverse square interaction and exchange in an external harmonic trap. This model is the well-known spin-Calogero model (sCM) and its fully nonlinear collective field theory description is known. We extend the field theory description to the presence of an external harmonic trap and obtain analytic results for statics and dynamics of the system. For instance, we find how the equilibrium density profile changes upon tuning the interaction strength. The results we obtain for equilibrium configurations are very similar to the ones obtained recently by Ma and Yang [1] for a model of fermions with short ranged interactions. Our main approximation is the neglect of the terms of higher order in spatial derivatives in equations of motion - gradientless approximation [2]. Within this approximation the hydrodynamic equations of motion can be written as a set of decoupled forced Riemann-Hopf equations for the dressed Fermi momenta of the model. This enables us to write analytical solutions for the dynamics of spin and charge. We describe the time evolution of the charge density when an initial non-equilibrium profile is created by cooling the gas with an additional potential in place and then suddenly removing the potential. We present our results as a simple "single-particle" evolution in the phase-space reminiscing a similar description of the dynamics of non-interacting one-dimensional fermions.

11 pages, 5 figures (figure typo corrected and references added)

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Cited by in corpus (8)

• Quantum quench and thermalization of one-dimensional Fermi gas via phase space hydrodynamics
• Real-time spin-charge separation in one-dimensional Fermi gases from generalized hydrodynamics
• Quantum Quench in Non-relativistic Fermionic Field Theory: Harmonic traps and 2d String Theory
• Hydrodynamics of local excitations after an interaction quench in 1D cold atomic gases
• Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory
• Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions
• Particle Propagator of Spin Calogero-Sutherland Model
• Heisenberg Symmetry and Collective Modes of One Dimensional Unitary Correlated Fermions