Optical conductivity beyond the electric-dipole approximation: a gauge- and translationally-invariant formulation


We express the Kubo formula for the spatially-dispersive optical conductivity in terms of three types of matrix elements between valence and conduction Bloch states: the first is the off-diagonal Berry connection matrix which, as is well known, describes electric-dipole interband transitions. The other two describe magnetic-dipole and electric-quadrupole transitions; they are, respectively, interband generalizations of the intrinsic orbital moment of a wavepacket [1], and of the quantum metric tensor [2]. This form of the spatially-dispersive conductivity is physically transparent due to the gauge invariance and origin independence of its individual terms; this is in contrast to the standard quantum multipole theory [3], where the individual terms are origin dependent and limited to bounded samples. The relation between the two formulations is established by considering the limit of a crystal composed of nonoverlapping molecules. Numerical results are presented for optical rotation and nonreciprocal directional dichroism in a tight-binding model of a chiral trigonal crystal.

[1] D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010)
[2] R. Cheng, Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum System: A pedagogical Introduction, arXiv:1012.1337v2 (2013)
[3] R. E. Raab and O. L. De Lange, Multipole theory in electromagnetism: classical, quantum, and symmetry aspects, with applications, Oxford University Press (2004)