Topological Magnetoelectric Effect
The chapter introduces the basic parameters that determine the topological magnetoelectric effect and discusses the anomalous quantum Hall effect in topological thin films.
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Topological Magnetoelectric Effect
The electrodynamics of a media is described by the Maxwell equations which relate physical fields, current and charge density [1]. In four-dimensional Minkowski space xi ¼ (ct, r) (i ¼ 0, 1, 2, 3, c is the speed of light in vacuum), with the metric tensor gij ¼ diag {1, 1, 1, 1, }, the inner product is defined as XiXi ¼ X2, where Xi ¼ gijX j and over repeated indices, summation is implied. The potentials are expressed as components of the four-vector Ai ¼ (φ/c, A), Ai ¼ (φ/c, A). The gauge-invariant electromagnetic field tensor is defined as F ik ¼ ∂i Ak ∂k Ai , ∂i
∂ : ∂xi
ð4:1Þ
Differentiating the potential Ai we obtain the electromagnetic tensor expressed through the physical fields 0
0 E x =c B E x =c 0 F ik ¼ B @ E y =c By Ez =c By
Ey =c Bz 0 Bx
1 Ez =c By C C, Bx A 0
ð4:2Þ
where E ¼ ∂A/∂t —φ, B ¼ — A are the electric field and magnetic flux density, respectively. It can be checked by direct calculations in (4.1) that Eiklm ∂k F lm ¼ 0, i ¼ 0,1,2,3,
ð4:3Þ
where Eiklm is the Levi-Civita symbol, which is equal to +1 (1) if an even (odd) number of pairwise permutations bring indexes to the sequence 0,1,2,3. From (4.2) and (4.3) we find the first pair of Maxwell equations in vector form:
© Springer Nature Switzerland AG 2020 V. Litvinov, Magnetism in Topological Insulators, https://doi.org/10.1007/978-3-030-12053-5_4
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4 Topological Magnetoelectric Effect
—B¼0 ∂B : —E¼ ∂t
ð4:4Þ
The second pair of Maxwell equations follows from the least action principle: conditions for minimum action by varying electromagnetic potentials which are treated as dynamical variables. The action should be invariant with respect to frame rotations in Minkowski space. The eigenvalues of Fik are determined by the characteristic equation P(λ) Det[Fik λgik] ¼ 0 and do not depend on the frame. This means that invariant combinations of Fik can be found as coefficients in the characteristic polynomial P(λ): inv ¼ 2 B2 E 2 =c2 F ik F ik inv ¼
4 EB 1 iklm E F ik F lm , c 2
ð4:5Þ
where Fik gilFlmgmk ¼ Fik(E, B). Taking into account both invariants (4.5), one can express the electromagnetic field action as follows: 1 Sf ¼ 4Z 0
ð
h i κ η iklm E F ik F lm , d4 x F ik F ik þ 2
ð4:6Þ
pffiffiffiffiffiffiffiffiffiffiffi where d4x ¼ drd(ct), Z 0 ¼ μ0 =ε0 , μ0, ε0, are the impedance, permeability, and permittivity of vacuum, respectively, κ is the normalization constant to be specified later, and η is the dimensionless so-called axion field. The action of charges interacting with the electromagnetic field is written as Sef ¼
1 c
ð d4 x Ai ji ,
ð4:7Þ
where the four-current is defined as ji ¼ (cρ, j); ρ, j are the charge and current density, respectively. The total action from (4.6) and (4.7) can be expressed through the Lagrangian density L: ð 1 η κ iklm E F ik F lm Ai ji : S ¼ S f þ Sef ¼ L drdt, L ¼ F ik F ik þ 4μ0 2
ð4:8Þ
Equation (4.8) is the starting point from which one can derive the second pair of Maxwell equations. Before we proceed with the derivation, let us c
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