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Hartree-Fock ground state of the composite fermion metal

Published Jan 1, 1995• 0 citations
OpenAlex

Abstract

Within the Hartree-Fock approximation the ground state of the composite fermion metal is found. We observe that the single-particle energy spectrum is dominated by the logarithmic interaction exchange term which leads to an infinite jump of the singleparticle energy at the Fermi momentum. It is shown that the Hartree-Fock result brings no corrections to the RPA Fermi velocity. 1

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Within the Hartree-Fock approximation the ground state of the composite fermion metal is found. We observe that the single-particle energy spectrum is dominated by the logarithmic interaction exchange term which leads to an infinite jump of the singleparticle energy at the Fermi momentum. It is shown that the Hartree-Fock result brings no corrections to the RPA Fermi velocity. 1
We study the Hartree-Fock approximation of graphene in infinite volume, with instantaneous Coulomb interactions. First we construct its translation-invariant ground state and we recover the well-known fact that, due to the exchange term, the effective Fermi velocity is logarithmically divergent at zero momentum. In a second step we prove the existence of a ground state in the presence of local defects and we discuss some properties of the linear response to an external electric field. All our results are non-perturbative.
Any exact eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a symmetry-broken function $\ensuremath{\Phi}$. For two particles, we exactly express $\ensuremath{\Phi}$ in terms of (single-particle) orbitals for all energy levels and any interparticle interaction. Especially for the ground state, $\ensuremath{\Phi}$ is given by the simple Hartree-Fock and Hartree Ans\"atze for fermions and bosons, respectively. Implications for several and many particles as well as a numerical example for interacting bosons are provided.
It is well known that the second-order ground-state correlation energy calculated in the usual random-phase approximation (RPA) is off by a factor of 2. It has also been noted that there is a factor-of-2 discrepancy in the correlation corrections to the Hartree-Fock occupation probabilities. The reason for these phenomena is traced to the treatment of the exchange terms in the interaction matrix element. An illustration based on a simple single multipole interaction shows that the factor of 2 arises from a coherent sum of small higher-order terms. Results based on the consideration of one or a few eigenmodes at a time are seen to be free of this difficulty. The correlation energy, occupation-probability corrections, and equations of motion are reexamined in the Beliaev-Zelevinsky boson expansion, as an illustration of clearly defined higher-order treatments.
The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ 0 (1) (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ 0(n)<ɛ FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m.