17 Axion Electrodynamics in Magnetoelectric Media
465
17.3 The Green Function Approach to the Electromagnetic
Response of Linear Isotropic Homogeneous
Magnetolectric Media
The knowledge of the Green function (GF) of an electromagnetic system allows the
calculation of its response to arbitrary external sources, providing a definite starting
point for the algebraic or numerical approximations which are required in most
cases. This method supersedes the image approach, frequently used in the literature,
which is appealing for interpreting results in terms of the superposition of images of
charges and currents. Nevertheless, it requires a good amount of educated guesses
which are far from obvious in many interesting cases. We will focus on calculating
the GFs for the cases when materials with piecewise constant MEP’s exhibit planar,
cylindrical and spherical symmetries. Certainly one could solve for the electric and
magnetic fields from the modified Maxwell equations together with the boundary
conditions, however, just as in ordinary electrodynamics, there might be occasions
where information about the sources is unknown and rather we are provided with data
of the 4-potential at some given boundaries. In these cases, the GF method provides
the general solution to such boundary-value problem (Dirichlet or Neumann) for
arbitrary sources.
As an important class of magnetoelectric media with constant MEP we single out
TIs, which serve to illustrate some of the phenomena discussed. Three dimensional
TIs are a class of topological materials that can host conducting helical surface states
each having the dispersion relation of a non-degenerate Dirac cone with a crossing
point at/close to the Fermi level. Nevertheless, TIs behave as insulators in the bulk
with a finite energy gap. The surface state is further topologically protected by timereversal symmetry and/or inversion symmetry, coupled with spin-momentum locking
properties. The latter means that the spin orientation of the electrons on the surface
Dirac cone is always locked perpendicularly to their momentum. A distinguishing
feature of 3D TIs among magnetoelectrics, is that the MEP θ is of topological nature
and arises from the bulk band structure. It is given by a non-Abelian Berry flux over
the Brillouin zone [19]. For 3D time reversal invariant insulators in a manifold without
boundaries there are only two possibilities: θ = 0 for normal insulators and θ = π
for TIs. In order to continuously connect both classes of insulators, time reversal
invariance should be broken at the interface. This can be achieved, for example, by
depositing a thin magnetic coating of a few nanometers at the interface. According to
the specific nature of the coating, the MEP of the TI will be modified to θ = π + 2nπ ,
where n is an integer [40].
From a macroscopic perspective we consider TIs as a class of magnetoelectric
media described by θ -ED and characterized by the choice of a constant MEP θ .
To illustrate the calculation of the GF it will be enough to consider the simplest
case where the (3 + 1) dimensional spacetime M can be split in such a way that
M = U × R, where U is a three-dimensional manifold and R is the temporal axis.
Moreover, the space U is partitioned in two region U 1 and U 2 , such that U 1 and
U 2 have a two-dimensional common interface . In this way U = U 1 ∪ U 2 and
465
17.3 The Green Function Approach to the Electromagnetic
Response of Linear Isotropic Homogeneous
Magnetolectric Media
The knowledge of the Green function (GF) of an electromagnetic system allows the
calculation of its response to arbitrary external sources, providing a definite starting
point for the algebraic or numerical approximations which are required in most
cases. This method supersedes the image approach, frequently used in the literature,
which is appealing for interpreting results in terms of the superposition of images of
charges and currents. Nevertheless, it requires a good amount of educated guesses
which are far from obvious in many interesting cases. We will focus on calculating
the GFs for the cases when materials with piecewise constant MEP’s exhibit planar,
cylindrical and spherical symmetries. Certainly one could solve for the electric and
magnetic fields from the modified Maxwell equations together with the boundary
conditions, however, just as in ordinary electrodynamics, there might be occasions
where information about the sources is unknown and rather we are provided with data
of the 4-potential at some given boundaries. In these cases, the GF method provides
the general solution to such boundary-value problem (Dirichlet or Neumann) for
arbitrary sources.
As an important class of magnetoelectric media with constant MEP we single out
TIs, which serve to illustrate some of the phenomena discussed. Three dimensional
TIs are a class of topological materials that can host conducting helical surface states
each having the dispersion relation of a non-degenerate Dirac cone with a crossing
point at/close to the Fermi level. Nevertheless, TIs behave as insulators in the bulk
with a finite energy gap. The surface state is further topologically protected by timereversal symmetry and/or inversion symmetry, coupled with spin-momentum locking
properties. The latter means that the spin orientation of the electrons on the surface
Dirac cone is always locked perpendicularly to their momentum. A distinguishing
feature of 3D TIs among magnetoelectrics, is that the MEP θ is of topological nature
and arises from the bulk band structure. It is given by a non-Abelian Berry flux over
the Brillouin zone [19]. For 3D time reversal invariant insulators in a manifold without
boundaries there are only two possibilities: θ = 0 for normal insulators and θ = π
for TIs. In order to continuously connect both classes of insulators, time reversal
invariance should be broken at the interface. This can be achieved, for example, by
depositing a thin magnetic coating of a few nanometers at the interface. According to
the specific nature of the coating, the MEP of the TI will be modified to θ = π + 2nπ ,
where n is an integer [40].
From a macroscopic perspective we consider TIs as a class of magnetoelectric
media described by θ -ED and characterized by the choice of a constant MEP θ .
To illustrate the calculation of the GF it will be enough to consider the simplest
case where the (3 + 1) dimensional spacetime M can be split in such a way that
M = U × R, where U is a three-dimensional manifold and R is the temporal axis.
Moreover, the space U is partitioned in two region U 1 and U 2 , such that U 1 and
U 2 have a two-dimensional common interface . In this way U = U 1 ∪ U 2 and