Axion electrodynamics in topological materials

Conventionally, metals and insulators have been distinguished by the existence of bandgaps. In 2005, a novel phase of matter that does not belong to either conventional metals or insulators, called the topological insulator, was discovered.^1–5 It is notable that topological insulators have bulk bandgaps but also have gapless boundary (edge or surface) states. Furthermore, a topological insulator phase and a trivial insulator phase cannot be connected adiabatically to each other. In other words, bulk bandgap closing is required for the transitions between topologically nontrivial and trivial phases. In addition, before the establishment of the concept of topological insulators, different phases of matter had usually been distinguished from each other by the order parameters that indicate spontaneous symmetry breaking. For example, magnetism can be understood as a consequence of spontaneous spin rotational symmetry breaking. However, from the viewpoint of symmetry analysis, time-reversal invariant topological insulators and time-reversal invariant band insulators cannot be distinguished. The ways to distinguish such topologically nontrivial and trivial insulator phases can be divided into two types (which, of course, give rise to equivalent results). One way is introducing a “topological invariant” such as $Z2$ invariant,^1,6–8 which are calculated from the Bloch-state wave function of the system. The other way is the “topological field theory,”⁹ which describes the responses of topological phases to external fields and is the focus of this Tutorial.

In the topological field theory, the responses of a topological phase to external fields are described by a topological term. In two spatial dimensions, it is well known that the quantum Hall effect of a time-reversal symmetry broken phase can be described by a Chern–Simons action with the quantized coefficient given by the first Chern number.^10,11 In three spatial dimensions, time-reversal symmetry plays an important role. The topological magnetoelectric effect described by the so-called $θ$ term⁹ is a hallmark response of three-dimensional (3D) time-reversal invariant topological insulators to external electric and magnetic fields. In the presence of time-reversal symmetry, the coefficient of the magnetoelectric effect $θ$ takes a quantized value $θ=π$ (mod $2π$⁠) for topological insulators, while $θ=0$ in trivial insulators. However, in systems with broken time-reversal symmetry, e.g., in magnetically ordered phases, the value of $θ$ can be arbitrary, i.e., can deviate from the quantized value $π$ or $0$⁠, which means that the value of $θ$ can even depend on space and time as $θ(r,t)$⁠. It should be noted that spatial-inversion symmetry breaking can also lead to the deviation of $θ$ from the quantized value $π$ or $0$⁠.

In the field theory literature, the phenomena described by the $θ$ term is termed the axion electrodynamics¹² because the $θ$ term has exactly the same form as the action describing the coupling between a hypothetical elementary particle, axion, and a photon. The axion was proposed about 40 years ago to solve the so-called strong CP problem in quantum chromodynamics.^13–15 By subsequent studies in particle physics and astrophysics, the axion is now considered as a candidate for dark matter.^16–19 However, regardless of intensive experimental searches, the axion has not yet been found. Since the coefficient of the $θ$ term, $θ(r,t)$⁠, is a field describing the axion, observing the magnetoelectric responses in materials whose effective action is described by a $θ$ term is equivalent to realizing the (dynamical) axion field in condensed matter.²⁰ So far, it has been shown theoretically that in a class of magnetic insulators such as magnetically doped topological insulators, the value of $θ(r,t)$ is proportional to the antiferromagnetic order parameter (i.e., the Néel field), i.e., the antiferromagnetic spin fluctuation is identical to a dynamical axion field.²⁰ In Fig. 1, a classification of 3D insulators in terms of the value of $θ$ is schematically shown.

The effective action of the form of the $θ$ term appears not only in insulator phases but also in semimetal phases. The key in the case of topological semimetals is the breaking of time-reversal or spatial-inversion symmetry, which can lead to nonzero and nonquantized expressions for $θ$⁠. For example, in a time-reversal broken Weyl semimetal with two Weyl nodes, its response to external electric and magnetic fields is described by a $θ$ term with $θ(r,t)=2(b⋅r−b0t)$⁠,^21–25 where $b$ is the distance between the two Weyl nodes in momentum space and $b0$ is the energy difference between the two nodes. In contrast, in the case of topological superconductors, their topological nature is captured only by thermal responses,^26–28 since charge and spin are not conserved. It has been heuristically suggested that the effective action of 3D time-reversal invariant topological superconductors may be described by an action which is analogous to the $θ$ term but is written in terms of gravitational fields corresponding to a temperature gradient and a mechanical rotation.^29,30

In this Tutorial, we overview theoretical and experimental studies on the axion electrodynamics in topological materials. In Sec. II, we start by deriving the topological magnetoelectric effect described by a $θ$ term in phenomenological and microscopic ways in 3D time-reversal invariant topological insulators. We also review recent experimental studies toward observations of the quantized magnetoelectric effect. In Sec. III, we review the basics and recent experimental realizations of the so-called axion insulators in which the value of $θ$ is quantized due to a combined symmetry (effective time-reversal symmetry), regardless of the breaking of time-reversal symmetry, focusing on MnBi$2$Te$4$ family of materials. In Sec. IV, we consider generic expressions for $θ$ in insulators and extend the derivation of the $θ$ term in a class of insulators with broken time-reversal and inversion symmetries whose realization requires antiferromagnetic orderings. In Sec. V, we describe emergent dynamical phenomena from the realization of the dynamical axion field in topological antiferromagnetic insulators. In Secs. VI and VII, we extend the study of the axion electrodynamics in condensed matter to Weyl semimetals and topological superconductors, respectively, whose effective action can be described by topological terms analogous to the $θ$ term. In Sec. VIII, we summarize this Tutorial and outlook future directions of this fascinating research field.

In this section, we describe the basics of the topological magnetoelectric effect, one of the intriguing properties characteristic to 3D topological insulators. We derive phenomenologically and microscopically the $θ$ term in 3D topological insulators, which is the low-energy effective action describing their responses to external electric and magnetic fields, i.e., the topological magnetoelectric effect. We also review recent theoretical and experimental studies toward observations of the topological magnetoelectric effect.

As has been briefly mentioned in Sec. I, topological phases can be characterized by their response to external fields. One of the noteworthy characters peculiar to 3D topological insulators is the topological magnetoelectric effect, which is described by the so-called $θ$ term.⁹ The $θ$ term is written as

where $h=2πℏ$ is the Planck’s constant, $e>0$ is the magnitude of the electron charge, $c$ is the speed of light, and $E$ and $B$ are external electric and magnetic fields, respectively. From the variation of this action with respect to $E$ and $B$⁠, we obtain the cross-correlated responses expressed by

with $P$ being electric polarization and $M$ being magnetization. We see that Eq. (2) clearly exhibits a linear magnetoelectric effect, as schematically illustrated in Fig. 2. Since $E⋅B$ is odd under time reversal (i.e., $E⋅B→−E⋅B$ under $t→−t$⁠), time-reversal symmetry requires that the action (1) is invariant under the transformation $θ→−θ$⁠. Then, it follows that in the presence of time-reversal symmetry $θ$ takes a quantized value $θ=π$ (mod $2π$⁠) for topological insulators, while $θ=0$ in trivial insulators. A simple and intuitive proof of this quantization has been given.³¹ However, in systems with broken time-reversal symmetry, e.g., in magnetically ordered phases, the value of $θ$ can be arbitrary, i.e., can deviate from the quantized value $π$ or $0$⁠,³² which means that the value of $θ$ can even depend on space and time as $θ(r,t)$⁠. A similar argument can be applied to spatial-inversion symmetry. Namely, $θ$ takes a quantized value $θ=π$ or $θ=0$ (mod $2π$⁠) in the presence of inversion symmetry,^33,34 and inversion symmetry breaking can also lead to the deviation of $θ$ from the quantized value, because $E⋅B$ is also odd under spatial inversion. Table I shows the constraints on the value of $θ$ by time-reversal and spatial-inversion symmetries.

The magnetoelectric effect is the generation of bulk electric polarization (magnetization) by an external magnetic (electric) field. The linear magnetoelectric coupling coefficient is generically described by

where $i,j=x,y,z$ indicates the spatial direction, $E$ and $B$ are external electric and magnetic fields, and $P$ and $M$ are the electric polarization and the magnetization. In general, both time-reversal and spatial-inversion symmetries of the system must be broken, since the occurrence of nonzero $P$ (⁠$M$⁠) breaks spatial-inversion (time-reversal) symmetry. This requirement is consistent with the constraints on the value of $θ$ by time-reversal and spatial-inversion symmetries (see Table I). Among several origins of the magnetoelectric effect, we are particularly interested in the orbital (i.e., electronic band) contribution to the linear magnetoelectric coupling of the form

where $δij$ is the Kronecker delta. Here, note that $θ$ is a dimensionless constant. Equation (4) implies the Lagrangian density $L=(e2θ/4π2ℏc)E⋅B$⁠, since the magnetization and polarization can be derived from the free energy of the system $F$ as $M=−∂F/∂B$ and $P=−∂F/∂E$⁠. Notably, the susceptibility of the topological magnetoelectric effect in Eq. (4) with $θ=π$ reads (in SI units)

which is rather large compared to those of prototypical magnetoelectric materials, e.g., the total linear magnetoelectric susceptibility $αxx=αyy=0.7ps/m$ of the well-known antiferromagnetic Cr$2$O$3$ at low temperatures.^35,36

It should be noted here that we need to take into account the presence of boundaries (i.e., surfaces) of a 3D topological insulator, when we consider the realization of the quantized magnetoelectric effect in a 3D topological insulator. This is because, as is mentioned just above, finite $P$ and $M$ require the breaking of both time-reversal and spatial-inversion symmetries of the whole system, whereas the bulk of the topological insulator has to respect both time-reversal and inversion symmetries. As we will see in the following, the occurrence of the quantized magnetoelectric effect is closely related to the (half-quantized) anomalous Hall effect on the surface, which requires a somewhat special setup that breaks both time-reversal and inversion symmetries as shown in Fig. 2. In this setup, time-reversal symmetry is broken due to the surface magnetization. Inversion symmetry is also broken because the magnetization directions on a side surface and the other side surface are opposite to each other (spatial inversion does not change the direction of spin).

Before deriving the quantized magnetoelectric effect in 3D topological insulators, we briefly consider the anomalous Hall effect on the surfaces in which the Hall conductivity takes a half-quantized value $e2/2h$⁠. Let us start with the effective Hamiltonian for the surface states of 3D topological insulators such as Bi$2$Se$3$⁠, which is described by 2D two-component massless Dirac fermions,³⁷ 

where $vF$ is the Fermi velocity of the surface state (i.e., the slope of the Dirac cone) and $σx,σy$ are the Pauli matrices for the spin degree of freedom. The energy eigenvalues of the Hamiltonian (6) are readily obtained as $Esurface(k)=±ℏvFkx2+ky2$ from a simple algebra $Hsurface2=ℏ2vF2(kx2+ky2)12×2$⁠. The Fermi velocity of the surface states in Bi$2$Se$3$ is experimentally observed as $vF≈5×105$ m/s.³⁸ 

Due to the spin-momentum locking, the surface states are robust against disorder, as long as time-reversal symmetry is preserved. Namely, the backscattering of surface electrons from $(k,↑)$ to $(−k,↑)$ are absent.³⁹ Theoretically, it has been shown that 2D two-component massless Dirac fermions cannot be localized in the presence of nonmagnetic disorder.^40,41 However, surface states are not robust against magnetic disorder that breaks time-reversal symmetry. This is because the surface Dirac fermions described by Eq. (6) can be massive by adding a term proportional to $σz$⁠, i.e., $mσz$⁠, which opens a gap of $2m$ in the energy spectrum. More precisely, such a mass term can be generated by considering the exchange interaction between the surface electrons and magnetic impurities^42–44 such that $Hexch.=J∑iSi⋅σδ(r−Ri)$⁠, where $Si$ is the impurity spin at position $Ri$⁠. Then, the homogeneous part of the impurity spins gives rise to the position-independent Hamiltonian,

where $nimp$ is the density of magnetic impurities and $S¯imp$ is the averaged spin of magnetic impurities. Adding Eq. (7) to the Hamiltonian (6) leads to a gapped spectrum

We see that $mx$ and $my$ do not open the gap but only shift the position of the Dirac cone in the momentum space.Let us consider a general $2×2$ Hamiltonian given by $H(k)=R(k)⋅σ$⁠. In the case of massive Dirac fermions, $R(k)$ is given by $R(k)=(vFky,−vFkx,mz)$⁠. The Hall conductivity of the system with the Fermi level being in the gap can be calculated by⁴⁵ 

where $R^=R(k)/|R(k)|$ is a unit vector. The integral is equivalent to the area where the unit vector $R^$ moves on the unit sphere, which, namely, gives the winding number of $R^$⁠. At $k=0$⁠, the unit vector $R^$ points to the north or south pole, that is, $R^=(0,0,sgn(mz))$⁠. At large $k$ with $|k|≫|mz|$⁠, $R^$ almost points to the horizontal directions. Hence, varying $k$⁠, $R^$ covers the half of the unit sphere, which gives $2π$⁠.

Equation (9) indicates that the anomalous Hall effect occurs on the surfaces of 3D topological insulators, when magnetic impurities are doped or a magnetic film is put on the surfaces.^44,46 The direction of the Hall current depends on the sign of $mz$⁠, i.e., the direction of the magnetization of magnetic impurities or proximitized magnetization. Actually, the surface quantum anomalous Hall effect has been observed experimentally.^47,48 The observed surface quantum anomalous Hall effect in a thin film of Cr-doped (Bi,Sb)$2$Te$3$ is shown in Fig. 3. Note that in those systems, the magnetization directions of top and bottom surfaces are the same, and thus the observed Hall conductivity is $2×e2/(2h)=e2/h$⁠. It can be seen from Fig. 3(b) that the Hall conductivity takes the quantized value when the chemical potential lies in the surface bandgap.

We have seen in Sec. II C that the surface states of 3D topological insulators can be gapped (i.e., the surface Dirac fermions can be massive) via the exchange interaction with magnetic impurities or proximitized magnetization which breaks time-reversal symmetry, giving rise to the surface half-quantized anomalous Hall effect. We show phenomenologically in the following that, as a consequence of the surface half-quantized anomalous Hall effect, the topological magnetoelectric effect [Eq. (2)] emerges in 3D topological insulators.

Let us consider a case where the side surface of a cylindrical 3D topological insulator is ferromagnetically ordered due to magnetic doping or the proximity effect,⁹ as shown in Fig. 2. The resulting surface Dirac fermions are massive. When an external electric field $E$ is applied parallel to the cylinder, the surface anomalous Hall current $jH$ is induced as

where $n^$ is a unit vector normal to the side surface. From the Ampère’s law, the magnetization $M$ with $|M|=|jH|/c$ (⁠$c$ is the speed of light) is obtained as [see Fig. 2(a)]

Similarly, when an external magnetic field $B$ is applied parallel to the cylinder, the circulating electric field $Eind$ normal to the magnetic field is induced as $∇×Eind=−∂B/∂t$⁠. Then, the induced electric field $Eind$ generates the surface anomalous Hall current parallel to the magnetic field as

On the other hand, a polarization current is equivalent to the time derivative of the electric polarization. Finally, the induced electric polarization $P$ is given by [see Fig. 2(b)]

Equations (11) and (13) clearly show the magnetoelectric effect. Here, recall that the magnetization and polarization can be derived from the free energy of the system $F$ as $M=−∂F/∂B$ and $P=−∂F/∂E$⁠. To satisfy the relations (11) and (13), the free energy must have the following form:⁹ 

where we have omitted $sgn(m)$ for simplicity, and $θ=π$⁠. The integrand can be regarded as the Hamiltonian density. The equivalent action is written as

where $d4x=dtd3r$⁠, $Fμν=∂μAν−∂νAμ$ with $Aμ=(A0,−A)$ being the electromagnetic four potential, and $εμνρλ$ is the Levi–Civitá symbol with the convention $ε0123=1$⁠. Here, the electric field and the magnetic field are given, respectively, by $E=−∇A0−∂A/∂t$ and $B=∇×A$⁠. Note that $e2/ℏc$ (⁠$≃1/137$⁠) is the fine-structure constant. Equation (15) is indeed the $θ$ term [Eq. (1)]. Under time-reversal (⁠$t→−t$⁠), electric and magnetic fields are transformed as $E→E$ and $B→−B$⁠, respectively. Similarly, under spatial inversion (⁠$r→−r$⁠), electric and magnetic fields are transformed as $E→−E$ and $B→B$⁠, respectively. Hence, the term $E⋅B$ is odd under time-reversal or spatial inversion. On the other hand, 3D topological insulators have time-reversal symmetry, which indicates that $Sθ$ remains unchanged under time-reversal. In other words, the value of $θ$ must be invariant under the transformation $θ→−θ$⁠. It follows that $θ=π$ (mod $2π$⁠) in time-reversal invariant topological insulators and $θ=0$ in normal (topologically trivial) insulators.

Note that $Sθ$ is a surface term when the value of $θ$ is constant, i.e., independent of spatial coordinate and time, since we can rewrite the integrand of $Sθ$ in a total derivative form,

which indicates that the topological magnetoelectric effect in the bulk is a consequence of the surface response to the electric and magnetic fields. However, as we shall see later, the presence of the $θ$ term that is dependent of spatial coordinate and/or time results in an electric current generation in the bulk.

Here, let us consider the inverse process of the derivation of the $θ$ term (15). Namely, we derive the surface anomalous Hall current from Eq. (15). We have seen in Eq. (16) that the integrand of the $θ$ term is a total derivative when the value of $θ$ is constant. For definiteness, let us see what happens at a given surface in the $z$ direction. Using Eq. (16) and integrating out with respect to $z$⁠, the surface term can be obtained from Eq. (15) as

where $d3x=dtdxdy$⁠. Recall that, in general, an electric current density $jν$ in the $ν$ direction can be obtained from the variation of an action with respect to the electromagnetic vector potential $Aν$⁠: $jν=δS/δAν$⁠. Without loss of generality, we may consider the current in the $x$ direction,

where $Ey=−∂yA0−∂tAy$ is the electric field in the $y$ direction. Since $θ=π$ in topological insulators, Eq. (18) clearly shows the surface half-quantized anomalous Hall effect.

More precisely, we should consider an electric current derived directly from the $θ$ term. Namely, we should consider the spatial dependence of $θ$ such that $θ=0$ in vacuum and $θ=π$ inside the topological insulator. Notice that the $θ$ term can be rewritten as

Then, the electric current density is obtained as

The magnetic-field induced term is the so-called chiral magnetic effect,⁴⁹ which will be mentioned later. For concreteness, we require that the region $z≤0$ (⁠$z>0$⁠) be the topological insulator (vacuum). The $z$ dependence of $θ(r,t)$ can be written in terms of the Heaviside step function as $θ(z)=π[1−Θ(z)]$⁠, since $θ=π$ (⁠$θ=0$⁠) inside (outside) the topological insulator. Then, we obtain $∂zθ=−πδ(z)$⁠, which gives rise to the half-quantized Hall conductivity at the topological insulator surface $z=0$⁠.

So far, we have derived the topological magnetoelectric effect [Eq. (2)] from a surface property of 3D topological insulators. In this section, we derive the $θ$ term microscopically from a low-energy effective model of 3D topological insulators. There are several ways to derive the $θ$ term microscopically. One way is to use the so-called Fujikawa’s method.^50,51 Another way is the dimensional reduction from (4+1)-dimensions to (3+1)-dimensions,⁹ which will be briefly mentioned in Sec. IV A. Here, we show the derivation of the $θ$ term based on Fujikawa’s method.

Let us start from the low-energy continuum model for prototypical 3D topological insulators such as Bi$2$Se$3$⁠. The bulk electronic structure of Bi$2$Se$3$ near the Fermi level is described by two $p$-orbitals $P1z+$ and $P2z−$ with $±$ denoting parity. Defining the basis $[|P1z+,↑⟩,|P1z+,↓⟩,|P2z−,↑⟩,|P2z−,↑⟩]$ and retaining the wave vector $k$ up to quadratic order, the low-energy effective Hamiltonian around the $Γ$ point is given by^37,52

where $k±=kx±iky$ and $M(k)=m0−B1kz2−B2k⊥2$⁠. The coefficients for Bi$2$Se$3$ estimated by a first-principles calculation read $m0=0.28$ eV, $A1=2.2$ eV$⋅$Å, $A2=4.1$ eV$$Å, $B1=10$ eV$$Å$2$⁠, and $B2=56.6$ eV$$Å$2$⁠.^37,52 Here, note that we have introduced a basis in Eq. (21) that is slightly different from that Refs. 37 and 52. The $4×4$ matrices $αμ$ are given by the so-called Dirac representation,

where the Clifford algebra ${αμ,αν}=2δμν1$ is satisfied. The above Hamiltonian is nothing but an anisotropic 3D Dirac Hamiltonian with a momentum-dependent mass.

Before proceeding to the derivation of the $θ$ term, it is informative to consider the lattice version of Eq. (21). Here, recall that the $Z2$ invariant,^1,6–8 which identifies whether a phase is topologically nontrivial or trivial, is calculated in lattice models. This means that we cannot directly show that the phase described by the effective Hamiltonian (21) represents a 3D topological insulator. From this viewpoint, we need to construct a lattice Hamiltonian from the continuum Hamiltonian (21). The simplest 3D lattice is the cubic lattice. We replace $ki$ and $ki2$ terms by $ki→sin⁡ki$ and $ki2→2(1−cos⁡ki)$⁠. Although this replacement is valid only when $ki≪1$⁠, as is shown below, it turns out that this replacement describes the topological insulator phase. We also simplify the coefficients to obtain the isotropic lattice Hamiltonian

where we have defined $ℏvF=A1=A2$ and $r=−2B1=−2B2$⁠. As is mentioned below, the Hamiltonian (23) is also called the Wilson–Dirac Hamiltonian,^53–55 which was originally introduced in lattice quantum chromodynamics.

In cubic lattices, the eight time-reversal invariant momenta $Λα$⁠, which are invariant under $ki→−ki$⁠, are given by $(0,0,0)$⁠, $(π/a,0,0)$⁠, $(0,π/a,0)$⁠, $(0,0,π/a)$⁠, $(π/a,π/a,0)$⁠, $(π/a,0,π/a)$⁠, $(0,π/a,π/a)$⁠, and $(π/a,π/a,π/a),$ where $a$ is the lattice constant. We can calculate the $Z2$ invariant of the system as^6,8

Indeed, the topological insulator phase with $0>m0/r>−2$ satisfies the above realistic value for Bi$2$Se$3$⁠; $m0/r∼−0.1$⁠, where we have assumed the value of the lattice constant as $a=3$ Å.

It should be noted here that the lattice Dirac Hamiltonian (23) is exactly the same as the Hamiltonian of the Wilson fermions, which was originally introduced in the lattice gauge theory to avoid the fermion doubling problem.⁵³ Namely, we can see that Eq. (23) around the $Γ$ point $(0,0,0)$ represents the usual (continuum) massive Dirac fermions with mass $m0$⁠, while Eq. (23) around other momentum points, e.g., $(π/a,0,0)$⁠, represent massive Dirac fermions with the mass $m0+2r$⁠.

Now, let us return to the continuum Hamiltonian (21) to obtain the $θ$ term. As we have seen in Eq. (24), the lattice Hamiltonian (23) describes a topological insulator when $0>m0/r>−2$⁠. Without loss of generality, we can set $m0<0$ and $r>0$⁠. Then, the Hamiltonian (21) with $m0<0$ and $r>0$⁠, which describes a topological insulator, around the $Γ$ point can be simplified by ignoring the terms second-order in $ki$ as

where $m0<0$⁠. Except for the negative mass $m0$⁠, this is the usual Dirac Hamiltonian. In the presence of an external electromagnetic vector potential $A$⁠, minimal coupling results in $k→k+eA$⁠, with $e>0$ being the magnitude of the electron charge. In the presence of an external electromagnetic scalar potential $A0$⁠, the energy density is modified as $ψ†H0ψ→ψ†(H0−eA0)ψ$⁠. Using these facts, the action of the system in the presence of an external electromagnetic four potential $Aμ=(A0,−A)$ is written in the usual relativistic form,⁵⁶ 

where $ψ†(r,t)$ is a fermionic field representing the basis of the Hamiltonian (21) and $ψ¯=ψ†γ0$⁠. Here, the gamma matrices $γμ$ are given by the so-called Dirac representation as

which satisfy the relation ${γμ,γν}=2gμν$ with $gμν=diag(+1,−1,−1,−1)$ being the metric tensor. It is convenient to study the system in the imaginary time notation, i.e., in Euclidean spacetime. Namely, we rewrite $t$⁠, $A0$⁠, and $γj$ as $t→−iτ$⁠, $A0→iA0$⁠, and $γj→iγj$ (⁠$j=1,2,3$⁠). The Euclidean action of the system is then written as

where we have used the fact that $m0=−m0(cos⁡π+iγ5sin⁡π)=−m0eiπγ5$⁠. Note that $γ0$ and $γ5$ are unchanged (⁠$γ0=γ0$ and $γ5=γ5$⁠), so that the anticommutation relation ${γμ,γν}=2δμν$ is satisfied. Note also that, in Euclidean spacetime, we do not distinguish between superscripts and subscripts.

Now, we are in a position to apply Fujikawa’s method^50,51 to the action (28). First, let us consider an infinitesimal chiral transformation defined by

where $ϕ∈[0,1]$⁠. Then, the partition function $Z$ is transformed as

The $θ$ term comes from the Jacobian defined by $D[ψ′,ψ¯′]=JD[ψ,ψ¯]$⁠. The action (28) is transformed as

The Jacobian is written as^50,51

Here, $Fμν=∂μAν−∂νAμ$⁠, and we have written $ℏ$ and $c$ explicitly. We repeat this procedure infinite times, i.e., integrate with respect to the variable $ϕ$ from $0$ to $1$⁠. Due to the invariance of the partition function, finally, we arrive at the following expression of $STIE$⁠:

where we have dropped the irrelevant surface term. The first term is the action of a topologically trivial insulator, since the mass $−m0$ is positive. The second term is the $θ$ term in the imaginary time, and we obtain Eq. (15) by substituting $τ=it$⁠.

As we have seen in Sec. II D, the experimental realization of the topological magnetoelectric effect in topological insulators requires that all the surface Dirac states are gapped by the magnetic proximity effect or magnetic doping, resulting in the zero anomalous Hall conductivity of the system. However, such an experimental setup is rather difficult to be realized. As an alternate route to realize the topological magnetoelectric effect, it has been proposed theoretically that the $ν=0$ quantum Hall state, which attributes to the difference between the Landau levels of the top and bottom surface Dirac states, can be utilized.^57,58 The $ν=0$ quantum Hall state has been experimentally observed in topological insulator (Bi$1−x$Sb$x$⁠)$2$Te$3$ films,⁵⁹ as shown in Fig. 4(a). The two-component Dirac fermions in a magnetic field are known to show the quantum Hall effect with the Hall conductivity,

where $n$ is an integer. Note that, as we have seen in Eq. (9), the $12$ contribution arises as a Berry phase effect. The total Hall conductivity contributed from the top and bottom surfaces of a topological insulator film in a magnetic field is then written as

The $ν=0$ quantum Hall state is realized when the Landau levels of the top and bottom surface states are $NT=−N−1$ and $NB=N$ (and vice versa), where $N$ is an integer.⁵⁷ This state corresponds to $nT=−N−1$ and $nB=N$ in Eq. (35), which can be achieved in the presence of an energy difference between the two surface states, as shown in Fig. 4(b). Here, recall that the electron density is given by $ne=σxyB/e$⁠, with $B$ being the magnetic field strength and $e$ being the elementary charge. Using this fact, the charge densities (⁠$ρ=−ene$⁠) at the top and bottom surfaces are obtained as $ρT=(N+12)Be2/h$ and $ρB=−(N+12)Be2/h$⁠, respectively. We consider the case of $N=0$⁠, which is experimentally relevant.⁵⁹ The induced electric polarization in a topological insulator film of thickness $d$ reads

which is indeed the topological magnetoelectric effect with the quantized coefficient $θ=π$⁠. Note that the case of $N≠0$⁠, which gives rise to $θ=(2N+1)π$⁠, still describes the topological magnetoelectric effect, since $θ=π$ modulo $2π$⁠. Another route to realize the topological magnetoelectric effect is a magnetic heterostructure in which the magnetization directions of the top and bottom magnetic insulators are antiparallel.^57,58 Several experiments have succeeded in fabricating magnetic heterostructures that exhibits a zero Hall plateau.^60–62 In Ref. 60, a magnetic heterostructure consisting of a magnetically doped topological insulator Cr-doped (Bi,Sb)$2$Te$3$ and a topological insulator (Bi,Sb)$2$Te$3$ was grown by molecular beam epitaxy. A zero Hall conductivity plateau was observed in this study as shown in Fig. 5, implying an axion insulator state. In Ref. 62, a magnetic heterostructure of a topological insulator (Bi,Sb)$2$Te$3$ sandwiched by two kinds of magnetically doped topological insulators V-doped (Bi,Sb)$2$Te$3$ and Cr-doped (Bi,Sb)$2$Te$3$ was grown by molecular beam epitaxy. Importantly, as shown in Fig. 6, the antiparallel magnetization alignment of the top and bottom magnetic layers was directly observed by magnetic force microscopy when the system exhibited a zero Hall resistivity plateau. Note, however, that the above experiments did not make a direct observation of the magnetoelectric effect, i.e., the electric polarization induced by a magnetic field or the magnetization induced by an electric field.

As has been known in particle physics^12,63 before the discovery of 3D topological insulators, the $θ$ term modifies the Maxwell’s equations. Since the Maxwell’s equations describe electromagnetic wave propagation in materials, the presence of the $θ$ term leads to unusual optical properties such as the quantized Faraday and Kerr rotations in topological insulators,^9,64,65 which can be viewed as a consequence of the topological magnetoelectric effect. To see this, let us start from the total action of an electromagnetic field $Aμ=(A0,−A)$ in the presence of a $θ$ term is given by

where $α=e2/ℏc≃1/137$ is the fine-structure constant and $Fμν=∂μAν−∂νAμ$ is the electromagnetic field tensor. The electric and magnetic fields are, respectively, given by $E=−∇A0−(1/c)∂A/∂t$ and $B=∇×A$⁠. Note that $E⋅B=(1/8)εμνρλFμνFρλ$ and $FμνFμν=2(B2/μ0−ε0E2)$⁠. Here, recall that the classical equation of motion for the field $Aμ$ is obtained from the Euler–Lagrange equation,

where $L$ is the Lagrangian density of the system. From Eqs. (37) and (38), one finds that the Maxwell’s equations are modified in the presence of a $θ$ term^9,12,63

The $∇θ$ terms in Eq. (39) play roles when there is a boundary, e.g., gives rise to the surface Hall current as we have seen in Eq. (20).

The modified Maxwell’s Eq. (39) can be solved under the boundary conditions (see Fig. 7). It is found that the Faraday and Kerr rotation angles are independent of the material (i.e., topological insulator thin film) parameters such as the dielectric constant and thickness.^64,65 Specifically, in the quantized limit, the Faraday and Kerr rotation angles are given, respectively, by^64,65

These quantized angles have been experimentally observed in the anomalous Hall state⁶⁶ and the quantum Hall state [Fig. 8(a)].^67,68 Also, as predicted in Ref. 64, a universal relationship in units of the fine-structure constant $α$ between the Faraday and Kerr rotation angles has been observed [Fig. 8(b)].^66,67

In Sec. II, we have seen that the topological magnetoelectric effect with the quantized coefficient $θ=π$ (mod $2π$⁠) occurs in 3D time-reversal invariant topological insulators. In general, the value of $θ$ is no longer quantized and becomes arbitrary in systems with broken time-reversal symmetry. However, in a class of 3D antiferromagnetic insulators, an “effective” time-reversal symmetry represented by a combination of time-reversal and a lattice translation is present, leading to the topological magnetoelectric effect with the quantized coefficient $θ=π$ (mod $2π$⁠). In this section, we review theoretical and experimental studies on such antiferromagnetic topological insulators, which are also called the axion insulators. Starting from the basics of the antiferromagnetic topological insulators, we focus on the MnBi$2$Te$4$ family of materials that are layered van der Waals compounds and have recently been experimentally realized.

Following Ref. 69, we consider a class of insulators in which time-reversal symmetry is broken but the combined symmetry of time-reversal and a lattice translation is preserved. We note here that the presence or absence of inversion symmetry does not affect their topological classification, although the presence of inversion symmetry greatly simplifies the evaluation of their topological invariants as in the case of time-reversal invariant topological insulators.⁸ Let us start from some general arguments on symmetry operations. The time-reversal operator $Θ$ for spin-1/2 systems is generically given by $Θ=iσyK$ with $Θ2=−1$⁠, where $σi$ are Pauli matrices and $K$ is complex conjugation operator. In the presence of time-reversal symmetry, the Bloch Hamiltonian of a system $H(k)$ satisfies

Recall that momentum is the generator of lattice translation. An operator that denotes a translation by a vector $x$ is given by $T(x)=e−ik⋅x$⁠. Then, the translation operator that moves a lattice by half a unit cell in the $a3$ direction is written as

where $a3$ is a primitive translation vector and $1$ is an identity operator that acts on the half of the unit cell.⁶⁹ One can see that $T1/22$ gives a translation by $a3$ because $T1/22=e−ik⋅a3$⁠. Now, we consider the combination of $Θ$ and $T1/2$ defined by $S=ΘT1/2$⁠. It follows that $S2=−e−ik⋅a3$⁠, which means that the operator $S$ is antiunitary like $Θ$⁠. Here, we have used the fact that $Θ$ and $T1/2$ are commute. Note, however, that $S2=−1$ only on the Brillouin zone plane satisfying $k⋅a3=0$⁠, while $Θ2=−1$⁠. When a system is invariant under the operation $S$⁠, the Bloch Hamiltonian $H(k)$ satisfies

which has the same property as time-reversal symmetry in Eq. (41). Therefore, the $Z2$ topological classification can also be applied in systems with the $S$ symmetry.^69,70Figure 9 shows a schematic illustration of an antiferromagnetic topological insulator protected by the $S=ΘT1/2$ symmetry. In this simple model, the unit cell consists of nonmagnetic equivalent $A1$ and $A2$ atomic layers and antiferromagnetically ordered $B1$ and $B2$ atomic layers. The half-uni-cell translation $T1/2$ moves the $B1$ layer to the $B2$ layer, and time-reversal $Θ$ changes a spin-up state into a spin-down state. Therefore, the system is obviously invariant under the $S=ΘT1/2$ transformation.