Exact first-order density matrix for a d-dimensional harmonically confined Fermi gas at finite temperature

Abstract

We present an exact closed form expression for the {\em finite temperature} first-order density matrix of a harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where exact expressions have been limited to quantities derived from the {\em diagonal} first-order density matrix. We compare our exact results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-N limit. As an interesting application, we derive a closed form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional parabolically confined quantum dot. We numerically test this exact result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation (LDA) to the exchange energy of an inhomogeneous 2D Fermi gas.

12 pages, 3 figures included in the text, RevTeX4. Text before Eq.(25) corrected. Additional equation following Eq.(25) has been added

References in corpus (10)

• Evaporative Cooling of a Two-Component Degenerate Fermi Gas
• Simple Analytical Particle and Kinetic Energy Densities for a Dilute Fermionic Gas in a d-Dimensional Harmonic Trap
• Exact particle and kinetic energy densities for one-dimensional confined gases of non-interacting fermions
• Non-Interacting Fermions in a One-Dimensional Harmonic Atom Trap: Exact One-Particle Properties at Zero Temperature
• Some exact results for a trapped quantum gas at finite temperature
• Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions
• Particle density and non-local kinetic energy density functional for two-dimensional harmonically confined Fermi vapors
• Kinetic energy functional for Fermi vapors in spherical harmonic confinement
• One-dimensional non-interacting fermions in harmonic confinement: equilibrium and dynamical properties
• Temperature dependence of density profiles for a cloud of non-interacting fermions moving inside a harmonic trap in one dimension

Cited by in corpus (12)

• Coherence properties of the two-dimensional Bose-Einstein condensate
• Density profile of a harmonically trapped ideal Fermi gas in arbitrary dimension
• Density gradients for the exchange energy of electrons in two dimensions
• Thomas-Fermi von Weizsäcker theory for a harmonically trapped, two-dimensional, spin-polarized dipolar Fermi gas
• Ideal Fermi gases in harmonic oscillator potential traps
• Exact results for a charged, harmonically trapped quantum gas at arbitrary temperature and magnetic field strength
• Harmonically trapped jellium
• Finite temperature analytical results for a harmonically confined gas obeying exclusion statistics in $d$-dimensions
• A manifestly Hermitian semiclassical expansion for the one-particle density matrix of a two-dimensional Fermi gas
• The Classical-Map Hyper-Netted-Chain (CHNC) technique for inhomogeneous electron systems. Application to quantum dots
• Evaluation of inverse integral transforms for undergraduate physics students
• Preserving the Hermiticity of the One-Body Density Matrix for a Non-Interacting Fermi Gas