The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. (English) Zbl 1197.82113

Summary: In a recent article [the authors and A. Deleurence, Commun. Math. Phys. 281, No. 1, 129–177 (2008; Zbl 1157.82042)], we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree-Fock model, the ground state electronic density matrix is decomposed as \({\gamma = \gamma^0_{\text{per}} + Q_{\nu,\varepsilon_{\text F}}}\), where \({\gamma^0_{\text{per}}}\) is the ground state density matrix of the host crystal and \({Q_{\nu,\varepsilon_{\text F}}}\) the modification of the electronic density matrix generated by a modification \(\nu \) of the nuclear charge of the host crystal, the Fermi level \(\varepsilon _{F}\) being kept fixed.
The purpose of the present article is twofold. First, we study in more detail the mathematical properties of the density matrix \({Q_{\nu,\varepsilon_{\text F}}}\) (which is known to be a self-adjoint Hilbert-Schmidt operator on \({L^2(\mathbb{R}^3)}\)). We show in particular that, if \(\int_{\mathbb{R}^3}\,\nu \neq 0\), \(Q_{\nu,\varepsilon_{\text F}}\) is not trace-class. Moreover, the associated density of charge is not in \({L^1(\mathbb{R}^3)}\) if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect \(\nu \), the linear and nonlinear terms of the resolvent expansion of \({Q_{\nu,\varepsilon_{\text F}}}\). Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the Adler-Wiser formula.


82D25 Statistical mechanics of crystals
82D37 Statistical mechanics of semiconductors
81V10 Electromagnetic interaction; quantum electrodynamics
35Q55 NLS equations (nonlinear Schrödinger equations)
78A30 Electro- and magnetostatics
47N50 Applications of operator theory in the physical sciences


Zbl 1157.82042
Full Text: DOI arXiv


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