Unified formalism for calculating polarization, magnetization, and more in a periodic insulator

Abstract

In this paper, we propose a unified formalism, using Green's functions, to integrate out the electrons in an insulator under uniform electromagnetic fields. We derive a perturbative formula for the Green's function in the presence of uniform magnetic or electric fields. Applying the formula, we derive the formula for the polarization, the orbital magnetization, and the orbital magnetopolarizability, without assuming time reversal symmetry. Specifically, we realize that the terms linear in the electric field can only be expressed in terms of the Green's functions in one extra dimension. This observation directly leads to the result that the coefficient of the θ-term in any dimensions is given by a Wess-Zumino-Witten like term, integrated in the extended space whose boundary is the original physical Brillouin zone, with the group element replaced by the Green's function. This generalizes an earlier result for the case of time reversal invariance. 1

Effect of the boundary on thermodynamic quantities such as magnetization

“…In particular, using our formalism described in Ref. 14 , any quantity that does not involve an extension of the Green's function to one extra dimension, such as the magnetization in zero electric field, is independent of the boundary. On the other hand, quantities that require an extension to extra dimension, such as the polarization and the trace of the magneto-electric tensor, will depend on the boundary.…”

Simplified Topological Invariants for Interacting Insulators

“…For general interacting systems (''interacting systems/insulators'' refer to systems/insulators with many-body interaction instead of systems/insulators interacting with each other), the topological order parameters can be defined as the physical response function for the quantum Hall effect [21] and the topological magneto-electric effect [9]. For actual evaluations of these physical response functions, we proposed earlier that the Green's function is an useful tool in topological insulators [22], and there is much recent interest focused in this direction [23][24][25][26]. However, our original formula for the topological order parameter [22] is rather complicated; more recently, a much simpler formula was obtained for the inversionsymmetric interacting topological insulators [27].…”

Noncommutative geometry for three-dimensional topological insulators

“…The latter yields the polarization created by application of a magnetic field and has a contribution from the 3D Chern-Simons invariant, or equivalently the θ-term. 57,59 with the summation convention implied over repeated indices and for any pairs α, α = 1, · · · , N or r, r ∈ Λ r or k, k ∈ Λ BZ .…”