17 Axion Electrodynamics in Magnetoelectric Media
461
experimental value (d n ) exp < 3 × 10
−26 e cm produces the naturalness problem: why
should θ ∼ 10
−10 be so small. This is known as the strong CP problem. A solution
was proposed in [9, 10] through the introduction of a new field, the axion a(x),
and a new symmetry U (1) PQ realized again in the spontaneously broken mode. The
vacuum expectation value of a(x) can be chosen to cancel the previous CP violating
term, but a new pseudo Goldstone boson must appear. Estimations of the coupling
constant and mass of this new particle reveal extremely low values, thus making
its detection very difficult. The abelian sector of the axion coupling in the QCD
Lagrangian, L aγ γ = −g aγ γ a E · B, was proposed in [11] as a possibility to detect
the axion through its coupling with strong electromagnetic fields available in the
laboratory. The addition of this term to the usual electromagnetic Lagrangian, gives
rise to what is now known as axion electrodynamics. Recently the axion is considered
as a candidate to describe dark matter. Reference [5] contains a recent review of axion
physics, including the efforts made towards its detection.
Another subject related to the θ -coupling is the photon sector of the Standard
Model Extension (SME), designed to study possible violations of the Lorentz and
CPT symmetries [12–14]. There one considers the effective couplings (k F ) κλμν F
κλ
F
μν and (k AF )
κ
κλμν A
λ F
μν where the constant tensors (k F ) κλμν and (k AF )
κ define
fixed directions in spacetime that break Lorentz symmetry. An alternative way to perform this breaking is by choosing spacetime dependent coupling parameters [15],
like (k F ) κλμν = θ(x)) κλμν and (k AF ) κ = ∂ κ θ(x), for example. Both cases yield a
version of the axion coupling described previously which can be used as effective
theories with real Lorentz symmetry breaking in matter. The point of this comment is to observe that many of the techniques and of the numerous high-precision
experiments proposed to test Lorentz symmetry breaking could adapt to the case of
matter-photon interaction.
Recently, an important additional application of the Pontryagin extended electrodynamics has been highlighted in condensed matter physics, where a non-dynamical
axion angle θ provides an effective field theory describing the electromagnetic
response of some materials such as (i) magnetoelectric media [16, 17], (ii) metamaterials when θ is a purely complex function [18], (iii) topological insulators (TIs)
when θ = (2n + 1)π , with n integer [19–22] and (iv) Weyl semimetals (WSM) when
θ(x, t) = 2b · x − 2b 0 t , where b is the separation in momentum space between the
Weyl nodes and b 0 is their separation in energy [23]. Lately, the study of topological
insulating and Weyl semimetal phases, either from a theoretical or an experimental
perspective has been actively pursued [22–24].
We will devote this chapter to the study of the interaction between magnetoelectric media and the electromagnetic field. To do so we will employ field theoretic
techniques predicated on the aforementioned model Lagrangian of Maxwell’s theory extended by the magnetoelectric term. The chapter is organized as follows. In
Sect. 17.2 we present the general framework of electrodynamics in media characterized by a parameter θ (to be called a θ -medium) and review some of the most
important features of magnetoelectric media regarding their electromagnetic properties. Section 17.3 contains a summary of our generalized Green’s function method
to construct the corresponding electromagnetic fields produced by charges, currents
461
experimental value (d n ) exp < 3 × 10
−26 e cm produces the naturalness problem: why
should θ ∼ 10
−10 be so small. This is known as the strong CP problem. A solution
was proposed in [9, 10] through the introduction of a new field, the axion a(x),
and a new symmetry U (1) PQ realized again in the spontaneously broken mode. The
vacuum expectation value of a(x) can be chosen to cancel the previous CP violating
term, but a new pseudo Goldstone boson must appear. Estimations of the coupling
constant and mass of this new particle reveal extremely low values, thus making
its detection very difficult. The abelian sector of the axion coupling in the QCD
Lagrangian, L aγ γ = −g aγ γ a E · B, was proposed in [11] as a possibility to detect
the axion through its coupling with strong electromagnetic fields available in the
laboratory. The addition of this term to the usual electromagnetic Lagrangian, gives
rise to what is now known as axion electrodynamics. Recently the axion is considered
as a candidate to describe dark matter. Reference [5] contains a recent review of axion
physics, including the efforts made towards its detection.
Another subject related to the θ -coupling is the photon sector of the Standard
Model Extension (SME), designed to study possible violations of the Lorentz and
CPT symmetries [12–14]. There one considers the effective couplings (k F ) κλμν F
κλ
F
μν and (k AF )
κ
κλμν A
λ F
μν where the constant tensors (k F ) κλμν and (k AF )
κ define
fixed directions in spacetime that break Lorentz symmetry. An alternative way to perform this breaking is by choosing spacetime dependent coupling parameters [15],
like (k F ) κλμν = θ(x)) κλμν and (k AF ) κ = ∂ κ θ(x), for example. Both cases yield a
version of the axion coupling described previously which can be used as effective
theories with real Lorentz symmetry breaking in matter. The point of this comment is to observe that many of the techniques and of the numerous high-precision
experiments proposed to test Lorentz symmetry breaking could adapt to the case of
matter-photon interaction.
Recently, an important additional application of the Pontryagin extended electrodynamics has been highlighted in condensed matter physics, where a non-dynamical
axion angle θ provides an effective field theory describing the electromagnetic
response of some materials such as (i) magnetoelectric media [16, 17], (ii) metamaterials when θ is a purely complex function [18], (iii) topological insulators (TIs)
when θ = (2n + 1)π , with n integer [19–22] and (iv) Weyl semimetals (WSM) when
θ(x, t) = 2b · x − 2b 0 t , where b is the separation in momentum space between the
Weyl nodes and b 0 is their separation in energy [23]. Lately, the study of topological
insulating and Weyl semimetal phases, either from a theoretical or an experimental
perspective has been actively pursued [22–24].
We will devote this chapter to the study of the interaction between magnetoelectric media and the electromagnetic field. To do so we will employ field theoretic
techniques predicated on the aforementioned model Lagrangian of Maxwell’s theory extended by the magnetoelectric term. The chapter is organized as follows. In
Sect. 17.2 we present the general framework of electrodynamics in media characterized by a parameter θ (to be called a θ -medium) and review some of the most
important features of magnetoelectric media regarding their electromagnetic properties. Section 17.3 contains a summary of our generalized Green’s function method
to construct the corresponding electromagnetic fields produced by charges, currents