460
A. Martín-Ruiz et al.
17.1 Introduction
Classical and quantum electrodynamics summarize all our understanding of the interaction between matter and radiation. Although these topics have been profusely studied in many different areas since their early discoveries, still today they constitute a
fruitful research discipline and an excellent arena with potential for new discoveries.
This is specially true when precision measurements are at hand and also when new
materials come into play whose novel properties, of ultimate quantum origin, result
in new possible forms of interaction between light and such materials. That is the
case with topological insulators, as well as other materials with topological order.
Interestingly enough, the interaction between matter characterized by topological order and external electromagnetic fields can be described by an extension of
Maxwell’s theory. In fact, in electrodynamics there is the possibility of writing two
quadratic gauge and Lorentz invariant terms: the first one is the usual electromagnetic
density L EM = (E
2
− B
2
)/8π which yields Maxwell’s equations, and the second one
is the magnetoelectric term L θ = θ E · B, where θ is a coupling field usually termed
the axion angle.
Many of the interesting properties of the latter can be recognized from its covariant form L θ = −(θ/8))
μνρλ F μν F ρλ , where
μνρλ is the Levi-Civita symbol and
F μν = ∂ μ A ν − ∂ ν A μ is the electromagnetic field strength written in terms of the
connection A μ . When θ is globally constant, the θ -term is a total derivative and has
no effect on Maxwell’s equations. This property, together with its invariance under
continuous changes in the connection A μ , qualify P = −(1/8))
μνρλ F μν F ρλ to be a
topological invariant. Actually, P is the simplest example of a Pontryagin density
[1], corresponding to the abelian group U (1). This structure, together with its generalization to nonabelian groups, has been relevant in diverse topics in high-energy
physics such as anomalies [2], the strong CP problem [3], topological field theories
[4] and axions [5], for example.
It is interesting to recall that the coupling to a Pontryagin density has a long
story in the development of Yang-Mills theories, with its origin dating back to a
problem arising because the axial symmetry of the QCD Lagrangian, in the massless
limit, would imply the existence of low mass nucleons with odd parity, which are not
found in nature [6]. The alternative realization of this symmetry in the spontaneously
broken mode is not possible either because the three lighter mesons in the spectrum
(the pions) do not match the required four pseudo Goldstone bosons. The absence
of a fourth pseudoscalar meson with a mass similar to that of the pions is what is
known as the U (1) A problem. A solution was proposed in [7, 8] grounded in a more
detailed study of the structure of the vacuum in a Yang-Mills theory, which introduces
a superposition of topologically nonequivalent vacua as the true ground state of the
system: the θ vacuum. This superposition eliminates the U (1) A problem but requires
the CP violating coupling of the θ -vacuum to the non-abelian Pontryagin density
in the QCD Lagrangian. Unfortunately, this addition generates a new problem. In
particular, the resulting effective pion-nucleon interaction predicts an electric dipole
moment for the neutron given by d n ∼ 3.2 × 10
−16
θ e cm. Comparison with the
A. Martín-Ruiz et al.
17.1 Introduction
Classical and quantum electrodynamics summarize all our understanding of the interaction between matter and radiation. Although these topics have been profusely studied in many different areas since their early discoveries, still today they constitute a
fruitful research discipline and an excellent arena with potential for new discoveries.
This is specially true when precision measurements are at hand and also when new
materials come into play whose novel properties, of ultimate quantum origin, result
in new possible forms of interaction between light and such materials. That is the
case with topological insulators, as well as other materials with topological order.
Interestingly enough, the interaction between matter characterized by topological order and external electromagnetic fields can be described by an extension of
Maxwell’s theory. In fact, in electrodynamics there is the possibility of writing two
quadratic gauge and Lorentz invariant terms: the first one is the usual electromagnetic
density L EM = (E
2
− B
2
)/8π which yields Maxwell’s equations, and the second one
is the magnetoelectric term L θ = θ E · B, where θ is a coupling field usually termed
the axion angle.
Many of the interesting properties of the latter can be recognized from its covariant form L θ = −(θ/8))
μνρλ F μν F ρλ , where
μνρλ is the Levi-Civita symbol and
F μν = ∂ μ A ν − ∂ ν A μ is the electromagnetic field strength written in terms of the
connection A μ . When θ is globally constant, the θ -term is a total derivative and has
no effect on Maxwell’s equations. This property, together with its invariance under
continuous changes in the connection A μ , qualify P = −(1/8))
μνρλ F μν F ρλ to be a
topological invariant. Actually, P is the simplest example of a Pontryagin density
[1], corresponding to the abelian group U (1). This structure, together with its generalization to nonabelian groups, has been relevant in diverse topics in high-energy
physics such as anomalies [2], the strong CP problem [3], topological field theories
[4] and axions [5], for example.
It is interesting to recall that the coupling to a Pontryagin density has a long
story in the development of Yang-Mills theories, with its origin dating back to a
problem arising because the axial symmetry of the QCD Lagrangian, in the massless
limit, would imply the existence of low mass nucleons with odd parity, which are not
found in nature [6]. The alternative realization of this symmetry in the spontaneously
broken mode is not possible either because the three lighter mesons in the spectrum
(the pions) do not match the required four pseudo Goldstone bosons. The absence
of a fourth pseudoscalar meson with a mass similar to that of the pions is what is
known as the U (1) A problem. A solution was proposed in [7, 8] grounded in a more
detailed study of the structure of the vacuum in a Yang-Mills theory, which introduces
a superposition of topologically nonequivalent vacua as the true ground state of the
system: the θ vacuum. This superposition eliminates the U (1) A problem but requires
the CP violating coupling of the θ -vacuum to the non-abelian Pontryagin density
in the QCD Lagrangian. Unfortunately, this addition generates a new problem. In
particular, the resulting effective pion-nucleon interaction predicts an electric dipole
moment for the neutron given by d n ∼ 3.2 × 10
−16
θ e cm. Comparison with the