One of the intriguing properties characteristic to three-dimensional topological materials is the topological magnetoelectric phenomena arising from a topological term called the θ term. Such magnetoelectric phenomena are often termed the axion electrodynamics since 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 and is now considered a candidate for dark matter. In this Tutorial, we overview theoretical and experimental studies on the axion electrodynamics in three-dimensional topological materials. Starting from the topological magnetoelectric effect in three-dimensional time-reversal invariant topological insulators, we describe the basic properties of static and dynamical axion insulators whose realizations require magnetic orderings. We also discuss the electromagnetic responses of Weyl semimetals with a focus on the chiral anomaly. We extend the concept of the axion electrodynamics in condensed matter to topological superconductors, whose responses to external fields can be described by a gravitational topological term analogous to the θ term.

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 Z 2 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  2 π) 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.

FIG. 1.

Schematic of a classification of 3D insulators in terms of time-reversal symmetry and the orbital magnetoelectric coupling coefficient θ. In the first classification process, 3D insulators are divided into two types: insulators with or without time-reversal symmetry. In the second classification process, 3D insulators with time-reversal symmetry are divided into types: topological insulators and normal (trivial) insulators. Topological insulators are characterized by the topological magnetoelectric effect with the quantized coefficient θ = π (mod  2 π). In the second classification process, 3D insulators with broken time-reversal symmetry are divided into two types: axion insulators and magnetic insulators. In axion insulators, time-reversal symmetry is broken but 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  2 π). In magnetic insulators, the value of θ is arbitrary, including θ = 0. In a class of magnetic insulators termed topological magnetic insulators, θ is proportional to their magnetic order parameters M such as the Néel vector (i.e., antiferromagnetic order parameter), and the fluctuation of the order parameter realizes a dynamical axion field δ θ ( r , t ) δ M ( r , t ) in condensed matter. Here, note that spatial-inversion symmetry must be broken in order for the value of θ to be arbitrary, i.e., in the magnetic insulators we have mentioned above, whereas its breaking is not required in the other three phases. See also Table I for the role of inversion symmetry.

FIG. 1.

Schematic of a classification of 3D insulators in terms of time-reversal symmetry and the orbital magnetoelectric coupling coefficient θ. In the first classification process, 3D insulators are divided into two types: insulators with or without time-reversal symmetry. In the second classification process, 3D insulators with time-reversal symmetry are divided into types: topological insulators and normal (trivial) insulators. Topological insulators are characterized by the topological magnetoelectric effect with the quantized coefficient θ = π (mod  2 π). In the second classification process, 3D insulators with broken time-reversal symmetry are divided into two types: axion insulators and magnetic insulators. In axion insulators, time-reversal symmetry is broken but 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  2 π). In magnetic insulators, the value of θ is arbitrary, including θ = 0. In a class of magnetic insulators termed topological magnetic insulators, θ is proportional to their magnetic order parameters M such as the Néel vector (i.e., antiferromagnetic order parameter), and the fluctuation of the order parameter realizes a dynamical axion field δ θ ( r , t ) δ M ( r , t ) in condensed matter. Here, note that spatial-inversion symmetry must be broken in order for the value of θ to be arbitrary, i.e., in the magnetic insulators we have mentioned above, whereas its breaking is not required in the other three phases. See also Table I for the role of inversion symmetry.

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TABLE I.

Constraints on the value of θ by time-reversal and spatial-inversion symmetries. The mark ✓ (×) indicates the presence (absence) of the symmetry. Here, the notation of time-reversal symmetry in this table includes an “effective” time-reversal symmetry represented by a combination of time-reversal and a lattice translation, as well as “true” time-reversal symmetry.

Time reversalInversionValue of θ (mod 2π)
✓ ✓ 0 or π 
✓ × 0 or π 
× ✓ 0 or π 
× × Arbitrary 
Time reversalInversionValue of θ (mod 2π)
✓ ✓ 0 or π 
✓ × 0 or π 
× ✓ 0 or π 
× × Arbitrary 

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 b 0 t ),21–25 where b is the distance between the two Weyl nodes in momentum space and b 0 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 2Te 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
(1)
where h = 2 π 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
(2)
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  2 π) 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  2 π) 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.
FIG. 2.

Schematic picture of the topological magnetoelectric effect in a 3D topological insulator. (a) Magnetization M induced by an external electric field E. j H is the anomalous Hall current on the side surface induced by the electric field. (b) Electric polarization P induced by an external magnetic field B. Surface states are gapped by magnetic impurities (or a proximitized ferromagnet) whose magnetization direction is perpendicular to the surface, as indicated by green arrows.

FIG. 2.

Schematic picture of the topological magnetoelectric effect in a 3D topological insulator. (a) Magnetization M induced by an external electric field E. j H is the anomalous Hall current on the side surface induced by the electric field. (b) Electric polarization P induced by an external magnetic field B. Surface states are gapped by magnetic impurities (or a proximitized ferromagnet) whose magnetization direction is perpendicular to the surface, as indicated by green arrows.

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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
(3)
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
(4)
where δ i j is the Kronecker delta. Here, note that θ is a dimensionless constant. Equation (4) implies the Lagrangian density L = ( e 2 θ / 4 π 2 c ) 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)
(5)
which is rather large compared to those of prototypical magnetoelectric materials, e.g., the total linear magnetoelectric susceptibility α x x = α y y = 0.7 ps / m of the well-known antiferromagnetic Cr 2O 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 e 2 / 2 h. Let us start with the effective Hamiltonian for the surface states of 3D topological insulators such as Bi 2Se 3, which is described by 2D two-component massless Dirac fermions,37 
(6)
where v F is the Fermi velocity of the surface state (i.e., the slope of the Dirac cone) and σ x , σ y are the Pauli matrices for the spin degree of freedom. The energy eigenvalues of the Hamiltonian (6) are readily obtained as E surface ( k ) = ± v F k x 2 + k y 2 from a simple algebra H surface 2 = 2 v F 2 ( k x 2 + k y 2 ) 1 2 × 2. The Fermi velocity of the surface states in Bi 2Se 3 is experimentally observed as v F 5 × 10 5 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 σ z, i.e., m σ z, which opens a gap of 2 m in the energy spectrum. More precisely, such a mass term can be generated by considering the exchange interaction between the surface electrons and magnetic impurities42–44 such that H exch . = J i S i σ δ ( r R i ), where S i is the impurity spin at position R i. Then, the homogeneous part of the impurity spins gives rise to the position-independent Hamiltonian,
(7)
where n imp 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
(8)
We see that m x and m y 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 ) = ( v F k y , v F k x , m z ). The Hall conductivity of the system with the Fermi level being in the gap can be calculated by45 
(9)
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 ( m z ) ). At large k with | k | | m z |, R ^ almost points to the horizontal directions. Hence, varying k, R ^ covers the half of the unit sphere, which gives 2 π.

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 m z, 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) 2Te 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 × e 2 / ( 2 h ) = e 2 / 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.

FIG. 3.

(a) Schematic illustration of an experimental setup to detect the quantum anomalous Hall effect in a ferromagnetically ordered topological insulator thin film. (b) Gate-voltage V g dependence of the Hall conductivity σ x y and the longitudinal conductivity σ x x in a thin film of Cr-doped (Bi,Sb) 2Te 3. Reproduced with permission from Chang et al., Science 340, 167 (2013). Copyright 2013 American Association for the Advancement of Science.

FIG. 3.

(a) Schematic illustration of an experimental setup to detect the quantum anomalous Hall effect in a ferromagnetically ordered topological insulator thin film. (b) Gate-voltage V g dependence of the Hall conductivity σ x y and the longitudinal conductivity σ x x in a thin film of Cr-doped (Bi,Sb) 2Te 3. Reproduced with permission from Chang et al., Science 340, 167 (2013). Copyright 2013 American Association for the Advancement of Science.

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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 j H is induced as
(10)
where n ^ is a unit vector normal to the side surface. From the Ampère’s law, the magnetization M with | M | = | j H | / c ( c is the speed of light) is obtained as [see Fig. 2(a)]
(11)
Similarly, when an external magnetic field B is applied parallel to the cylinder, the circulating electric field E ind normal to the magnetic field is induced as × E ind = B / t. Then, the induced electric field E ind generates the surface anomalous Hall current parallel to the magnetic field as
(12)
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)]
(13)
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 
(14)
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
(15)
where d 4 x = d t d 3 r, F μ ν = μ A ν ν A μ with A μ = ( A 0 , A ) being the electromagnetic four potential, and ε μ ν ρ λ is the Levi–Civitá symbol with the convention ε 0123 = 1. Here, the electric field and the magnetic field are given, respectively, by E = A 0 A / t and B = × A. Note that e 2 / c ( 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 S θ 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  2 π) in time-reversal invariant topological insulators and θ = 0 in normal (topologically trivial) insulators.
Note that S θ 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 S θ in a total derivative form,
(16)
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
(17)
where d 3 x = d t d x d y. Recall that, in general, an electric current density j ν in the ν direction can be obtained from the variation of an action with respect to the electromagnetic vector potential A ν: j ν = δ S / δ A ν. Without loss of generality, we may consider the current in the x direction,
(18)
where E y = y A 0 t A y 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
(19)
Then, the electric current density is obtained as
(20)
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 z θ = π δ ( 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.

1. Effective Hamiltonian for 3D topological insulators

Let us start from the low-energy continuum model for prototypical 3D topological insulators such as Bi 2Se 3. The bulk electronic structure of Bi 2Se 3 near the Fermi level is described by two p-orbitals P 1 z + and P 2 z with ± denoting parity. Defining the basis [ | P 1 z + , , | P 1 z + , , | P 2 z , , | P 2 z , ] and retaining the wave vector k up to quadratic order, the low-energy effective Hamiltonian around the Γ point is given by37,52
(21)
where k ± = k x ± i k y and M ( k ) = m 0 B 1 k z 2 B 2 k 2. The coefficients for Bi 2Se 3 estimated by a first-principles calculation read m 0 = 0.28 eV, A 1 = 2.2 eV Å, A 2 = 4.1 eV Å, B 1 = 10 eV Å 2, and B 2 = 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 α μ are given by the so-called Dirac representation,
(22)
where the Clifford algebra { α μ , α ν } = 2 δ μ ν 1 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 Z 2 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 k i and k i 2 terms by k i sin k i and k i 2 2 ( 1 cos k i ). Although this replacement is valid only when k i 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
(23)
where we have defined v F = A 1 = A 2 and r = 2 B 1 = 2 B 2. 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 Λ α, which are invariant under k i k i, 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 Z 2 invariant of the system as6,8
(24)
Indeed, the topological insulator phase with 0 > m 0 / r > 2 satisfies the above realistic value for Bi 2Se 3; m 0 / 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 m 0, while Eq. (23) around other momentum points, e.g., ( π / a , 0 , 0 ), represent massive Dirac fermions with the mass m 0 + 2 r.

2. Fujikawa’s method

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 > m 0 / r > 2. Without loss of generality, we can set m 0 < 0 and r > 0. Then, the Hamiltonian (21) with m 0 < 0 and r > 0, which describes a topological insulator, around the Γ point can be simplified by ignoring the terms second-order in k i as
(25)
where m 0 < 0. Except for the negative mass m 0, this is the usual Dirac Hamiltonian. In the presence of an external electromagnetic vector potential A, minimal coupling results in k k + e A, with e > 0 being the magnitude of the electron charge. In the presence of an external electromagnetic scalar potential A 0, the energy density is modified as ψ H 0 ψ ψ ( H 0 e A 0 ) ψ. Using these facts, the action of the system in the presence of an external electromagnetic four potential A μ = ( A 0 , A ) is written in the usual relativistic form,56 
(26)
where ψ ( r , t ) is a fermionic field representing the basis of the Hamiltonian (21) and ψ ¯ = ψ γ 0. Here, the gamma matrices γ μ are given by the so-called Dirac representation as
(27)
which satisfy the relation { γ μ , γ ν } = 2 g μ ν with g μ ν = 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, A 0, and γ j as t i τ, A 0 i A 0, and γ j i γ j ( j = 1 , 2 , 3). The Euclidean action of the system is then written as
(28)
where we have used the fact that m 0 = m 0 ( cos π + i γ 5 sin π ) = m 0 e i π γ 5. Note that γ 0 and γ 5 are unchanged ( γ 0 = γ 0 and γ 5 = γ 5), so that the anticommutation relation { γ μ , γ ν } = 2 δ μ ν 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 method50,51 to the action (28). First, let us consider an infinitesimal chiral transformation defined by
(29)
where ϕ [ 0 , 1 ]. Then, the partition function Z is transformed as
(30)
The θ term comes from the Jacobian defined by D [ ψ , ψ ¯ ] = J D [ ψ , ψ ¯ ]. The action (28) is transformed as
(31)
The Jacobian is written as50,51
(32)
Here, F μ ν = μ A ν ν A μ, and we have written 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 S TI E:
(33)
where we have dropped the irrelevant surface term. The first term is the action of a topologically trivial insulator, since the mass m 0 is positive. The second term is the θ term in the imaginary time, and we obtain Eq. (15) by substituting τ = i t.

1. Utilizing topological insulator thin films

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 xSb x) 2Te 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,
(34)
where n is an integer. Note that, as we have seen in Eq. (9), the 1 2 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
(35)
The ν = 0 quantum Hall state is realized when the Landau levels of the top and bottom surface states are N T = N 1 and N B = N (and vice versa), where N is an integer.57 This state corresponds to n T = N 1 and n B = 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 n e = σ x y B / e, with B being the magnetic field strength and e being the elementary charge. Using this fact, the charge densities ( ρ = e n e) at the top and bottom surfaces are obtained as ρ T = ( N + 1 2 ) B e 2 / h and ρ B = ( N + 1 2 ) B e 2 / 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
(36)
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 = ( 2 N + 1 ) π, still describes the topological magnetoelectric effect, since θ = π modulo 2 π. 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) 2Te 3 and a topological insulator (Bi,Sb) 2Te 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) 2Te 3 sandwiched by two kinds of magnetically doped topological insulators V-doped (Bi,Sb) 2Te 3 and Cr-doped (Bi,Sb) 2Te 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.
FIG. 4.

(a) Quantum Hall effect in a topological insulator (Bi 1 xSb x) 2Te 3 thin film. (b) Schematic illustration of the Landau levels of the top and bottom surface states in the presence of an energy difference between the two surfaces. Reproduced with permission from Yoshimi et al., Nat. Commun. 6, 6627 (2015). Copyright 2015 Springer Nature.

FIG. 4.

(a) Quantum Hall effect in a topological insulator (Bi 1 xSb x) 2Te 3 thin film. (b) Schematic illustration of the Landau levels of the top and bottom surface states in the presence of an energy difference between the two surfaces. Reproduced with permission from Yoshimi et al., Nat. Commun. 6, 6627 (2015). Copyright 2015 Springer Nature.

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FIG. 5.

(a) Schematic illustration of the magnetic heterostructure. Red arrows indicate the magnetization directions. (b) The observed Hall conductivity as a function of an external magnetic field. Reproduced with permission from Mogi et al., Nat. Mater. 16, 516 (2017). Copyright 2017 Springer Nature.

FIG. 5.

(a) Schematic illustration of the magnetic heterostructure. Red arrows indicate the magnetization directions. (b) The observed Hall conductivity as a function of an external magnetic field. Reproduced with permission from Mogi et al., Nat. Mater. 16, 516 (2017). Copyright 2017 Springer Nature.

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FIG. 6.

Magnetic field dependence of (a) Hall resistivity and (b) magnetic domain contrasts. (c)–(j) Magnetic force microscopy images of the magnetic domains. Red and blue represent, respectively, upward and downward parallel magnetization alignment regions, while green represents antiparallel magnetization alignment regions. Reproduced with permission from Xiao et al., Phys. Rev. Lett. 120, 056801 (2018). Copyright 2018 American Physical Society.

FIG. 6.

Magnetic field dependence of (a) Hall resistivity and (b) magnetic domain contrasts. (c)–(j) Magnetic force microscopy images of the magnetic domains. Red and blue represent, respectively, upward and downward parallel magnetization alignment regions, while green represents antiparallel magnetization alignment regions. Reproduced with permission from Xiao et al., Phys. Rev. Lett. 120, 056801 (2018). Copyright 2018 American Physical Society.

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2. Faraday and Kerr rotations

As has been known in particle physics12,63 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 A μ = ( A 0 , A ) in the presence of a θ term is given by
(37)
where α = e 2 / c 1 / 137 is the fine-structure constant and F μ ν = μ A ν ν A μ is the electromagnetic field tensor. The electric and magnetic fields are, respectively, given by E = A 0 ( 1 / c ) A / t and B = × A. Note that E B = ( 1 / 8 ) ε μ ν ρ λ F μ ν F ρ λ and F μ ν F μ ν = 2 ( B 2 / μ 0 ε 0 E 2 ). Here, recall that the classical equation of motion for the field A μ is obtained from the Euler–Lagrange equation,
(38)
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 term9,12,63
(39)
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, by64,65
(40)
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
FIG. 7.

Schematic figure of a measurement of the quantized Faraday and Kerr rotations in a topological insulator thin film. Reproduced with permission from Maciejko et al., Phys. Rev. Lett. 105, 166803 (2010). Copyright 2010 American Physical Society.

FIG. 7.

Schematic figure of a measurement of the quantized Faraday and Kerr rotations in a topological insulator thin film. Reproduced with permission from Maciejko et al., Phys. Rev. Lett. 105, 166803 (2010). Copyright 2010 American Physical Society.

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FIG. 8.

(a) Magnetic field dependence of the Faraday rotation angle. From Dziom et al., Nat. Commun. 8, 15197 (2017). Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY) License. (b) Evolution of the scaling function f ( θ F , θ K ) = cot θ F cot θ K cot 2 θ F 2 cot θ F cot θ K 1 as a function of dc Hall conductance towards the universal relationship f ( θ F , θ K ) = α. From Okada et al., Nat. Commun. 7, 12245 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 8.

(a) Magnetic field dependence of the Faraday rotation angle. From Dziom et al., Nat. Commun. 8, 15197 (2017). Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY) License. (b) Evolution of the scaling function f ( θ F , θ K ) = cot θ F cot θ K cot 2 θ F 2 cot θ F cot θ K 1 as a function of dc Hall conductance towards the universal relationship f ( θ F , θ K ) = α. From Okada et al., Nat. Commun. 7, 12245 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

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In Sec. II, we have seen that the topological magnetoelectric effect with the quantized coefficient θ = π (mod  2 π) 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  2 π). 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 2Te 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 = i σ y K with Θ 2 = 1, where σ i 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
(41)
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 i k x. Then, the translation operator that moves a lattice by half a unit cell in the a 3 direction is written as
(42)
where a 3 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 T 1 / 2 2 gives a translation by a 3 because T 1 / 2 2 = e i k a 3. Now, we consider the combination of Θ and T 1 / 2 defined by S = Θ T 1 / 2. It follows that S 2 = e i k a 3, which means that the operator S is antiunitary like Θ. Here, we have used the fact that Θ and T 1 / 2 are commute. Note, however, that S 2 = 1 only on the Brillouin zone plane satisfying k a 3 = 0, while Θ 2 = 1. When a system is invariant under the operation S, the Bloch Hamiltonian H ( k ) satisfies
(43)
which has the same property as time-reversal symmetry in Eq. (41). Therefore, the Z 2 topological classification can also be applied in systems with the S symmetry.69,70 Figure 9 shows a schematic illustration of an antiferromagnetic topological insulator protected by the S = Θ T 1 / 2 symmetry. In this simple model, the unit cell consists of nonmagnetic equivalent A 1 and A 2 atomic layers and antiferromagnetically ordered B 1 and B 2 atomic layers. The half-uni-cell translation T 1 / 2 moves the B 1 layer to the B 2 layer, and time-reversal Θ changes a spin-up state into a spin-down state. Therefore, the system is obviously invariant under the S = Θ T 1 / 2 transformation.
FIG. 9.

Schematic illustration of an antiferromagnetic topological insulator protected by the S = Θ T 1 / 2 symmetry.

FIG. 9.

Schematic illustration of an antiferromagnetic topological insulator protected by the S = Θ T 1 / 2 symmetry.

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Next, let us consider the resulting surface states. Since S 2 = 1 on the Brillouin zone plane satisfying k a 3 = 0, the 2D subsystem on the ( k 1 , k 2 ) plane is regarded as a quantum spin-Hall system with time-reversal symmetry. This means that the k 1 or k 2 dependence of the surface spectra must be gapless because the k a 3 = 0 line of the surface states is the boundary of the 2D subsystem (the k a 3 = 0 plane) in the bulk Brillouin zone. In other words, at the surfaces that are parallel to a 3, which preserve the S symmetry, there exist an odd number of gapless surface states (as in the case of a strong time-reversal invariant topological insulator). On the other hand, at the surfaces that are perpendicular to a 3, which break the S symmetry, such a topological protection of the surface states no longer exists, and the surface states can have gapped spectra.

As we have seen above, the presence of S symmetry results in a realization of a new 3D topological insulator. This implies that such topological insulators exhibit a quantized magnetoelectric effect described by a θ term, as in the case of time-reversal invariant 3D topological insulators. To see this, recall that the magnetoelectric effect resulting from a θ term is expressed as P = θ e 2 / ( 4 π 2 c ) B, and M = θ e 2 / ( 4 π 2 c ) E, where P and M are the electric polarization and the magnetization, respectively. Under time-reversal Θ, the coefficient θ changes sign θ θ, because P P and E E, while M M and B B. On the other hand, the lattice translation T 1 / 2 does not affect θ.69 Combining these, the S operation implies the transformation such that θ θ + 2 π n with n being an integer. Then, it follows that θ = 0 or θ = π modulo 2 π.

1. Electronic structure of MnBi2Te4 bulk crystals

With the knowledge of antiferromagnetic topological insulators with the S symmetry, here we review recent experimental realizations of the antiferromagnetic topological insulator state in MnBi 2Te 4.71–81 The crystal structure of MnBi 2Te 4 is shown in Fig. 10. The septuple layer consisting of Te–Bi–Te–Mn–Te–Bi–Te is stacked along the [0001] direction by van der Waals forces. A theoretical calculation of the exchange coupling constants between Mn atoms shows that the intralayer coupling in each Mn layer is ferromagnetic, while the interlayer coupling between neighboring Mn layers is antiferromagnetic.71 The magnetic ground state is thus considered to be antiferromagnetic with the Néel vector pointing the out-of-plane direction (i.e., the z direction), which is called A-type AFM- z. The Néel temperature is reported to be about 25 K.71,74,79,81 The unit cell of the antiferromagnetic insulator state consists of two septuple layers (Fig. 10), where τ 1 / 2 c is the half-cell translation vector along the c axis that connects nearest spin-up and spin-down Mn atomic layers. It can be easily seen that this interlayer antiferromagnetism between the Mn atonic layers preserves the S = Θ τ 1 / 2 c symmetry, indicating that the system is a topological antiferromagnetic insulator, which we have discussed in Sec. III A. Interestingly, the bulk bandgap is estimated to be about 0.2 eV,71,72 which is comparable to that of the time-reversal invariant topological insulator Bi 2Se 3.

FIG. 10.

Crystal and magnetic structure of the antiferromagnetic topological insulator state in MnBi 2Te 4. The unit cell consists of two septuple layers. τ 1 / 2 c is the half-cell translation vector along the c axis that connects nearest spin-up and spin-down Mn atomic layers. From Hao et al., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 10.

Crystal and magnetic structure of the antiferromagnetic topological insulator state in MnBi 2Te 4. The unit cell consists of two septuple layers. τ 1 / 2 c is the half-cell translation vector along the c axis that connects nearest spin-up and spin-down Mn atomic layers. From Hao et al., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

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The A-type AFM- z state is invariant under spatial inversion P 1 with the inversion center located at the Mn atomic layer in each septuple layers. Importantly, P 2 Θ symmetry, the combination of spatial inversion P 2 with the inversion center located between two septuple layers and time-reversal Θ, is also preserved. The presence of P 2 Θ symmetry leads to doubly degenerate bands even in the absence of time-reversal symmetry.73,82,83 Here, following Refs. 72 and 97, we derive the low-energy effective Hamiltonian of the A-type AFM- z state. P 2 Θ symmetry requires that
(44)
since momentum k changes sign under both P 2 and Θ. As in the case of Bi 2Se 3 [Eq. (21)], the low-energy effective Hamiltonian of the nonmagnetic state of MnBi 2Te 4 around the Γ point is written in the basis of [ | P 1 z + , , | P 1 z + , , | P 2 z , , | P 2 z , ], where the states | P 1 z + , ↑↓ and | P 2 z , ↑↓ come from the p z orbitals of Bi and Te, respectively.72 In this basis, P 2 = τ z 1 and Θ = 1 i σ y K, where τ i and σ i act on the orbital and spin spaces, respectively, and K is complex conjugation operator. P 2 Θ symmetry constrains the possible form of the 4 × 4 Bloch Hamiltonian H ( k ) = i , j d i j ( k ) τ i σ j. It follows that the following five matrices and the identity matrix are allowed by P 2 Θ symmetry:
(45)
due to the property ( P 2 Θ ) ( τ i σ j ) ( P 2 Θ ) 1 = τ i σ j. Note that these five matrices anticommute with each other, leading to doubly degenerate energy eigenvalues. Using these five matrices, the low-energy effective Hamiltonian around the Γ point is written as72,97
(46)
where M ( k ) = M + B 1 k z 2 + B 2 ( k x 2 + k y 2 ). The mass m 5 is induced by the antiferromagnetic order. One can see that the Hamiltonian (46) is invariant under both P 2 and Θ when m 5 = 0. Indeed, the surface states of the lattice model constructed from Eq. (46) in a slab geometry in the z direction exhibit the half-quantized anomalous Hall conductivity σ x y = ± sgn ( m 5 ) e 2 / 2 h, implying the axion insulator state.97 

The surface states of antiferromagnetic MnBi 2Te 4 are somewhat complicated. Theoretical studies have predicted that the (0001) surface state (i.e., at the surface perpendicular to the z axis) which breaks the S symmetry of the A-type AFM- z state is gapped,71,72 as indicated by the property of antiferromagnetic topological insulators (see Sec. III A). The first experimental study reported that the (0001) surface state is gapped.71 However, subsequent studies reported that it is gapless.77–79,84,85 Figure 11(a) shows an ARPES measurement of the bulk and surface states, in which the surface state is clearly gapless Dirac cone at the (0001) surface. Among possible spin configurations that are allowed by symmetry, Ref. 77 proposed that the gapless surface state is protected by the mirror symmetry M x, while the S symmetry is broken at the surface. (Note that the mirror symmetry M x is broken in the A-type AFM- z state.) In other words, A-type AFM with the magnetic moments along the x axis (i.e., the in-plane direction), whose bulk and surface spectra obtained by a first-principles calculation is shown in Fig. 11(b), might be realized in MnBi 2Te 4 instead of the A-type AFM- z shown in Fig. 11(c). These observations of the gapless surface states imply the occurrence of a surface-mediated spin reconstruction.

FIG. 11.

(a) Bulk and surface spectra of MnBi 2Te 4 obtained by an ARPES measurement. Bulk and surface spectra of MnBi 2Te 4 obtained by a first-principles calculation, which assumes (b) A-type AFM with the magnetic moments along the x axis and (c) A-type AFM with the magnetic moments along the z axis. From Hao et al., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 11.

(a) Bulk and surface spectra of MnBi 2Te 4 obtained by an ARPES measurement. Bulk and surface spectra of MnBi 2Te 4 obtained by a first-principles calculation, which assumes (b) A-type AFM with the magnetic moments along the x axis and (c) A-type AFM with the magnetic moments along the z axis. From Hao et al., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

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As pointed in Ref. 72, it should be noted here that the antiferromagnetic order in MnBi 2Te 4 is essentially different from such an antiferromagnetic order in Fe-doped Bi 2Se 3 which has been proposed to realize a dynamical axion field.20 In the latter case, time-reversal Θ and inversion symmetries are both broken, allowing the deviation of the value of θ from π. The antiferromagnetic fluctuation contributes to the dynamical axion field at linear order in the Néel field. In contrast, in MnBi 2Te 4, an effective time-reversal S symmetry and inversion symmetry are both preserved, keeping the quantization θ = π and making no contribution to the dynamical axion field at linear order in the Néel field.

2. Transport properties of MnBi2Te4 thin films

Due to the intralayer ferromagnetism and interlayer antiferromagnetism of the Mn layers, the layered van der Waals crystal MnBi 2Te 4 exhibit interesting properties in its few-layer thin films. In even-septuple-layer films, P 2 and Θ symmetries are both broken, but P 2 Θ symmetry is preserved.73 As we have seen above, the presence of P 2 Θ symmetry leads to doubly degenerate bands. On the other hand, in odd-septuple-layer films, P 1 symmetry is preserved, but Θ and P 1 Θ symmetries are both broken, leading to spin-split bands.73 Consequently, the Chern number is zero in even-septuple-layer films as required by the P 2 Θ symmetry, while the Chern number in odd-septuple-layer films can be nonzero. Indeed, first-principles calculations show that there exist gapless chiral edge states in odd-septuple-layer films, whereas there do not in even-septuple-layer films.73,86 It should be noted that the zero-Chern-number state with σ x y = 0 is realized by the combination of half-quantized anomalous Hall states with opposite conductivities σ x y = ± e 2 / 2 h at the top and bottom surfaces, as shown in Fig. 12(a). In other words, this state is an axion insulator exhibiting a topological magnetoelectric effect with the quantized coefficient θ = π (see Sec. II D for a phenomenological derivation of the topological magnetoelectric effect). In contrast, even-septuple-layer films have the quantized anomalous Hall conductivity σ x y = ± e 2 / h that results from the half-quantized anomalous Hall conductivity σ x y = ± e 2 / 2 h of the same sign at the top and bottom surfaces, giving rise to the Chern number C = ± 1 as shown in Fig. 12(b).

Experimental observations that are consistent with theoretical predictions have been made. Figure 13 shows the resistivity measurement in a six-septuple-layer MnBi 2Te 4 film,87 in which an axion insulator behavior with a zero Hall plateau at the zero magnetic field and a Chern insulator behavior with the quantized Hall resistivity h / e 2 in a strong magnetic field were clearly observed. Also, the change in the Chern number between C = ± 1 was observed in response to the change in the magnetic field direction. Figure 14 shows the resistivity measurement in a five-septuple-layer MnBi 2Te 4 film,88 in which a quantum anomalous Hall effect with the quantized Hall resistivity h / e 2 was clearly observed.

FIG. 12.

Schematic illustration of (a) an axion insulator state realized in an even-septuple-layer MnBi 2Te 4 film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi 2Te 4 film. In even-septuple-layer (odd-septuple-layer) films, the anomalous Hall conductivities of the top and bottom surfaces are opposite (the same) to each other, resulting in the total anomalous Hall conductivity σ x y = 0 ( σ x y = ± e 2 / h), or equivalently, the Chern number C = 0 ( C = ± 1). From Li et al., Sci. Adv. 5, eaaw5685 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 12.

Schematic illustration of (a) an axion insulator state realized in an even-septuple-layer MnBi 2Te 4 film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi 2Te 4 film. In even-septuple-layer (odd-septuple-layer) films, the anomalous Hall conductivities of the top and bottom surfaces are opposite (the same) to each other, resulting in the total anomalous Hall conductivity σ x y = 0 ( σ x y = ± e 2 / h), or equivalently, the Chern number C = 0 ( C = ± 1). From Li et al., Sci. Adv. 5, eaaw5685 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

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FIG. 13.

Resistivity measurement in a six-septuple-layer MnBi 2Te 4 film, showing (a) an axion insulator behavior with a zero Hall plateau at zero magnetic field and (b) a Chern insulator behavior with the quantized Hall resistivity h / e 2 in a magnetic field of 9 T. Reproduced with permission from Liu et al., Nat. Mater. 19, 522 (2020). Copyright 2020 Springer Nature.

FIG. 13.

Resistivity measurement in a six-septuple-layer MnBi 2Te 4 film, showing (a) an axion insulator behavior with a zero Hall plateau at zero magnetic field and (b) a Chern insulator behavior with the quantized Hall resistivity h / e 2 in a magnetic field of 9 T. Reproduced with permission from Liu et al., Nat. Mater. 19, 522 (2020). Copyright 2020 Springer Nature.

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FIG. 14.

Resistivity measurement in a five-septuple-layer MnBi 2Te 4 film, showing a quantum anomalous Hall effect with the quantized transverse resistivity h / e 2 at the zero magnetic field. Reproduced with permission from Deng et al., Science 367, 895 (2020). Copyright 2020 American Association for the Advancement of Science.

FIG. 14.

Resistivity measurement in a five-septuple-layer MnBi 2Te 4 film, showing a quantum anomalous Hall effect with the quantized transverse resistivity h / e 2 at the zero magnetic field. Reproduced with permission from Deng et al., Science 367, 895 (2020). Copyright 2020 American Association for the Advancement of Science.

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Taking advantage of the nature of van der Waals materials, the layered van der Waals heterostructures of (MnBi 2Te 4) m(Bi 2Te 3) n can be synthesized. Here, it is well known that Bi 2Te 3 is a time-reversal invariant topological insulator.37 So far, MnBi 4Te 7 ( m = n = 1)89–93 and MnBi 6Te 10 ( m = 1 and n = 2)89,93,94 have been experimentally realized. Figure 15 shows schematic illustrations of MnBi 4Te 7 and MnBi 6Te 10 and their STEM images. In MnBi 4Te 7, a quintuple layer of Bi 2Te 3 and a septuple layer of MnBi 2Te 4 stack alternately. In MnBi 6Te 10, two quintuple layers of Bi 2Te 3 are sandwiched by septuple layers of MnBi 2Te 4. As in the case of MnBi 2Te 4, interlayer antiferromagnetism (between Mn layers) develops with a Néel temperature T N = 13 K in MnBi 4Te 789,90,93 and T N = 11 K in MnBi 6Te 10,93 and this antiferromagnetic insulator state is protected by the S = Θ T 1 / 2 symmetry, which indicates that MnBi 4Te 7 and MnBi 6Te 10 are also antiferromagnetic topological insulators. It was reported that, due to the gradual weakening of the antiferromagnetic exchange coupling associated with the increasing separation distance between Mn layers, a competition between antiferromagnetism and ferromagnetism occurs at low temperature 5 K.89,90 A magnetic phase diagram of MnBi 4Te 7 is shown in Fig. 16. Also, two distinct types of topological surface states are realized depending on the Bi 2Te 3 quintuple-layer termination or the MnBi 2Te 4 septuple-layer termination.91,92 ARPES studies showed that the Bi 2Te 3 quintuple-layer termination gives rise to gapped surface states, while the MnBi 2Te 4 septuple-layer termination gives rise to gapless surface states.91,92 Note that these terminations break the S symmetry, which implies in principle gapped surface states (see Sec. III A). It is suggested that the gap opening in the Bi 2Te 3 quintuple-layer termination can be explained by the magnetic proximity effect from the MnBi 2Te 4 septuple layer beneath and that the gaplessness in MnBi 2Te 4 septuple-layer termination can be explained by the restoration of time-reversal symmetry at the septuple-layer surface due to disordered spin.92 On the other hand, an ARPES study of MnBi 6Te 10 observed a gapped Dirac surface state in the MnBi 2Te 4 septuple-layer termination.94 

Since the bulk crystals of MnBi 4Te 7 and MnBi 6Te 10 are realized by van der Waals forces, various heterostructures in the 2D limit, which are made from the building blocks of the MnBi 2Te 4 septuple layer and the Bi 2Te 3 quintuple layer, can be obtained by exfoliation. A theoretical calculation shows that such 2D heterostructures exhibit the quantum spin-Hall effect without time-reversal symmetry and the quantum anomalous Hall effect.95 Theoretically, it is suggested that (MnBi 2Te 4)(Bi 2Te 3) n is a higher-order topological insulator hosting surface states with a Möbius twist.96 In contrast to MnBi 2Te 4 in which the value of θ is quantized to be π, it is suggested that the antiferromagnetic insulator phases of Mn 2Bi 6Te 11 (with m = 2 and n = 1)97 and Mn2Bi 2Te 598 in which the S symmetry is absent, break both time-reversal and inversion symmetries, realizing a dynamical axion field.

FIG. 15.

Schematic illustrations of (a) MnBi 4Te 7 and (b) MnBi 6Te 10. STEM images of (c) MnBi 4Te 7 and (d) MnBi 6Te 10, showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi 2Te 3 and a septuple layer of MnBi 2Te 4, respectively. From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 15.

Schematic illustrations of (a) MnBi 4Te 7 and (b) MnBi 6Te 10. STEM images of (c) MnBi 4Te 7 and (d) MnBi 6Te 10, showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi 2Te 3 and a septuple layer of MnBi 2Te 4, respectively. From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

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FIG. 16.

Magnetic phase diagram of MnBi 4Te 7 as functions of temperature and out-of-plane magnetic field, showing a complex competition between antiferromagnetism (AFM) and ferromagnetism (FM). From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 16.

Magnetic phase diagram of MnBi 4Te 7 as functions of temperature and out-of-plane magnetic field, showing a complex competition between antiferromagnetism (AFM) and ferromagnetism (FM). From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

Close modal

EuIn 2As 2 and EuSn 2As 2 have also been considered a candidate class of materials for antiferromagnetic topological insulators with inversion symmetry.99 Different from MnBi 2Te 4 which is a layered van der Waals material, EuIn 2As 2 has a three-dimensional crystal structure as shown in Fig. 17. EuSn 2As 2 has a very similar crystal and magnetic structure to EuIn 2As 2. Two metastable magnetic structures with the magnetic moments parallel to the b axis (AFM b) and the c axis (AFM c) have been known in EuIn 2As 2 and EuSn 2As 2.100,101 As in the case of MnBi 2Te 4, the antiferromagnetic insulator phases of EuIn 2As 2 and EuSn 2As 2 are protected by the S = Θ T 1 / 2 symmetry, with the half-unit-cell translation vector connecting four Eu atoms along the c axis. Indeed, ARPES measurements in EuIn 2As 2102 and EuSn 2As 278 suggests that they are antiferromagnetic topological insulators. Theoretically, it is suggested that antiferromagnetic EuIn 2As 2 (both AFM b and AFM c) is at the same time a higher-order topological insulator with gapless chiral hinge states lying within the gapped surface states.99 

We have seen in Sec. II that time-reversal symmetry and inversion symmetry impose the constraint on the coefficient θ of the topological magnetoelectric effect such that θ = π in 3D topological insulators and θ = 0 in 3D normal insulators. In this section, first, we derive a generic expression for θ which is given in terms of the Bloch-state wave function. Then, we show explicitly that the value of θ can be arbitrary in a class of antiferromagnetic insulators with broken time-reversal and inversion symmetries, taking a microscopic tight-binding model called the Fu–Kane–Mele–Hubbard (FKMH) model as an example.

FIG. 17.

Crystal and magnetic structures of EuIn 2As 2. There are two metastable magnetic structures where the magnetic moments align parallel to (a) the b axis and (b) the c axis. Reproduced with permission from Xu et al., Phys. Rev. Lett. 122, 256402 (2019). Copyright 2019 American Physical Society.

FIG. 17.

Crystal and magnetic structures of EuIn 2As 2. There are two metastable magnetic structures where the magnetic moments align parallel to (a) the b axis and (b) the c axis. Reproduced with permission from Xu et al., Phys. Rev. Lett. 122, 256402 (2019). Copyright 2019 American Physical Society.

Close modal
It is known that the chiral anomaly in (1+1) dimensions can be derived from the dimensional reduction from the (2+1)D Chern–Simons action. A similar way of deriving the effective action of (3+1)D time-reversal invariant topological insulators from the dimensional reduction from the (4+1)D Chern–Simons action was considered in Ref. 9. To see this, let k w be the momentum in the fourth dimension and ( k x , k y , k z ) be the momentum in 3D spatial dimensions. The second Chern number in 4D momentum space ( k x , k y , k z , k w ) is given by9,103,104
(47)
where
(48)
Here, | u α is the periodic part of the Bloch wave function of the occupied band α. By substituting the explicit expression for f i j (48) into Eq. (47), we obtain
(49)
where j , k , l = 1 , 2 , 3 indicate the 3D spatial direction. Here, note that ε 4 j k l = ε j k l 4 ε j k l due to the convention ε 1234 = 1. On the other hand, the corresponding topological action in (4+1) dimension ( x , y , z , w ) is given by
(50)
which can be rewritten as
(51)
where we have used the identity ε 4 ν ρ σ τ = ε ν ρ σ τ and defined θ ( r , t ) ν ( 2 ) ϕ. Here, ϕ = d w A 4 ( r , w , t ) can be regarded as the flux due to the extra dimension. In analogy with the (1+1)D case in which the first Chern number is given by ν ( 1 ) = d ϕ P / ϕ with P the electric polarization, Eq. (49) indicates a relation between the generalized polarization P 3 and the Chern number ν ( 2 ). Then, it follows that P 3 = ν ( 2 ) ϕ / 2 π. Finally, we arrive at a general expression for θ,9,32
(52)
where i , j , k = 1 , 2 , 3, d 3 k = d k x d k y d k z, and the integration is done over the Brillouin zone of the system. Equation (52) can be derived more rigorously and microscopically, starting from a generic Bloch Hamiltonian and its wave function.105,106 Figure 18 shows a numerically calculated value of θ using Eq. (52) and other equivalent expressions for θ in the Fu–Kane–Mele model on a diamond lattice with a staggered Zeeman field that breaks both time-reversal and inversion symmetries.32 One can see that the value of θ is no longer quantized once time-reversal symmetry is broken and varies continuously between θ = 0 corresponding to the case of a normal insulator and θ = π corresponding to the case of a topological insulator.
FIG. 18.

Numerically obtained value of θ in the Fu–Kane–Mele model on a diamond lattice. Here, β = tan 1 ( | h | / δ t 1 ) with h ( = U n ) being a staggered Zeeman field in the [111] direction of the diamond lattice, and δ t 1 being the hopping strength anisotropy due to the lattice distortion in the [111] direction. When β = π ( β = 0), the system is a topological (normal) insulator. Reproduced with permission from Essin et al., Phys. Rev. Lett. 102, 146805 (2009). Copyright 2009 American Physical Society.

FIG. 18.

Numerically obtained value of θ in the Fu–Kane–Mele model on a diamond lattice. Here, β = tan 1 ( | h | / δ t 1 ) with h ( = U n ) being a staggered Zeeman field in the [111] direction of the diamond lattice, and δ t 1 being the hopping strength anisotropy due to the lattice distortion in the [111] direction. When β = π ( β = 0), the system is a topological (normal) insulator. Reproduced with permission from Essin et al., Phys. Rev. Lett. 102, 146805 (2009). Copyright 2009 American Physical Society.

Close modal
A generic expression for θ [Eq. (52)] is applicable to the arbitrary band structure. However, some techniques (such as choosing a gauge for the Berry connection A) are required to calculate numerically. On the other hand, it has been shown that there exists an explicit expression for θ that can be calculated easily from the Bloch Hamiltonian of a certain class of insulators with broken time-reversal and inversion symmetries,20 which calculation does not rely on a specific choice of gauge. Here, we consider a generic 4 × 4 Bloch Hamiltonian of the form
(53)
with matrices α i satisfying the Clifford algebra { α i , α j } = 2 δ i j 1. Here, the matrix α 4 is invariant under both time-reversal and spatial inversion. Specifically, it has been known that the antiferromagnetic insulator phases of 3D correlated systems with spin–orbit coupling, such as Bi 2Se 3 doped with magnetic impurities such as Fe20 and 5 d transition-metal oxides with the corundum structure,109 can be described by Eq. (53). More recently, it has been suggested that van der Waals layered antiferromagnets such as Mn 2Bi 6Te 1197 and Mn2Bi 2Te 598 can also be described by Eq. (53). In such systems, we can calculate the value of θ using the following expression:20,109
(54)
where i , j , k , l = 1 , 2 , 3 , 5, | R | = i = 1 5 R i 2, and the integration is done over the Brillouin zone.

1. Four-band Dirac model

Let us derive a simpler expression for θ in systems whose effective continuum Hamiltonian is given by a massive Dirac Hamiltonian. We particularly consider a generic Dirac Hamiltonian with a symmetry-breaking mass term of the form
(55)
which can be derived by expanding Eq. (53) around some momentum points X and retaining only the terms linear in q = k X. Here, the matrix α 4 is invariant under both time-reversal and spatial inversion and the matrix α 5 = α 1 α 2 α 3 α 4 breaks both time-reversal and inversion symmetries. In other words, the system has both time-reversal and inversion symmetries when m 5 = 0. For concreteness, we require that the system be a time-reversal invariant topological insulator when m 0 < 0, as we have considered in Eq. (25). The action of the system in the presence of an external electromagnetic potential A μ is given by [see also Eq. (26)]
(56)
where t is real time, ψ ( r , t ) is a four-component spinor, ψ ¯ = ψ γ 0, m = ( m 0 ) 2 + ( m 5 ) 2, cos θ = m 0 / m , sin θ = m 5 / m , and we have used the fact that α 4 = γ 0, α 5 = i γ 0 γ 5 and α j = γ 0 γ j ( j = 1 , 2 , 3). Here, the gamma matrices satisfy the identities { γ μ , γ 5 } = 0 and { γ μ , γ ν } = 2 g μ ν with g μ ν = diag ( 1 , 1 , 1 , 1 ) ( μ , ν = 0 , 1 , 2 , 3). One can see that the action (56) is identical to Eq. (28), except for the generic value of θ in the exponent. By applying Fujikawa’s method to the action (56), the θ term is obtained as110,111
(57)
where
(58)
Here, the first term in Eq. (58) is 0 or π, which describes whether the system is topologically trivial or nontrivial. The second term in Eq. (58) describes the deviation from the quantized value due to the m 5 mass. Note that tan 1 ( m 5 / m 0 ) m 5 / m 0, i.e., the deviation is proportional to m 5 when m 5 m 0.

2. Fu–Kane–Mele–Hubbard model on a diamond lattice

In Eq. (58), we have seen that the m 5 mass term that breaks both time-reversal and inversion symmetries generates a deviation of the value of θ from the quantized value π or 0. Here, following Ref. 110, we discuss a microscopic origin of this m 5 mass term and derive an expression for θ of the form of Eq. (58) in a 3D correlated system with spin–orbit coupling. To this end, we start with the Fu–Kane–Mele–Hubbard (FKMH) model on a diamond lattice, whose tight-binding Hamiltonian is given by6,8,110,111
(59)
where c i σ is an electron creation operator at a site i with spin σ ( =↑ , ), n i σ = c i σ c i σ, and a is the lattice constant of the fcc lattice. d i j 1 and d i j 2 are the two vectors that connect two sites i and j on the same sublattice. σ = ( σ x , σ y , σ z ) are the Pauli matrices for the spin degree of freedom. The first through third terms in Eq. (59) represent the nearest-neighbor hopping, the next-nearest-neighbor spin–orbit coupling, and the on-site repulsive electron–electron interactions, respectively.
In the mean-field approximation, the interaction term is decomposed as U i n i n i U i [ n i n i + n i n i n i × n i c i c i c i c i c i c i × c i c i + c i c i c i c i ]. The spin–orbit coupling breaks spin SU(2) symmetry and, therefore, the directions of the spins are coupled to the lattice structure. Hence, we should parameterize the antiferromagnetic ordering between the two sublattices A and B [see Fig. 19(a)] in terms of the spherical coordinate ( n , θ , φ ),
(60)
where S i μ = 1 2 c i μ α σ α β c i μ β ( μ = A , B ) with i denoting the i th unit cell. It is convenient to express the mean-field Hamiltonian in terms of the 4 ×4 α matrices that anticommute with each other. We can define the basis c k [ c k A , c k A , c k B , c k B ] T with the wave vector k in the first Brillouin zone of the fcc lattice [see Fig. 19(b)]. Then, the single-particle Hamiltonian H MF ( k ) [ H MF k c k H MF ( k ) c k] is written in the form of Eq. (53),6,8 where the alpha matrices α i are given by the so-called chiral representation,
(61)
which satisfies { α i , α j } = 2 δ i j 1 with α 5 = α 1 α 2 α 3 α 4. In the present basis, the time-reversal operator and spatial-inversion (parity) operator are given by T = 1 ( i σ 2 ) K ( K is the complex conjugation operator) and P = τ 1 1, respectively. We have introduced the hopping strength anisotropy δ t 1 due to the lattice distortion along the [111] direction. Namely, we have set such that t i j = t + δ t 1 for the [111] direction, and t i j = t for the other three directions. When δ t 1 = 0, the system is a semimetal, i.e., the energy bands touch at the three points X r = 2 π ( δ r x , δ r y , δ r z ) ( r = x , y , z) with δ x x = δ y y = δ z z = 1 (and otherwise zero) indicating a Kronecker delta. Finite δ t 1 opens a gap of 2 | δ t 1 | at the X r points.
FIG. 19.

(a) Schematic illustration of the antiferromagnetic order between the two sublattices (denoted by red and blue) in the FKMH model. (b) The first Brillouin zone of an fcc lattice. Around the X r points with r = x , y , z (represented by green circles), massive Dirac Hamiltonians are derived.

FIG. 19.

(a) Schematic illustration of the antiferromagnetic order between the two sublattices (denoted by red and blue) in the FKMH model. (b) The first Brillouin zone of an fcc lattice. Around the X r points with r = x , y , z (represented by green circles), massive Dirac Hamiltonians are derived.

Close modal
It is notable that, in the ground state characterized by the antiferromagnetic order parameter (60), the Dirac Hamiltonians around the X r points acquire another mass induced by α 5 that breaks both time-reversal and inversion symmetries. In the strongly spin–orbit coupled case when the condition U n f 2 λ ( f = 1 , 2 , 3) is satisfied, we can derive the Dirac Hamiltonians around the X ~ r points, which are slightly deviated from the X r points,110 
(62)
Here, the subscript f can be regarded as the “flavor” of Dirac fermions. This Hamiltonian (62) has the same form as Eq. (55), which means that Fujikawa’s method can be applied to derive the θ term in the FKMH model. It follows that110 
(63)
Here, note that this expression for θ is valid only when the symmetry-breaking mass U n f ( f = 1 , 2 , 3) is small so that the condition U n f 2 λ is satisfied. In other words, the Dirac Hamiltonian of the form (62) must be derived as the effective Hamiltonian of the system.

A comparison of the analytical result [Eq. (63)] with a numerical result obtained from Eq. (52) in Ref. 32 has been made.110 In the numerical result (Fig. 18), in which the Néel vector is set to be in the [111] direction as n x = n y = n z h / U, the value of θ has a linear dependence on β h / δ t 1 when U n f / δ t 1 1 (i.e., around β = 0 or β = π). Thus, the analytical result [Eq. (63)] is in agreement with the numerical result when the deviation from the quantized value ( 0 or π) is small, since in Eq. (63), tan 1 ( U n f / δ t 1 ) U n f / δ t 1 when U n f / δ t 1 1.

In real materials, there are two contributions to the linear magnetoelectric coupling: electronic and ionic (i.e., lattice) contributions. These contributions can be further decomposed in to spin and orbital parts. Among the electronic contribution, Eq. (52) represents on an electronic orbital contribution to the isotropic linear magnetoelectric coupling. Here, note that there exist two additional electronic orbital (but non-topological) contributions to the isotropic linear magnetoelectric coupling.105,106 Cr 2O 3 is an antiferromagnetic insulator with broken time-reversal and inversion symmetries and is well known as a material that exhibits a linear magnetoelectric effect with α x x = α y y and α z z. Figure 20 shows the value of θ in Cr 2O 3 obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-space mesh Δ k.36 The value of θ extrapolated in the Δ k = 0 limit is θ = 1.3 × 10 3, which corresponds to α i i = 0.01 ps / m ( i = x , y , z). This value is about two orders of magnitude smaller than the experimentally observed value (i.e., full response) of the linear magnetoelectric tensor in Cr 2O 3. The values of θ in other conventional magnetoelectrics have also been evaluated in Ref. 36 as θ = 0.9 × 10 4 in BiFeO 3 and θ = 1.1 × 10 4 in GdAlO 3, which are both very small compared to the quantized value π. As a different approach, it has been proposed that the value of θ may be extracted from experimental observed parameters.107,108

What are the conditions for larger values of θ in real materials? It was also shown in Ref. 36, the value of θ in Cr 2O 3 is approximately proportional to the spin–orbit coupling strength, which implies that materials with strong spin–orbit coupling can have large values of θ. In addition, as we have seen in Secs. IV A and IV B, the breaking of both time-reversal and inversion symmetries are necessary to induce the deviation of θ from the quantized values π or 0. The value of θ changes continuously from π [see Fig. 18 and Eq. (58)]. Therefore, a system that lies near a topological insulator phase such as magnetically doped topological insulators can be one of good candidate systems. It is notable that if a material has a large value of θ ( π ), then it will exhibit a significantly large magnetoelectric effect of α i i = e 2 θ / [ ( 4 π 2 c ) ( c μ 0 2 ) ] 24 ps / m.

FIG. 20.

Value of θ in Cr 2O 3 obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-space mesh. The line indicates the second-order polynomial extrapolation to an infinitely dense mesh ( Δ k 0). Reproduced with permission from Coh et al., Phys. Rev. B 83, 085108 (2011). Copyright 2011 American Physical Society.

FIG. 20.

Value of θ in Cr 2O 3 obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-space mesh. The line indicates the second-order polynomial extrapolation to an infinitely dense mesh ( Δ k 0). Reproduced with permission from Coh et al., Phys. Rev. B 83, 085108 (2011). Copyright 2011 American Physical Society.

Close modal
So far, we have seen the “static” expressions for θ in insulators. In other words, we have not considered what happens in a system with a θ term when the system is excited by external forces. In general, the total value of θ can be decomposed into the sum of the static part (the ground-state value) θ 0 and the dynamical part δ θ ( r , t ) as