Abstract
Optical spectra provide a versatile tool for studying the electronic properties of matter. In addition, the absolute spectral weight of an optical spectrum reveals optical sum-rules, which are one of the most powerful tools of experimental and theoretical physics providing access to deeply rooted quantities such as the effective mass of the charge carriers and their kinetic energy. The formalism for the optical conductivity of correlated electrons is presented in this chapter for general values of the inverse wavelength \(q\) and general band dispersion \(\epsilon _k\) of the electrons. The corresponding sumrule is found to have a characteristic \(q\)-dependence for the nearest-neighbour tight binding model, causing in this case a vanishing of spectral weight for \(q\) at the Brillouin-zone boundary, i.e. for \(qa=\pi \). These findings are of possible importance for \(k\)-resolved infrared spectroscopy, a technique which is in full development at the moment.
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Notes
- 1.
Equation (9.50) is obtained if one represents the current operators as commutators of the hamiltonian with the dipole operator defined in (9.68). The expectation value of the hamiltonian is used to cancel out the factor \(\omega _{mn}\) in the denominator of the expression. In the final step the commutator of the dipole operator and the current operator is calculated, which completes the derivation of (9.50).
- 2.
One can pose the question whether the corresponding expression for the current density satisfies the continuity equation. On a fundamental level this relation expresses the conservation of the number of particles. This condition corresponds to the local constraint \(\nabla \cdot {\varvec{J}} + \partial \rho /\partial t=0\) in continuous space. Although there is no obvious way to define a quantity equivalent to \(\nabla \cdot \varvec{J}\) for a lattice, the situation is in fact somewhat simpler. It is sufficient to verify that removal of an electron from a given site \(\varvec{r}_m\) is always compensated by the creation of an electron elsewhere in the lattice. Since the current operator in (9.23) swaps electrons between different sites, the conservation of particle number is therefor built in the definition of the current operator
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Acknowledgments
It is a pleasure to thank Adrian Kantian, Christophe Berthod, Alexey Kuzmenko and Gianni Blatter for their comments. This work was supported by the SNSF through Grants No. 200020-140761 and 200020-135085, and the National Center of Competence in Research (NCCR) Materials with Novel Electronic PropertiesMaNEP.
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van der Marel, D. (2015). Optical Properties of Correlated Electrons. In: Avella, A., Mancini, F. (eds) Strongly Correlated Systems. Springer Series in Solid-State Sciences, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44133-6_9
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