Orbital Magnetization in Solids: Boundary contributions as a non-Hermitian effect

K. Kyriakou, K. Moulopoulos

Published 2018-11-06Version 1

The theory of orbital magnetization is reconsidered through careful definition of additional quantities that incorporate a non-Hermitian effect due to anomalous operators that break the domain of definition of the Hermitian Hamiltonian. As a result, boundary contributions to the observable are rigorously and analytically taken into account. In this framework, we first extend the standard velocity operator definition in order to incorporate an anomaly of the position operator that is inherent in band theory, which results to an explicit boundary velocity contribution. Using the extended velocity, we define the electrons' intrinsic orbital circulation which is an intensive quantity of periodic systems that properly counts the circulating micro-currents embodied in the wavefunctions' structure (bulk and boundary contributions). A connection between the nth band electrons' collective intrinsic circulation and the Wannier-based local and itinerant circulation contributions used in the modern theory of orbital magnetization is made. We develop a quantum mechanical formalism for the orbital magnetization of extended and periodic topological solids (insulators or semimetals) without any Wannier-localization approximation or heuristic extension [Caresoli, Thonhauser, Vanderbilt and Resta, Phys. Rev. B 74, 024408 (2006)]. It is rigorously shown that, as a result of the non-Hermitian effect, an emerging covariant derivative enters the one-band (adiabatically deformed) approximation k-space expression for the orbital magnetization. In the corresponding many-band (unrestricted) k-space formula, the non-Hermitian effect contributes an additional boundary quantity which is expected to give locally (in momentum space) giant contributions whenever band crossings occur along with Hall voltage due to imbalance of electron accumulation at the opposite boundaries of the material.