Berry phase polarization and orbital magnetization responses of insulators: Formulas for generalized polarizabilities and their application

Abstract: Condensed matter physics is often concerned with determining the response of a solid to an external stimulus. This paper revisits and extends the microscopic formalism for calculating response coefficients -- here referred to as (generalized) polarizabilities -- in crystalline electronic insulators. The main focus is on the Berry phase polarization and orbital magnetization, for which we obtain general formulas describing the linear response to an arbitrary (but static and uniform) perturbation. The response of an arbitrary lattice-periodic observable (e.g. spin, layer pseudospin) to electric and magnetic fields is also examined, and serves as a basis for mircoscopically establishing Maxwell relations between conjugate generalized polarizabilities. We furthermore introduce and examine the notion of Berry curvature or Hall vector polarizability, i.e., the response of the Berry curvature to a general perturbation, and show how it relates to Berry phase polarization and orbital magnetization responses. For all polarizabilities considered, we obtain simplified formulas applicable to two- and four-band models, expressed directly in terms of the Hamiltonian and the perturbation. Three specific applications of these formulas are discussed: (i) a computation of the magnetoelectric polarizabilities of model antiferromagnets in one and two dimensions; (ii) a general proof of (quasi)topological signatures in the polarizabilities of Dirac fermions in two dimensions; (iii) a calculation of the strain-induced Berry curvature polarizability in an altermagnet.