14 research outputs found
Orbital Approximation for the Reduced Bloch Equations: Fermi-Dirac Distribution for Interacting Fermions and Hartree-Fock Equation at Finite Temperature
In this paper, we solve a set of hierarchy equations for the reduced
statistical density operator in a grand canonical ensemble for an identical
many-body fermion system without or with two-body interaction. We take the
single-particle approximation, and obtain an eigen-equation for the
single-particle states. For the case of no interaction, it is an eigen-equation
for the free particles, and solutions are therefore the plane waves. For the
case with two-body interaction, however, it is an equation which is the
extension of usual Hartre-Fock equation at zero temperature to the case of any
finite temperature. The average occupation number for the single-particle
states with mean field interaction is also obtained, which has the same
Fermi-Dirac distribution from as that for the free fermion gas. The derivation
demonstrates that even for an interacting fermion system, only the lowest
orbitals, where is the number of particles, are occupied at zero
temperature. In addition, their practical applications in such fields as
studying the temperature effects on the average structure and electronic
spectra for macromolecules are discussed.Comment: Modify the last paragraph regarding the applications of the equations
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Second Quantized Reduced Bloch Equations and the Exact Solutions for Pairing Hamiltonian
In this article, we present a set of hierarchy Bloch equations for the
reduced density operators in either canonical or grand canonical ensembles in
the occupation number representation. They provide a convenient tool for
studying the equilibrium quantum statistical mechanics for some model systems.
As an example of their applications, we solve the equations for the model
system with a pairing Hamiltonian. With the aid of its symplectic group
symmetry, we obtain the statistical reduced density matrices with different
orders. As a special instance for the solutions, we also get the reduced
density matrices of the ground state for a superconductor