464
A. Martín-Ruiz et al.
The electromagnetic fields E and B are expressed in terms of the electromagnetic
potentials and A as
E = −
1
c
∂A
∂t
− ∇,
B = ∇ × A,
(17.6)
which solve the homogeneous equations in (17.1). An important consequence of
(17.4) is the so-called magnetoelectric effect (MEE), summarized in the appearance
of the following effective field dependent charge and current densities
ρ θ =
α
4π 2 ∇θ · B,
J θ = −
cα
4π 2 ∇θ × E −
α
4π 2
∂θ
∂t
B,
(17.7)
evidencing the ability of the magnetic (electric) fields to produce charge (current) densities, respectively. This effect is one of the most remarkable physical consequences
of the additional θ coupling. It was predicted in [26] and subsequently observed in
[30]. For an updated review of the MEE see, for example, [31]. A universal topological magnetoelectric effect has recently been measured in TIs [32]. Many additional
consequences of the MEE have been highlighted using different approaches. For
example, electric charges close to the interface between two θ -media induce image
magnetic monopoles (and vice versa) [33–36]. Also, the propagation of electromagnetic waves across a θ -boundary have been studied finding that a non trivial Faraday
rotation of the polarizations appears [29, 34, 35, 37]. The shifting of the spectral
lines in hydrogen-like ions placed in front of a planar TI, as well as the modifications
to the Casimir Polder potential in the non-retarded approximation were studied in
[38]. The classical dynamics of a Rydberg hydrogen atom near a planar TI has also
been investigated [39].
We observe that the dynamical modifications in (17.4) depend on the spatial and
temporal gradients of the MEP, as required because the Pontryagin density P is a
total derivative. In this way, the coupling with θ does not affect the equations of
motion when the MEP is globally constant. The explicit dependence on θ , instead of
∂ μ θ , of the constitutive relations may erroneously induce the belief that a globally
constant θ could produce dynamical effects. Nevertheless, from this perspective one
would identify additional polarization P θ = −σ B and magnetization M θ = −σ E
with σ = θα/4π
2 . When θ is constant, the calculation of the effective sources
ρ eff = ∇ · P θ = σ ∇ · B,
J eff =
1
c
∂P θ
∂t
+ ∇ × M θ = −σ
1
c
∂B
∂t
+ ∇ × E
,
(17.8)
yields identically zero due to the homogeneous Maxwell equations.
A. Martín-Ruiz et al.
The electromagnetic fields E and B are expressed in terms of the electromagnetic
potentials and A as
E = −
1
c
∂A
∂t
− ∇,
B = ∇ × A,
(17.6)
which solve the homogeneous equations in (17.1). An important consequence of
(17.4) is the so-called magnetoelectric effect (MEE), summarized in the appearance
of the following effective field dependent charge and current densities
ρ θ =
α
4π 2 ∇θ · B,
J θ = −
cα
4π 2 ∇θ × E −
α
4π 2
∂θ
∂t
B,
(17.7)
evidencing the ability of the magnetic (electric) fields to produce charge (current) densities, respectively. This effect is one of the most remarkable physical consequences
of the additional θ coupling. It was predicted in [26] and subsequently observed in
[30]. For an updated review of the MEE see, for example, [31]. A universal topological magnetoelectric effect has recently been measured in TIs [32]. Many additional
consequences of the MEE have been highlighted using different approaches. For
example, electric charges close to the interface between two θ -media induce image
magnetic monopoles (and vice versa) [33–36]. Also, the propagation of electromagnetic waves across a θ -boundary have been studied finding that a non trivial Faraday
rotation of the polarizations appears [29, 34, 35, 37]. The shifting of the spectral
lines in hydrogen-like ions placed in front of a planar TI, as well as the modifications
to the Casimir Polder potential in the non-retarded approximation were studied in
[38]. The classical dynamics of a Rydberg hydrogen atom near a planar TI has also
been investigated [39].
We observe that the dynamical modifications in (17.4) depend on the spatial and
temporal gradients of the MEP, as required because the Pontryagin density P is a
total derivative. In this way, the coupling with θ does not affect the equations of
motion when the MEP is globally constant. The explicit dependence on θ , instead of
∂ μ θ , of the constitutive relations may erroneously induce the belief that a globally
constant θ could produce dynamical effects. Nevertheless, from this perspective one
would identify additional polarization P θ = −σ B and magnetization M θ = −σ E
with σ = θα/4π
2 . When θ is constant, the calculation of the effective sources
ρ eff = ∇ · P θ = σ ∇ · B,
J eff =
1
c
∂P θ
∂t
+ ∇ × M θ = −σ
1
c
∂B
∂t
+ ∇ × E
,
(17.8)
yields identically zero due to the homogeneous Maxwell equations.