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Self-consistency conditions in a symmetry-breaking external field

Yu. B. Ivanov

Published Feb 1, 1980• 0 citations
OpenAlex

Abstract

Self-consistency conditions are derived for the mass operator, the density matrix, and the two-particle interaction in systems which are situated in a (not necessarily weak) symmetry-breaking field.

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We consider a two-dimensional gas of interacting fermions in presence of an external constant magnetic field: the system is extended and homogeneous, and thus assumed to be invariant under magnetic translations. Working within the Hartree-Fock approximation, we analyze the system directly in the thermodynamic limit by solving a self-consistent fixed-point equation for the one-particle density matrix. We prove that, provided that the interactions among fermions are sufficiently weak, there exists a unique one-particle density matrix that solves the self-consistency condition. By choosing the Fermi-Dirac distribution as the function in the fixed-point equation, this approach can describe both positive and zero-temperature cases. At zero temperature and when the chemical potential of the non-interacting system lies in a spectral gap of the free Landau operator, our self-consistent solution is an orthogonal projection (an "interacting" effective Fermi projection). We prove that its integrated density of states varies linearly with the external magnetic field, provided the interaction is weak enough: the slope of this variation is quantized and independent of the interaction. According to the Středa formula, this can be seen as yet another expression of the universality of the quantum Hall effect in weakly interacting systems, at least within the Hartree-Fock approximation.
The Kohn-Sham (KS) equations determine, in a self-consistent way, the particle density of an interacting fermion system at thermal equilibrium. We consider a situation when the KS equations are known to have a unique solution at high temperatures and this solution is a uniform particle density. We show that, at zero temperature, there are stable solutions that are not uniform. We provide the general principles behind this phenomenon, namely the conditions when it can be observed and how to construct these non-uniform solutions. Two concrete examples are provided, including fermions on the sphere which are shown to crystallize in a structure that resembles the C60 molecule. 1
The one-dimensional many-fermion system with weak attractive interaction, which is known to have self-consistent Hartree solutions corresponding to either uniform or spatially periodic mass density, is discussed as a physical model and as a mathematical problem. The Hartree problem for sinusoidal mass density leads to the self-consistent Mathieu problem, which is analyzed for the case where the first Brillouin zone boundary does not necessarily coincide with the Fermi surface (no energy gap between ground and single-particle excited states). The Mathieu equation is solved in the weak-binding approximation in the vicinity of the two lowest band gaps, the self-consistency condition is analyzed in detail, and the N-particle energy is calculated for various cases. The results suggest that the periodic state will not have lower energy than the uniform state unless the first gap in the one-particle spectrum lies near the Fermi surface. The fact that self-consistency can be obtained for a considerable range of periodicities suggests that the periodic solutions may be of importance in some many-fermion systems.