462
A. Martín-Ruiz et al.
and boundary conditions in systems satisfying the following coordinate conditions:
(i) the coordinates can be chosen such that the interface between two media, with
different θ , is defined by setting to a constant only one of coordinates and (ii) the
Laplacian is separable in such coordinates. At the end of this section the particularly
simple case of planar symmetry is discussed. There the reader is referred to the analogous extensions to cylindrical and spherical coordinates. As a specific application
of our methods to the case of a planar interface, the Casimir effect between two
metallic plates with a topological insulator between them is considered in Sect. 17.4.
In Sect. 17.5, we study the problem of Vavilov-Cherenkov radiation produced by
an electric charge propagating perpendicularly to the planar interface between two
different magnetoelectric media to find that, besides the usual forward-directed radiation, there is a backward-directed emission of radiation, a so-called reversed VavilovCherenkov radiation (RVCR), solely due to the magnetoelectric nature of the media.
As yet another application, in Sect. 17.6 we show how a static electric charge induces
unexpected electromagnetic fields when placed near the surface of a Weyl semimetal
and provide two experimental proposals, feasible for present-day sensitivities, that
could be performed to measure the effects associated with the topological character
of the WSM. Finally in Sect. 17.7 we present some concluding remarks, provide a
summary of important results and highlight the benefits of our approach for solving
the electromagnetic response of magnetoelectric media.
Our conventions are taken from [25], where F μν = ∂ μ A ν − ∂ ν A μ , ˜
F
μν
=
μναβ
F αβ /2, F
i0
= E
i , F
i j
= −
i jk B
k and ˜
F
i0
= B
i , ˜
F
i j
=
i jk E
k . Also V = (V
i
) =
(V x , V y , V z ) for any vector V. The metric is (+, −, −, −) and
0123
= +1 =
123 .
17.2 Nondynamical Axion Electrodynamics
In this section we discuss the basic features arising from adding to Maxwell’s electrodynamics the coupling of the abelian Pontryagin density to a pseudoscalar field
θ(x), leading to a theory that we call θ -electrodynamics (θ -ED), retaining the name
of axion-electrodynamics for the case where the axion field θ becomes dynamical.
We call the parameter θ(x) the magnetoelectric polarizability (MEP) of the medium,
which we consider in the same footing as its permittivity ε and permeability μ. The
nature of the MEP depends on the type of magnetoelectric material under consideration and it is ultimately related to the magnetic symmetries of the substance [26,
27] and/or to the properties of its band structure [19–21]. It can be calculated from a
Kubo-type response formula, once a microscopic model Hamiltonian for the material is adopted. The permittivity tensor is usually understood by the Drude-Lorentz
type of single resonance oscillator model [28].
Magnetoelectric media [16, 17] are naturally existing materials like antiferromagnets [30], topological insulators (TIs) [19–22] and Weyl semimetals [23, 24],
for example. Leaving aside the remarkable microscopic properties of different magnetoelectric media, we will concentrate on θ -ED as the effective macroscopic theory
A. Martín-Ruiz et al.
and boundary conditions in systems satisfying the following coordinate conditions:
(i) the coordinates can be chosen such that the interface between two media, with
different θ , is defined by setting to a constant only one of coordinates and (ii) the
Laplacian is separable in such coordinates. At the end of this section the particularly
simple case of planar symmetry is discussed. There the reader is referred to the analogous extensions to cylindrical and spherical coordinates. As a specific application
of our methods to the case of a planar interface, the Casimir effect between two
metallic plates with a topological insulator between them is considered in Sect. 17.4.
In Sect. 17.5, we study the problem of Vavilov-Cherenkov radiation produced by
an electric charge propagating perpendicularly to the planar interface between two
different magnetoelectric media to find that, besides the usual forward-directed radiation, there is a backward-directed emission of radiation, a so-called reversed VavilovCherenkov radiation (RVCR), solely due to the magnetoelectric nature of the media.
As yet another application, in Sect. 17.6 we show how a static electric charge induces
unexpected electromagnetic fields when placed near the surface of a Weyl semimetal
and provide two experimental proposals, feasible for present-day sensitivities, that
could be performed to measure the effects associated with the topological character
of the WSM. Finally in Sect. 17.7 we present some concluding remarks, provide a
summary of important results and highlight the benefits of our approach for solving
the electromagnetic response of magnetoelectric media.
Our conventions are taken from [25], where F μν = ∂ μ A ν − ∂ ν A μ , ˜
F
μν
=
μναβ
F αβ /2, F
i0
= E
i , F
i j
= −
i jk B
k and ˜
F
i0
= B
i , ˜
F
i j
=
i jk E
k . Also V = (V
i
) =
(V x , V y , V z ) for any vector V. The metric is (+, −, −, −) and
0123
= +1 =
123 .
17.2 Nondynamical Axion Electrodynamics
In this section we discuss the basic features arising from adding to Maxwell’s electrodynamics the coupling of the abelian Pontryagin density to a pseudoscalar field
θ(x), leading to a theory that we call θ -electrodynamics (θ -ED), retaining the name
of axion-electrodynamics for the case where the axion field θ becomes dynamical.
We call the parameter θ(x) the magnetoelectric polarizability (MEP) of the medium,
which we consider in the same footing as its permittivity ε and permeability μ. The
nature of the MEP depends on the type of magnetoelectric material under consideration and it is ultimately related to the magnetic symmetries of the substance [26,
27] and/or to the properties of its band structure [19–21]. It can be calculated from a
Kubo-type response formula, once a microscopic model Hamiltonian for the material is adopted. The permittivity tensor is usually understood by the Drude-Lorentz
type of single resonance oscillator model [28].
Magnetoelectric media [16, 17] are naturally existing materials like antiferromagnets [30], topological insulators (TIs) [19–22] and Weyl semimetals [23, 24],
for example. Leaving aside the remarkable microscopic properties of different magnetoelectric media, we will concentrate on θ -ED as the effective macroscopic theory