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 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.

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 2π). 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 2π). 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 δθ(r,t)δM(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 2π). 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 2π). 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 δθ(r,t)δM(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(brb0t),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 MnBi2Te4 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

Sθ=dtd3rθe24π2cEB,
(1)

where h=2π 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

P=θe24π2cB,M=θe24π2cE,
(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 EB is odd under time reversal (i.e., EBEB under tt), time-reversal symmetry requires that the action (1) is invariant under the transformation θθ. 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 EB 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. jH 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. jH 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

αij=MjEi|B=0=PiBj|E=0,
(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

αij=e2θ4π2cδij,
(4)

where δij is the Kronecker delta. Here, note that θ is a dimensionless constant. Equation (4) implies the Lagrangian density L=(e2θ/4π2c)EB, 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)

e24πc1μ02c24.3ps/m,
(5)

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 Cr2O3 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 Bi2Se3, which is described by 2D two-component massless Dirac fermions,37 

Hsurface(k)=vF(kyσxkxσy)=vF(k×ez)σ,
(6)

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)=±vFkx2+ky2 from a simple algebra Hsurface2=2vF2(kx2+ky2)12×2. The Fermi velocity of the surface states in Bi2Se3 is experimentally observed as vF5×105 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 impurities42–44 such that Hexch.=JiSiσδ(rRi), where Si is the impurity spin at position Ri. Then, the homogeneous part of the impurity spins gives rise to the position-independent Hamiltonian,

Hexch.=JnimpS¯impσmσ,
(7)

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

Esurface(k)=±(vFkx+my)2+(vFkymx)2+mz2.
(8)

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 

σxy=e2h14πdkxdkyR^(R^kx×R^ky)=sgn(mz)e22h,
(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(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)2Te3 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.

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 Vg dependence of the Hall conductivity σxy and the longitudinal conductivity σxx in a thin film of Cr-doped (Bi,Sb)2Te3. 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 Vg dependence of the Hall conductivity σxy and the longitudinal conductivity σxx in a thin film of Cr-doped (Bi,Sb)2Te3. 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 jH is induced as

jH=sgn(m)e22hn^×E,
(10)

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)]

M=sgn(m)e22hcE.
(11)

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

jH=sgn(m)e22hBt.
(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)]

P=sgn(m)e22hcB.
(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 

F=d3re22hcEB=d3rθe24π2cEB,
(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

Sθ=d4xθe24π2cEB=d4xθe232π2cεμνρλFμνFρλ,
(15)

where d4x=dtd3r, Fμν=μAννAμ 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=A0A/t and B=×A. Note that e2/c (1/137) is the fine-structure constant. Equation (15) is indeed the θ term [Eq. (1)]. Under time-reversal (tt), electric and magnetic fields are transformed as EE and BB, respectively. Similarly, under spatial inversion (rr), electric and magnetic fields are transformed as EE and BB, respectively. Hence, the term EB 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 θθ. 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,

εμνρλFμνFρλ=4εμνρλμ(AνρAλ),
(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

Ssurface=d3xθe28π2cεzνρλAνρAλ,
(17)

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,

jx=δSsurfaceδAx=θe24π2cεzxρλρAλ=θe24π2cEy,
(18)

where Ey=yA0tAy 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

Sθ=dtd3re28π2εμνρλ[μθ(r,t)]AνρAλ.
(19)

Then, the electric current density is obtained as

jx=δSθδAx=e24π2[tθ(r,t)Bxzθ(r,t)Ey].
(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 z0 (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[1Θ(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.

1. Effective Hamiltonian for 3D topological insulators

Let us start from the low-energy continuum model for prototypical 3D topological insulators such as Bi2Se3. The bulk electronic structure of Bi2Se3 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 by37,52

Heff(k)=[M(k)0A1kzA2k0M(k)A2k+A1kzA1kzA2kM(k)0A2k+A1kz0M(k)]=A2kxα1+A2kyα2+A1kzα3+M(k)α4,
(21)

where k±=kx±iky and M(k)=m0B1kz2B2k2. The coefficients for Bi2Se3 estimated by a first-principles calculation read m0=0.28 eV, A1=2.2 eVÅ, A2=4.1 eVÅ, B1=10 eVÅ2, and B2=56.6 eVÅ2.37,52 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,

αj=[0σjσj0],α4=[1001],
(22)

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 kisinki and ki22(1coski). Although this replacement is valid only when ki1, 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

Heff(k)=vF(α1sinkx+α2sinky+α3sinkz)+[m0+ri=x,y,z(1coski)]α4,
(23)

where we have defined vF=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 kiki, 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 as6,8

(1)ν=α=18sgn[m0+ri=x,y,z(1cosΛαi)]={1(0>m0/r>2,4>m0/r>6)+1(m0/r>0,2>m0/r>4,6>m0/r).
(24)

Indeed, the topological insulator phase with 0>m0/r>2 satisfies the above realistic value for Bi2Se3; m0/r0.1, 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.

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>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

HTI(k)=vFkα+m0α4,
(25)

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 kk+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 ψH0ψψ(H0eA0)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 

STI=dtd3rψ{i(tieA0)[HTI(k+eA)]}ψ=dtd3rψ¯[iγμ(μieAμ)m0]ψ,
(26)

where ψ(r,t) is a fermionic field representing the basis of the Hamiltonian (21) and ψ¯=ψγ0. Here, the gamma matrices γμ are given by the so-called Dirac representation as

γ0=α4=[1001],γj=α4αj=[0σjσj0],γ5=iγ0γ1γ2γ3=[0110],
(27)

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 tiτ, A0iA0, and γjiγj (j=1,2,3). The Euclidean action of the system is then written as

STIE=iSTI=dτd3rψ¯[γμ(μieAμ)m0eiπγ5]ψ,
(28)

where we have used the fact that m0=m0(cosπ+iγ5sinπ)=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 method50,51 to the action (28). First, let us consider an infinitesimal chiral transformation defined by

ψψ=eiπdϕγ5/2ψ,ψ¯ψ¯=ψ¯eiπdϕγ5/2,
(29)

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

Z=D[ψ,ψ¯]eSTIE[ψ,ψ¯]Z=D[ψ,ψ¯]eSTIE[ψ,ψ¯].
(30)

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

STIE=dτd3rψ¯[γμ(μieAμ)m0eiπ(1dϕ)γ5]ψ+i2πdτd3rdϕμ(ψ¯γμγ5ψ).
(31)

The Jacobian is written as50,51

J=exp[idτd3rdϕπe232π2cεμνρλFμνFρλ].
(32)

Here, Fμν=μAννAμ, 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 STIE:

STIE=dτd3rψ¯[γμ(μieAμ)m0]ψ+idτd3rπe232π2cεμνρλFμνFρλ,
(33)

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.

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 (Bi1xSbx)2Te3 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,

σxy=(n+12)e2h,
(34)

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

σxy=(nT+nB+1)e2hνe2h.
(35)

The ν=0 quantum Hall state is realized when the Landau levels of the top and bottom surface states are NT=N1 and NB=N (and vice versa), where N is an integer.57 This state corresponds to nT=N1 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

P=12d[dρT+(d)ρB]=e22hB,
(36)

which is indeed the topological magnetoelectric effect with the quantized coefficient θ=π. Note that the case of N0, 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)2Te3 and a topological insulator (Bi,Sb)2Te3 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)2Te3 sandwiched by two kinds of magnetically doped topological insulators V-doped (Bi,Sb)2Te3 and Cr-doped (Bi,Sb)2Te3 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 (Bi1xSbx)2Te3 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 (Bi1xSbx)2Te3 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 Aμ=(A0,A) in the presence of a θ term is given by

S=dtd3rα4π2θEB116πdtd3rFμνFμν,
(37)

where α=e2/c1/137 is the fine-structure constant and Fμν=μAννAμ 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 EB=(1/8)εμνρλFμνFρλ and FμνFμν=2(B2/μ0ε0E2). Here, recall that the classical equation of motion for the field Aμ is obtained from the Euler–Lagrange equation,

δSδAμ=LAμν(L(νAμ))=0,
(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

E=4πρ2α(θ2π)B,×E=1cBt,B=0,×B=4πcJ+1cEt+2αc[t(θ2π)B+c(θ2π)×E].
(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

θF=tan1(α)α,θK=tan1(1/α)π2.
(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(θF,θK)=cotθFcotθKcot2θF2cotθFcotθK1 as a function of dc Hall conductance towards the universal relationship f(θF,θK)=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(θF,θK)=cotθFcotθKcot2θF2cotθFcotθK1 as a function of dc Hall conductance towards the universal relationship f(θF,θK)=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 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 MnBi2Te4 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

ΘH(k)Θ1=H(k).
(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)=eikx. Then, the translation operator that moves a lattice by half a unit cell in the a3 direction is written as

T1/2=e(i/2)ka3[0110],
(42)

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=eika3. Now, we consider the combination of Θ and T1/2 defined by S=ΘT1/2. It follows that S2=eika3, 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 ka3=0, while Θ2=1. When a system is invariant under the operation S, the Bloch Hamiltonian H(k) satisfies

SH(k)S1=H(k),
(43)

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.

FIG. 9.

Schematic illustration of an antiferromagnetic topological insulator protected by the S=ΘT1/2 symmetry.

FIG. 9.

Schematic illustration of an antiferromagnetic topological insulator protected by the S=ΘT1/2 symmetry.

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Next, let us consider the resulting surface states. Since S2=1 on the Brillouin zone plane satisfying ka3=0, the 2D subsystem on the (k1,k2) plane is regarded as a quantum spin-Hall system with time-reversal symmetry. This means that the k1 or k2 dependence of the surface spectra must be gapless because the ka3=0 line of the surface states is the boundary of the 2D subsystem (the ka3=0 plane) in the bulk Brillouin zone. In other words, at the surfaces that are parallel to a3, 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 a3, 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=θe2/(4π2c)B, and M=θe2/(4π2c)E, where P and M are the electric polarization and the magnetization, respectively. Under time-reversal Θ, the coefficient θ changes sign θθ, because PP and EE, while MM and BB. On the other hand, the lattice translation T1/2 does not affect θ.69 Combining these, the S operation implies the transformation such that θθ+2πn with n being an integer. Then, it follows that θ=0 or θ=π modulo 2π.

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 MnBi2Te4.71–81 The crystal structure of MnBi2Te4 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 25K.71,74,79,81 The unit cell of the antiferromagnetic insulator state consists of two septuple layers (Fig. 10), where τ1/2c 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=Θτ1/2c 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 Bi2Se3.

FIG. 10.

Crystal and magnetic structure of the antiferromagnetic topological insulator state in MnBi2Te4. The unit cell consists of two septuple layers. τ1/2c 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 MnBi2Te4. The unit cell consists of two septuple layers. τ1/2c 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 P1 with the inversion center located at the Mn atomic layer in each septuple layers. Importantly, P2Θ symmetry, the combination of spatial inversion P2 with the inversion center located between two septuple layers and time-reversal Θ, is also preserved. The presence of P2Θ 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. P2Θ symmetry requires that

(P2Θ)H(k)(P2Θ)1=H(k),
(44)

since momentum k changes sign under both P2 and Θ. As in the case of Bi2Se3 [Eq. (21)], the low-energy effective Hamiltonian of the nonmagnetic state of MnBi2Te4 around the Γ point is written in the basis of [|P1z+,,|P1z+,,|P2z,,|P2z,], where the states |P1z+,↑↓ and |P2z,↑↓ come from the pz orbitals of Bi and Te, respectively.72 In this basis, P2=τz1 and Θ=1iσyK, where τi and σi act on the orbital and spin spaces, respectively, and K is complex conjugation operator. P2Θ symmetry constrains the possible form of the 4×4 Bloch Hamiltonian H(k)=i,jdij(k)τiσj. It follows that the following five matrices and the identity matrix are allowed by P2Θ symmetry:

τxσx,τxσy,τxσz,τy1,τz1,
(45)

due to the property (P2Θ)(τiσj)(P2Θ)1=τiσj. 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

H(k)=τx(A2kyσxA2kxσy+m5σz)+A1kzτy+M(k)τz,
(46)

where M(k)=M+B1kz2+B2(kx2+ky2). The mass m5 is induced by the antiferromagnetic order. One can see that the Hamiltonian (46) is invariant under both P2 and Θ when m5=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 σxy=±sgn(m5)e2/2h, implying the axion insulator state.97 

The surface states of antiferromagnetic MnBi2Te4 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.77–79,84,85Figure 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 Mx, while the S symmetry is broken at the surface. (Note that the mirror symmetry Mx 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 MnBi2Te4 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 MnBi2Te4 obtained by an ARPES measurement. Bulk and surface spectra of MnBi2Te4 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 MnBi2Te4 obtained by an ARPES measurement. Bulk and surface spectra of MnBi2Te4 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 MnBi2Te4 is essentially different from such an antiferromagnetic order in Fe-doped Bi2Se3 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 MnBi2Te4, 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 MnBi2Te4 exhibit interesting properties in its few-layer thin films. In even-septuple-layer films, P2 and Θ symmetries are both broken, but P2Θ symmetry is preserved.73 As we have seen above, the presence of P2Θ symmetry leads to doubly degenerate bands. On the other hand, in odd-septuple-layer films, P1 symmetry is preserved, but Θ and P1Θ 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 P2Θ 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 σxy=0 is realized by the combination of half-quantized anomalous Hall states with opposite conductivities σxy=±e2/2h 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 σxy=±e2/h that results from the half-quantized anomalous Hall conductivity σxy=±e2/2h 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 MnBi2Te4 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/e2 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 MnBi2Te4 film,88 in which a quantum anomalous Hall effect with the quantized Hall resistivity h/e2 was clearly observed.

FIG. 12.

Schematic illustration of (a) an axion insulator state realized in an even-septuple-layer MnBi2Te4 film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi2Te4 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 σxy=0 (σxy=±e2/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 MnBi2Te4 film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi2Te4 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 σxy=0 (σxy=±e2/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 MnBi2Te4 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/e2 in a magnetic field of 9T. 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 MnBi2Te4 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/e2 in a magnetic field of 9T. 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 MnBi2Te4 film, showing a quantum anomalous Hall effect with the quantized transverse resistivity h/e2 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 MnBi2Te4 film, showing a quantum anomalous Hall effect with the quantized transverse resistivity h/e2 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 (MnBi2Te4)m(Bi2Te3)n can be synthesized. Here, it is well known that Bi2Te3 is a time-reversal invariant topological insulator.37 So far, MnBi4Te7 (m=n=1)89–93 and MnBi6Te10 (m=1 and n=2)89,93,94 have been experimentally realized. Figure 15 shows schematic illustrations of MnBi4Te7 and MnBi6Te10 and their STEM images. In MnBi4Te7, a quintuple layer of Bi2Te3 and a septuple layer of MnBi2Te4 stack alternately. In MnBi6Te10, two quintuple layers of Bi2Te3 are sandwiched by septuple layers of MnBi2Te4. As in the case of MnBi2Te4, interlayer antiferromagnetism (between Mn layers) develops with a Néel temperature TN=13K in MnBi4Te789,90,93 and TN=11K in MnBi6Te10,93 and this antiferromagnetic insulator state is protected by the S=ΘT1/2 symmetry, which indicates that MnBi4Te7 and MnBi6Te10 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 5K.89,90 A magnetic phase diagram of MnBi4Te7 is shown in Fig. 16. Also, two distinct types of topological surface states are realized depending on the Bi2Te3 quintuple-layer termination or the MnBi2Te4 septuple-layer termination.91,92 ARPES studies showed that the Bi2Te3 quintuple-layer termination gives rise to gapped surface states, while the MnBi2Te4 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 Bi2Te3 quintuple-layer termination can be explained by the magnetic proximity effect from the MnBi2Te4 septuple layer beneath and that the gaplessness in MnBi2Te4 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 MnBi6Te10 observed a gapped Dirac surface state in the MnBi2Te4 septuple-layer termination.94 

Since the bulk crystals of MnBi4Te7 and MnBi6Te10 are realized by van der Waals forces, various heterostructures in the 2D limit, which are made from the building blocks of the MnBi2Te4 septuple layer and the Bi2Te3 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 (MnBi2Te4)(Bi2Te3)n is a higher-order topological insulator hosting surface states with a Möbius twist.96 In contrast to MnBi2Te4 in which the value of θ is quantized to be π, it is suggested that the antiferromagnetic insulator phases of Mn2Bi6Te11 (with m=2 and n=1)97 and Mn2Bi2Te598 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) MnBi4Te7 and (b) MnBi6Te10. STEM images of (c) MnBi4Te7 and (d) MnBi6Te10, showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi2Te3 and a septuple layer of MnBi2Te4, 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) MnBi4Te7 and (b) MnBi6Te10. STEM images of (c) MnBi4Te7 and (d) MnBi6Te10, showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi2Te3 and a septuple layer of MnBi2Te4, respectively. From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

Close modal
FIG. 16.

Magnetic phase diagram of MnBi4Te7 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 MnBi4Te7 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

EuIn2As2 and EuSn2As2 have also been considered a candidate class of materials for antiferromagnetic topological insulators with inversion symmetry.99 Different from MnBi2Te4 which is a layered van der Waals material, EuIn2As2 has a three-dimensional crystal structure as shown in Fig. 17. EuSn2As2 has a very similar crystal and magnetic structure to EuIn2As2. Two metastable magnetic structures with the magnetic moments parallel to the b axis (AFMb) and the c axis (AFMc) have been known in EuIn2As2 and EuSn2As2.100,101 As in the case of MnBi2Te4, the antiferromagnetic insulator phases of EuIn2As2 and EuSn2As2 are protected by the S=ΘT1/2 symmetry, with the half-unit-cell translation vector connecting four Eu atoms along the c axis. Indeed, ARPES measurements in EuIn2As2102 and EuSn2As278 suggests that they are antiferromagnetic topological insulators. Theoretically, it is suggested that antiferromagnetic EuIn2As2 (both AFMb and AFMc) 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 EuIn2As2. 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 EuIn2As2. 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 kw be the momentum in the fourth dimension and (kx,ky,kz) be the momentum in 3D spatial dimensions. The second Chern number in 4D momentum space (kx,ky,kz,kw) is given by9,103,104

ν(2)=132π2d4kεijkltr[fijfkl],
(47)

where

fij=iAjjAii[Ai,Aj],Ajαβ=iuα|kj|uβ.
(48)

Here, |uα is the periodic part of the Bloch wave function of the occupied band α. By substituting the explicit expression for fij(48) into Eq. (47), we obtain

ν(2)=18π2d4kkw{ε4jkltr[AjkAl23iAjAkAl]}dkwP3(kw)kw,
(49)

where j,k,l=1,2,3 indicate the 3D spatial direction. Here, note that ε4jkl=εjkl4εjkl due to the convention ε1234=1. On the other hand, the corresponding topological action in (4+1) dimension (x,y,z,w) is given by

S=ν(2)24π2dtd4xεμνρστAμνAρσAτ,
(50)

which can be rewritten as

S=ν(2)8π2dtd3xdwε4νρστA4νAρσAτ=132π2dtd3xθ(r,t)ενρστFνρFστ,
(51)

where we have used the identity ε4νρστ=ενρστ and defined θ(r,t)ν(2)ϕ. Here, ϕ=dwA4(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)=dϕP/ϕ with P the electric polarization, Eq. (49) indicates a relation between the generalized polarization P3 and the Chern number ν(2). Then, it follows that P3=ν(2)ϕ/2π. Finally, we arrive at a general expression for θ,9,32

θ=14πBZd3kεijktr[AijAk23iAiAjAk],
(52)

where i,j,k=1,2,3, d3k=dkxdkydkz, 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.105,106Figure 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=tan1(|h|/δt1) with h(=Un) being a staggered Zeeman field in the [111] direction of the diamond lattice, and δt1 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=tan1(|h|/δt1) with h(=Un) being a staggered Zeeman field in the [111] direction of the diamond lattice, and δt1 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

H(k)=i=15Ri(k)αi,
(53)

with matrices αi satisfying the Clifford algebra {αi,αj}=2δij1. Here, the matrix α4 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 Bi2Se3 doped with magnetic impurities such as Fe20 and 5d 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 Mn2Bi6Te1197 and Mn2Bi2Te598 can also be described by Eq. (53). In such systems, we can calculate the value of θ using the following expression:20,109

θ=14πBZd3k2|R|+R4(|R|+R4)2|R|3εijklRiRjkxRkkyRlkz,
(54)

where i,j,k,l=1,2,3,5, |R|=i=15Ri2, 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

H(q)=qxα1+qyα2+qzα3+m0α4+m5α5,
(55)

which can be derived by expanding Eq. (53) around some momentum points X and retaining only the terms linear in q=kX. Here, the matrix α4 is invariant under both time-reversal and spatial inversion and the matrix α5=α1α2α3α4 breaks both time-reversal and inversion symmetries. In other words, the system has both time-reversal and inversion symmetries when m5=0. For concreteness, we require that the system be a time-reversal invariant topological insulator when m0<0, as we have considered in Eq. (25). The action of the system in the presence of an external electromagnetic potential Aμ is given by [see also Eq. (26)]

S=dtd3rψ¯(r,t)[iγμ(μieAμ)meiθγ5]ψ(r,t),
(56)

where t is real time, ψ(r,t) is a four-component spinor, ψ¯=ψγ0, m=(m0)2+(m5)2, cosθ=m0/m, sinθ=m5/m, and we have used the fact that α4=γ0, α5=iγ0γ5 and αj=γ0γj (j=1,2,3). Here, the gamma matrices satisfy the identities {γμ,γ5}=0 and {γμ,γν}=2gμν with gμν=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

Sθ=dtd3re22πhθEB,
(57)

where

θ=π2[1sgn(m0)]tan1(m5m0).
(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 m5 mass. Note that tan1(m5/m0)m5/m0, i.e., the deviation is proportional to m5 when m5m0.

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

In Eq. (58), we have seen that the m5 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 m5 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

H=i,j,σtijciσcjσ+i4λa2i,jciσ(dij1×dij2)cj+Uinini,
(59)

where ciσ is an electron creation operator at a site i with spin σ(=↑,), niσ=ciσciσ, and a is the lattice constant of the fcc lattice. dij1 and dij2 are the two vectors that connect two sites i and j on the same sublattice. σ=(σx,σy,σz) 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 UininiUi[nini+ninini×nicicicicicici×cici+cicicici]. 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),

SiA=SiB=(nsinθcosφ,nsinθsinφ,ncosθ)n1ex+n2ey+n3ez(n),
(60)

where Siμ=12ciμασαβciμβ(μ=A,B) with i denoting the ith 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 ck[ckA,ckA,ckB,ckB]T with the wave vector k in the first Brillouin zone of the fcc lattice [see Fig. 19(b)]. Then, the single-particle Hamiltonian HMF(k) [HMFkckHMF(k)ck] is written in the form of Eq. (53),6,8 where the alpha matrices αi are given by the so-called chiral representation,

αj=[σj00σj],α4=[0110],α5=[0ii0],
(61)

which satisfies {αi,αj}=2δij1 with α5=α1α2α3α4. In the present basis, the time-reversal operator and spatial-inversion (parity) operator are given by T=1(iσ2)K (K is the complex conjugation operator) and P=τ11, respectively. We have introduced the hopping strength anisotropy δt1 due to the lattice distortion along the [111] direction. Namely, we have set such that tij=t+δt1 for the [111] direction, and tij=t for the other three directions. When δt1=0, the system is a semimetal, i.e., the energy bands touch at the three points Xr=2π(δrx,δry,δrz) (r=x,y,z) with δxx=δyy=δzz=1 (and otherwise zero) indicating a Kronecker delta. Finite δt1 opens a gap of 2|δt1| at the Xr 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 Xr 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 Xr 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 Xr points acquire another mass induced by α5 that breaks both time-reversal and inversion symmetries. In the strongly spin–orbit coupled case when the condition Unf2λ (f=1,2,3) is satisfied, we can derive the Dirac Hamiltonians around the X~r points, which are slightly deviated from the Xr points,110 

HMF(X~r+q)=qxα1+qyα2+qzα3+δt1α4+Unfα5.
(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 

θ=π2[1+sgn(δt1)]f=1,2,3tan1(Unfδt1).
(63)

Here, note that this expression for θ is valid only when the symmetry-breaking mass Unf (f=1,2,3) is small so that the condition Unf2λ 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 nx=ny=nzh/U, the value of θ has a linear dependence on βh/δt1 when Unf/δt11 (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), tan1(Unf/δt1)Unf/δt1 when Unf/δt11.

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 Cr2O3 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 αxx=αyy and αzz. Figure 20 shows the value of θ in Cr2O3 obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-space mesh Δk.36 The value of θ extrapolated in the Δk=0 limit is θ=1.3×103, which corresponds to αii=0.01ps/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 Cr2O3. The values of θ in other conventional magnetoelectrics have also been evaluated in Ref. 36 as θ=0.9×104 in BiFeO3 and θ=1.1×104 in GdAlO3, 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 Cr2O3 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 αii=e2θ/[(4π2c)(cμ02)]24ps/m.

FIG. 20.

Value of θ in Cr2O3 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 (Δk0). Reproduced with permission from Coh et al., Phys. Rev. B 83, 085108 (2011). Copyright 2011 American Physical Society.

FIG. 20.

Value of θ in Cr2O3 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 (Δk0). 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) θ0 and the dynamical part δθ(r,t) as

θ(r,t)=θ0+δθ(r,t).
(64)

The dynamical part δθ(r,t) is often referred to as the dynamical axion field,20 since the θ term has exactly the same form as the action describing the coupling between a hypothetical elementary particle, axion, and a photon. Namely, θ(r,t) in condensed matter can be regarded as a (pseudoscalar) field for axion quasiparticles. In this section, first, we derive the action of axion quasiparticles in topological antiferromagnetic insulators. Then, we consider the consequences of the realization of the dynamical axion field in the condensed matter.

Here, following Refs. 20 and 111, we derive the action of axion quasiparticles in topological antiferromagnetic insulators whose effective Hamiltonian is given by a massive Dirac Hamiltonian (55), which is applicable to magnetically doped Bi2Se3 and the Fu–Kane–Mele–Hubbard model as we have seen. In this case, the presence of the mass term m5α5 that breaks time-reversal and inversion symmetries results in nonquantized values of θ. Here, let us consider the fluctuation of m5 (which corresponds to the fluctuation of the Néel field) denoted by m5+δm5, and derive the action for δm5. For this purpose, it is convenient to adopt a perturbative method. The action of the antiferromagnetic insulator phase in the presence of an external electromagnetic potential Aμ is written as [see Eq. (56)]

S=dtd3rψ¯(r,t)[iγμDμm0+iγ5(m5+δm5)]ψ(r,t),
(65)

where Dμ=μieAμ with e>0 being the magnitude of the electron charge. By integrating out the fermionic field ψ, we obtain the effective action Weff for δm5 and Aμ as

Z=D[ψ,ψ¯]eiSeiWeff[δm5,Aμ]=exp{Trln[G01(1+G0V)]}=exp[Tr(lnG01)n=11nTr(G0V)n].
(66)

In order to obtain the action of the low-energy spin-wave excitation, i.e., the antiferromagnetic magnon, we set the Green’s function of the unperturbed part as G0=(iγμμm0+iγ5m5)1, and the perturbation term as V=eγμAμ+iγ5δm5. Note that we have used that iγμDμm0+iγ5(m5+δm5)=G01+V. In the random phase approximation, the leading-order terms read

iWeff[δm5,Aμ]=12Tr(G0iγ5δm5)2+Tr[(G0eγμAμ)2(G0iγ5δm5)],
(67)

where the first and second terms on the right-hand side correspond to a bubble-type diagram and a triangle-type diagram, respectively (see Fig. 21).

FIG. 21.

Schematic of (a) a bubble-type Feynman diagram and (b) a triangle-type Feynman diagram. The solid lines, wavy lines, and double lines indicate the Green’s function G0, the electromagnetic field A, and the Néel field δm5, respectively.

FIG. 21.

Schematic of (a) a bubble-type Feynman diagram and (b) a triangle-type Feynman diagram. The solid lines, wavy lines, and double lines indicate the Green’s function G0, the electromagnetic field A, and the Néel field δm5, respectively.

Close modal

To compute the traces of the gamma matrices we use the following identities: tr(γμ)=tr(γ5)=0, tr(γμγν)=4gμν, tr(γμγνγ5)=0, and tr(γμγνγργσγ5)=4iεμνρσ. The first term in Eq. (67) is given explicitly by

W1=d4q(2π)4Π(q)δm5(q)δm5(q)iJdtd3r[(tδm5)2(viiδm5)2m2(δm5)2].
(68)

Here, J, vi, and m are the stiffness, velocity, and mass of the spin-wave excitation mode, which are given, respectively, by20 

J=2Π(q)q02|q0=BZd3k(2π)3i=14Ri216|R|5,
(69)
Jm2=Π(q)|q0=m52BZd3k(2π)314|R|3,
(70)

where |R|=a=15Ra2 and q0 indicates the limit of both q00 and q0. The second term in Eq. (67) is the so-called triangle anomaly, which gives the θ term. The final result is112,113

W2=idtd3re24π2[δm5(r,t)m0]EB,
(71)

from which we find that the fluctuation of the m5α5 mass term behaves just as a dynamical axion field.

For concreteness, let us consider the antiferromagnetic insulator phase of Bi2Se3 family doped with magnetic impurities such as Fe.20 In this case, the direction of the Néel field n in the ground state is along the z axis: m5=(2/3)Unz and nx=ny=0, where U is the on-site electron–electron interaction strength. Defining δθ(r,t)=δm5(r,t)/m0=(2/3)Uδnz/m0 and substituting this into Eqs. (68) and (71), we, finally, arrive at the action of the axion quasiparticle,

Saxion=g2Jdtd3r[(tδθ)2(viiδθ)2m2δθ2]+dtd3re24π2δθ(r,t)EB,
(72)

where g2=m02. Finally, we mention briefly the case of the FKMH model. We find from Eq. (62) that there exist three m5,fα5 mass terms with m5,f=Unf (f=1,2,3). Namely, all the three spatial components of the Néel field n is contained in the kinetic part of the action of the axion field, which means that the kinetic part is described by the nonlinear sigma model for antiferromagnets.114 This is interesting because an effective action of an antiferromagnet is naturally derived although our original action (65) does not explicitly indicate that the mass m5 corresponds to a component of the Néel field.

In the following, we consider the consequences of the realization of a dynamical axion field in condensed matter. Among several theoretical studies on the emergent phenomena from a dynamical axion field,20,111,115–118 we particularly focus on three studies on the responses of topological antiferromagnetic insulators with a dynamical axion field δθ(r,t) to external electric and magnetic fields.

1. Axionic polariton

It has been proposed that the presence of a dynamical axion field can lead to a new type of polariton, the axionic polariton.20 To see this, we start with the total action involving an axion field δθ [Eq. (72)] and an electromagnetic field Aμ=(A0,A), which is given by

S=g2Jdtd3r[(μδθ)(μδθ)m2δθ2]+dtd3rα4π2δθEB116πdtd3rFμνFμν,
(73)

where α=e2/c1/137 is the fine-structure constant and Fμν=μAννAμ is the electromagnetic field tensor. Note that EB=(1/8)εμνρλFμνFρλ and FμνFμν=2(B2/μ0ε0E2). Here, recall that the classical equation of motion for a field ϕ is generically obtained from the Euler–Lagrange equation,

δSδϕ=Lϕμ(L(μϕ))=0,
(74)

where L is the Lagrangian density of the system. We consider the case of a constant magnetic field B=B0. Then, the equations of motion for the axion and electromagnetic fields are obtained from Eq. (74) as

2Et2c22EαπεB02δθt2=0,2δθt2v22δθ+m2δθα8π2g2JB0E=0,
(75)

where c is the speed of light in the media and ε is the dielectric constant. Neglecting the dispersion of the axion field compared to the electric field E, the dispersion of the electric field, i.e., the axionic polariton, ω±(k), is given by20 

2ω±(k)=c2k2+m2+b2±(c2k2+m2+b2)24c2k2m2,
(76)

with b2=α2B02/8π3εg2J. The photon dispersion in the absence of the axion field is just ω(k)=ck. In the presence of the axion field, the photon dispersion ω±(k) has two branches separated by a gap between m and m2+b2. As shown in Fig. 22, this gap gives rise to a total reflection of incident light in the case when the incident light frequency is in the gap. The point is the tunability of the axionic polariton gap by the external magnetic field B0.

2. Dynamical chiral magnetic effect and anomalous Hall effect

Next, we consider an electric current response in insulators with a dynamical axion field. To this end, we rewrite the θ term in the Chern–Simons form, which procedure becomes possible when a dynamical axion field is realized

Sθ=dtd3re28π2εμνρλ[μθ(r,t)]AνρAλ.
(77)

Then, the induced four-current density jν can be obtained from the variation of the above action with respect to the four potential Aν as jν=δSθ/δAν=(e2/4π2)εμνρλ[μθ(r,t)]ρAλ. The induced electric current density and charge density are given by12 

j(r,t)=δSθδA=e24π2[θ˙(r,t)B+θ(r,t)×E],ρ(r,t)=δSθδA0=e24π2θ(r,t)B,
(78)

where θ˙=θ(r,t)/t. The magnetic-field induced current is the so-called chiral magnetic effect, which was first studied in nuclear physics.49 The electric-field induced current is the anomalous Hall effect, since it is perpendicular to the electric field. Note that the electric current [Eq. (78)] is a bulk current that can flow in insulators:111 the magnetic-field induced and electric-field induced currents are, respectively, understood as a polarization current P/t=e2/(4π2)θ˙B and a magnetization current ×M=e2/(4π2)θ×E, where P and M are directly obtained from the θ term [see Eq. (2)]. The electric current given by Eq. (78) has been studied in the antiferromagnetic insulator phase of the FKMH model.111 As we have seen in Eq. (63), the dynamical axion field can be realized in the FKMH model by the fluctuation of the antiferromagnetic order parameter, i.e., by the antiferromagnetic spin excitation.

The magnetic-field induced current in Eq. (78), i.e., the dynamical chiral magnetic effect, emerges due to the time dependence of the antiferromagnetic order parameter. The simplest situation is the antiferromagnetic resonance. The dynamics of the sublattice magnetizations SiA=mA and SiB=mB can be phenomenologically described by119 

m˙A=mA×{ωJmB+[gμBB+ωA(mAen0)]en0},m˙B=mB×{ωJmA+[gμBB+ωA(mBen0)]en0},
(79)

where ωJ and ωA are the exchange field and anisotropy field, respectively. Here, we have considered the case where a microwave (i.e., ac magnetic field) of frequency ωrf is irradiated and a static magnetic field B=Ben0 is applied along the easy axis of the antiferromagnetic order. In the antiferromagnetic resonance state that is realized when ωrf=ω±, the antiferromagnetic order parameter is described as the precession around the easy axis,119 

n±(t)[mA(t)mB(t)]/2n0en0+δn±eiω±t,
(80)

where ω±=gμBB±(2ωJ+ωA)ωA are the resonance frequencies. Schematic illustration of the dynamics of mA and mB in the antiferromagnetic resonance state is shown in Fig. 23(a). Substituting the solution (80) into the first term in Eq. (78), a simplified expression for the dynamical chiral magnetic effect is obtained around the phase boundary where Unf/M01 as111 

jCME(t)=e24π2UD1M0Ba=±ωaδnasin(ωat+α),
(81)

where D1 is a constant and δn± is a Lorentzian function of ωrf. Equation (81) means that an alternating current is induced by the antiferromagnetic resonance. The maximum value of the dynamical chiral magnetic effect (81)|jCME|max=e24π2U|D1||M0|Bω±δn± is estimated as |jCME|max1×104A/m2, which is experimentally observable. It should be noted that there is no energy dissipation due to Joule heat in the dynamical chiral magnetic effect, unlike the conventional transport regime under electric fields.

FIG. 22.

Axionic polariton phenomenon. (a) In the absence of a static magnetic field, the incident light can transmit through the media. (b) In the presence of a static magnetic field parallel to the electric field of light, a total reflection of incident light occurs when the incident light frequency is in the gap. Reproduced with permission from Li et al., Nat. Phys. 6, 284 (2010). Copyright 2010 Springer Nature.

FIG. 22.

Axionic polariton phenomenon. (a) In the absence of a static magnetic field, the incident light can transmit through the media. (b) In the presence of a static magnetic field parallel to the electric field of light, a total reflection of incident light occurs when the incident light frequency is in the gap. Reproduced with permission from Li et al., Nat. Phys. 6, 284 (2010). Copyright 2010 Springer Nature.

Close modal
FIG. 23.

Schematic figures of (a) an antiferromagnetic resonance state and (b) a 1D antiferromagnetic domain wall.

FIG. 23.

Schematic figures of (a) an antiferromagnetic resonance state and (b) a 1D antiferromagnetic domain wall.

Close modal

The electric-field induced current in Eq. (78), i.e., the anomalous Hall effect, emerges due to the spatial dependence of the antiferromagnetic order parameter. As an example, we consider a 1D antiferromagnetic spin texture of length L along the Z direction, an orientational domain wall.120,121 As shown in Fig. 23(b), the antiferromagnetic order parameter n(r)=[mA(r)mB(r)]/2 at the two edges has a relative angle δ, resulting in θ(Z=0)=θ0 and θ(Z=L)=θ0+δ in the original spherical coordinate. A simplified expression for the anomalous Hall effect is obtained around the phase boundary where Unf/M01 as111 

JAHEX=0LdZjAHEX(Z)=e24π2UD2M0EY,
(82)

where D2(δ)=f[nf(θ0+δ)nf(θ0)] is a constant and a static electric filed E is applied perpendicular to the antiferromagnetic order as E=EYeY. The Hall conductivity is estimated as σXY=e24π2UD2M01×102e2/h, which is experimentally observable. Note that D2=0 when δ=0, which means that this anomalous Hall effect does not arise in uniform ground states.

3. Inverse process of the dynamical chiral magnetic effect

In Eq. (81), we have seen that ac current is generated by the antiferromagnetic resonance. It is natural to consider the inverse process of the dynamical chiral magnetic effect, i.e., a realization of the antiferromagnetic resonance induced by the ac electric field.116 To this end, we study a continuum model of an antiferromagnet whose free energy is given by122,123

F0=d3r[a2m2+A2i=x,y,z(in)2K2nz2Hm],
(83)

where a and A are the homogeneous and inhomogeneous exchange constants, respectively, and K is the easy-axis anisotropy along the z direction. n and m are the Néel vector and small net magnetization satisfying the constraint nm=0 with |n|=1 and |m|1. The fourth term is the Zeeman coupling with H=gμBB being an external magnetic field. For concreteness, we consider the antiferromagnetic insulator phase of the FKMH model (see Sec. IV B2). The θ term can be written in the free energy form [see also Eq. (14)]

Fθ=e24π23Un0M0d3r(ne[111])EB,
(84)

where we have used the fact that f=1,2,3nf=3ne[111], with e[111] being the unit vector along the [111] direction of the original diamond lattice in the FKMH model.

Phenomenologically, the antiferromagnetic spin dynamics can be described by the Landau–Lifshitz–Gilbert equation. From the total free energy of the system FAF=F0+Fθ, the effective fields for n and m are given by fn=δFAF/δn and fm=δFAF/δm. The Landau–Lifshitz–Gilbert equation is given by116,123

n˙=(γfmG1m˙)×n,m˙=(γfnG2n˙)×n+(γfmG1m˙)×m+τSP,
(85)

where γ=1/, G1 and G2 are dimensionless Gilbert-damping parameters, and τSP=GSP(n˙×n+m˙×m) is the additional damping torque with a spin pumping parameter GSP.124,126 Let us consider a case where an ac electric field Eac(t)=Eaceiω0tez and a static magnetic field B=Bez are both applied along the easy axis. Assuming the dynamics of the Néel field n(t)=ez+δn(t) and the net magnetization m(t)=δm(t) and solving the above Landau–Lifshitz–Gilbert equation, it is shown that the antiferromagnetic resonance can be realized by the ac electric field Eac(t). The resonance frequencies are116 

ω±=ωH±ωaωK,
(86)

where ωH=γgμBB, ωa=γa, and ωK=γK. The essential point is the coupling of the Néel field and the electric field through the θ term, as is readily seen in Eq. (84). Note that these resonance frequencies are not dependent on the parameters of the θ term. This is because the θ term acts only as the driving force to cause the resonance.

As shown in Fig. 24, in the resonance state, a pure dc spin current Js generated by the spin pumping is injected into the attached heavy-metal layer through the interface.124 The spin current is converted into an electric voltage across the transverse direction via the inverse spin-Hall effect:125VSP(ω0)αSHJs(ω0), where αSH is the spin-Hall angle. For example, in the case of B=0.1T and Eac=1V/m with possible (typical) values of the parameters, the magnitude of VSP in the resonance state is found to be VSP(ω±)10μV,116 which is experimentally observable. Furthermore, it should be noted that the above value of the ac electric field, Eac=1V/m, is small. Namely, from the viewpoint of lower energy consumption, the spin current generation using topological antiferromagnets with the θ term has an advantage compared to conventional “current-induced” methods that require such high-density currents as 1010A/m2.127 

FIG. 24.

Schematic figure of the electric-field induced antiferromagnetic resonance and its detection. An ac electric field Eac(t) induces the antiferromagnetic resonance. A dc pure spin current Js generated by the spin pumping into the attached heavy metal (HM) such as Pt can be detected through the inverse spin-Hall effect (ISHE) as a direct current Jc (i.e., the voltage VSP).

FIG. 24.

Schematic figure of the electric-field induced antiferromagnetic resonance and its detection. An ac electric field Eac(t) induces the antiferromagnetic resonance. A dc pure spin current Js generated by the spin pumping into the attached heavy metal (HM) such as Pt can be detected through the inverse spin-Hall effect (ISHE) as a direct current Jc (i.e., the voltage VSP).

Close modal

So far, we have focused on the axion electrodynamics in 3D insulators. In this section, we overview topological responses of Weyl semimetals to external electric and magnetic fields, which are described by the θ term. Although a number of novel phenomena have been proposed theoretically and observed experimentally in Weyl semimetals,128–130 we here focus on the very fundamental two effects, the anomalous Hall effect and chiral magnetic effect, starting from the derivation of the θ term. We also discuss the negative magnetoresistance effect that arises as a consequence of the condensed-matter realization of the chiral anomaly.

The Weyl semimetals have nondegenerate gapless linear dispersions around band-touching points (Weyl nodes). The low-energy effective Hamiltonian around a Weyl node is written as

HWeyl(k)=QvFkσ,
(87)

where Q=±1 indicates the chirality, vF is the Fermi velocity, and σi are Pauli matrices. The two energy eigenvalues are ±vFkx2+ky2+kz2. In contrast to 2D Weyl fermions such as those on the topological insulator surfaces, the 3D Weyl fermions described by Eq. (87) cannot acquire the mass, i.e., cannot be gapped, since all the three Pauli matrices are already used. This indicates the stableness of a single Weyl node. Because the sum of the chiralities of the Weyl nodes (or equivalently the monopoles in momentum space) in a system must be zero, the simplest realization of a Weyl semimetal is one with two Weyl nodes of opposite chiralities. Note that the minimal number of Weyl nodes in Weyl semimetals with broken inversion symmetry is four,131 while it is two in Weyl semimetals with broken time-reversal symmetry.

For concreteness, we consider a 4×4 continuum model Hamiltonian for two-node Weyl semimetals with broken time-reversal symmetry,129,132,134,135

H0(k)=vF(τzkσ+Δτx+bσ),
(88)

where τi and σi are the Pauli matrices for Weyl-node and spin degrees of freedom, respectively, and Δ is the mass of 3D Dirac fermions. The term bσ represents a magnetic interaction such as the exchange interaction between conduction electrons and magnetic impurities or the Zeeman coupling with an external magnetic field. Note that the Hamiltonian with b=0 describes a topological or normal insulator depending on the sign of Δ [see Eq. (25)]. Therefore, the above Hamiltonian (88) can be regarded as a model Hamiltonian describing a magnetically doped (topological or normal) insulator. Without loss of generality, we may set b=(0,0,b). In this case the Weyl semimetal phase is realized when |b/Δ|>1, and the Weyl nodes are located at (0,0,±b2Δ2).129 

Here, we outline the derivation of the θ term from the microscopic four-band model (88). In order to describe a more generic Weyl semimetal, we add the term μ5τz to the Hamiltonian, which generates a chemical potential difference 2μ5 between the two Weyl nodes. Note that this term breaks inversion symmetry. We also set Δ=0 for simplicity, so that the momentum-space distance between the Weyl nodes are 2b. Figure 25 shows a schematic illustration of the Weyl semimetal we consider. The action of the system in the presence of external electric and magnetic fields with the four potential Aμ=(A0,A) is given by [see also Eq. (26)]

S=dtd3rψ{i(tieA0)[H0(k+eA)μ5τz]}ψ=dtd3rψ¯iγμ(μieAμibμγ5)ψ,
(89)

where e>0, ψ is a four-component spinor, γ¯=ψγ0, γ0=τx, γj=τxτzσj=iτyσj, γ5=iγ0γ1γ2γ3=τz, and bμ=(μ5,b). Now, we apply Fujikawa’s method50,51 to the action. The procedure is the same as that in the case of topological insulators presented in Sec. II E2. Performing an infinitesimal gauge transformation for infinite times such that

ψψ=eidϕθ(r,t)γ5/2ψ,ψ¯ψ¯=ψ¯eidϕθ(r,t)γ5/2,
(90)

with θ(r,t)=2xμbμ=2(brμ5t) and ϕ[0,1], the action of the system becomes21 

S=dtd3rψ¯[iγμ(μieAμ)]ψ+e22π2dtd3r(brμ5t)EB,
(91)

where the first term represents the (trivial) action of massless Dirac fermions and the second term is nothing else but a θ term [Eq. (1)] with θ(r,t)=2(brμ5t). It should be noted that nonzero, nonquantized expression for θ is due to the time-reversal symmetry breaking by b and the inversion symmetry breaking by μ5.

FIG. 25.

Schematic illustration of a Weyl semimetal with two Weyl nodes. 2b and 2μ5 are the momentum-space distance and the chemical potential difference between the Weyl nodes, respectively. Q±=±1 are the chiralities of the Weyl nodes.

FIG. 25.

Schematic illustration of a Weyl semimetal with two Weyl nodes. 2b and 2μ5 are the momentum-space distance and the chemical potential difference between the Weyl nodes, respectively. Q±=±1 are the chiralities of the Weyl nodes.

Close modal

Next, let us consider the consequences of the presence of a θ term in Weyl semimetals. As we have also seen in the case of insulators with a dynamical axion field, an electric current is induced in the presence of a θ term. The induced electric current density and charge density are given by12 

j(r,t)=δSθδA=e24π2[θ˙(r,t)B+θ(r,t)×E],ρ(r,t)=δSθδA0=e24π2θ(r,t)B.
(92)

In the present case of θ(r,t)=2(brμ5t), we readily obtain a static current of the form

j=e22π2(b×Eμ5B),
(93)

in the ground state. The electric-field induced and magnetic-field induced terms are the anomalous Hall effect and chiral magnetic effect, respectively.21–25,49,129,133,134

To understand the occurrence of the anomalous Hall effect in Weyl semimetals [the first term in Eq. (93)], let us consider a 2D plane in momentum space, which is perpendicular to the vector b. For clarity, we set b=(0,0,b) and Δ=0. In this case, performing a canonical transformation, Eq. (88) can be rewritten in a block-diagonal form with two 2×2 Hamiltonians given by129 

H±(k)=vF(kxσx+kyσy)+m±(kz)σz,
(94)

with m±(kz)=vF(b±|kz|). The two Weyl nodes are located at (0,0,±b). It can be seen readily that m+(kz) is always positive and that m(kz) is positive when bkzb and otherwise negative. As we have seen in Eq. (9), the Hall conductivity of 2D massive Dirac fermions of the form (94) is given by σxy±(kz)=sgn[m±(kz)]e2/2h. Therefore, we find that the total 2D Hall conductivity is nonzero in the region bkzb and otherwise zero, which gives the 3D Hall conductivity as

σxy3D=bbdkz2π[σxy+(kz)+σxy(kz)]=be2πh.
(95)

This value is exactly the same as that of the first term in Eq. (93). The expression for the anomalous Hall conductivity can be generalized straightforwardly to the case of multi-node Weyl semimetals.136 The anomalous Hall conductivity in two-node Weyl semimetals [Eq. (95)] is robust against disorder in the sense that the vertex correction in the ladder-diagram approximation is absent as long as the chemical potential lies sufficiently close to the Weyl nodes.137,138

The chiral magnetic effect in Weyl semimetals [the second term in Eq. (93)] looks like a peculiar phenomenon. The chiral magnetic effect indicates that a direct current is generated along a static magnetic field even in the absence of electric fields, when there exists a chemical potential difference δμ=2μ5 between the two Weyl nodes. If the static chiral magnetic effect exists in real materials, there will be substantial possible applications. The existence of the static chiral magnetic effect is, however, ruled out in crystalline solids as discussed in Ref. 134, which is also consistent with our understanding that static magnetic fields do not generate equilibrium currents. As shall be discussed in detail below, the chiral magnetic effect can be realized under nonequilibrium circumstances, i.e., when the system is driven from equilibrium, for example, by the combined effect of electric and magnetic fields, which has been experimentally observed as the negative magnetoresistance in Weyl semimetals. Another possible situation for realizing the chiral magnetic effect is applying the oscillating (low-frequency) magnetic field.139–142 A related current generation by the oscillating magnetic field is the gyrotropic magnetic effect (natural optical activity),141,142 which is governed by the orbital magnetic moment of the Bloch electrons on the Fermi surface. This is in contrast to the chiral magnetic effect which is driven by the chiral anomaly and governed by the Berry curvature.22 Finally, we note that the dynamical chiral magnetic effect in topological antiferromagnetic insulators shown in Sec. V B2 is also one of the dynamical realizations of the chiral magnetic effect.

As we have seen above, the chiral magnetic effect does not occur in equilibrium. This means that a chemical potential difference between Weyl nodes δμ=2μ5 needs to be generated dynamically in order for the chiral magnetic effect to be realized in Weyl semimetals. In the case of Weyl semimetals, such a chemical potential difference can be generated by the so-called chiral anomaly. The chiral anomaly in Weyl semimetals is referred to as the electron number nonconservation in a given Weyl cone under parallel electric and magnetic fields, in which the rate of pumping of electrons is given by22,138,143

Nit=Qie24π22cEB,
(96)

where i is a valley (Weyl node) index and

Qi=d3k2πf0(εkm)εkmvkmΩkm
(97)

is the chirality of the valley. Here, εkm is the energy of Bloch electrons with momentum k in band m in a given valley i, f0(εkm) is the Fermi distribution function, vkm is the group velocity, and Ωkm is the Berry curvature. The difference of the total electron number between the Weyl nodes leads to the difference of the chemical potential between the Weyl nodes δμ. As shown in Fig. 26, this electron pumping can also be understood by the electron flow through the zeroth Landau level connecting Weyl nodes of opposite chiralities induced by a magnetic field. It should be noted here that electron pumping also occurs in parallel temperature gradient and magnetic field,144,145

Nit=eBT4π22cd3k2πεkmμTf0(εkm)εkmvkmΩkm,
(98)

which can be termed the thermal chiral anomaly. Here, T is the (unperturbed) temperature and μ is the chemical potential.

FIG. 26.

Electron pumping due to the chiral anomaly in a Weyl semimetal under parallel electric and magnetic fields along the z direction.

FIG. 26.

Electron pumping due to the chiral anomaly in a Weyl semimetal under parallel electric and magnetic fields along the z direction.

Close modal

A phenomenon manifested by the chiral anomaly is a negative magnetoresistance (or equivalently positive magnetoconductance) quadratic in the magnetic field for parallel electric and magnetic fields in Weyl and Dirac semimetals.137,138,143,144 Here, note that the usual magnetoresistance due to Lorentz force is positive. For concreteness, we consider the case of electric and magnetic fields along the z direction. The positive quadratic magnetoconductivity arising from the chiral anomaly reads137,138,143,144

σzz(Bz2)=e24π2c2(eBz)2vF3μ2τinter,
(99)

where μ is the equilibrium chemical potential and τinter is the intervalley scattering time. This unusual magnetoconductivity holds in the low-field limit Bz0, since it is derived from a semiclassical approach where the Landau quantization can be neglected. Expression (99) is understood as coming from j(EBτinter)B, which indicates that it is a consequence of the chiral magnetic effect [the second term in Eq. (93)]. It has been shown that the vertex correction in the ladder-diagram approximation is absent in the positive quadratic magnetoconductivity [Eq. (99)].138 Such an unusual negative magnetoresistance has recently been experimentally observed in the Dirac semimetals Na3Bi,146 Cd3As2,147,148 and ZrTe5,149 and in the Weyl semimetals TaAs150 and TaP.151 As shown in Fig. 27, the observed conductance is positive and proportional to B2 in the low-field limit as expected from Eq. (99). Also, we can see that the enhancement of the conductance is largest when the angle between the applied current and magnetic field is zero (i.e., when they are parallel), which is in agreement with the theoretical prediction. However, it must be noted here that those experimental observations of the negative magnetoresistance is now generally understood to be an artifact of “current jetting,”130 which can be large in high-mobility semimetals. The point is that disentangling precisely the intrinsic quantum effect of the chiral anomaly from the extrinsic classical effect of current jetting is not easy in experiments,152 although its presence is manifested theoretically.

FIG. 27.

Magnetic field dependence of the longitudinal conductance in the Dirac semimetal Na3Bi. The conductance shows a quadratic dependence on the magnetic field strength when the angle ϕ between the applied current and magnetic field is small, as expected from Eq. (99). Reproduced with permission from Xiong et al., Science 350, 413 (2015). Copyright 2015 American Association for the Advancement of Science.

FIG. 27.

Magnetic field dependence of the longitudinal conductance in the Dirac semimetal Na3Bi. The conductance shows a quadratic dependence on the magnetic field strength when the angle ϕ between the applied current and magnetic field is small, as expected from Eq. (99). Reproduced with permission from Xiong et al., Science 350, 413 (2015). Copyright 2015 American Association for the Advancement of Science.

Close modal

In this section, we discuss topological responses of 3D topological superconductors and superfluids that can regarded as the thermodynamic analog of the axion electromagnetic responses of topological insulators and Weyl semimetals. A well-known example of 3D topological superfluids is the superfluid 3He B phase.153 The topological nature of such topological superconductors and superfluids will manifest itself in thermal transport properties, such as the quantization of the thermal Hall conductivity,26 since charge and spin are not conserved while energy is still conserved.

The systematic classification of topologically nontrivial insulators and superconductors has been established in terms of symmetries and dimensionality and has clarified that topologically nontrivial superconductors and superfluids with time-reversal symmetry are also realized in three dimensions.153–155 From the bulk-boundary correspondence, there exist topologically protected gapless surface states in topological superconductors. In particular, the superconductivity infers that the gapless surface states are their own antiparticles and thus Majorana fermions.153 Because of the fact that Majorana fermions are charge neutral objects, an electric-transport study such as quantum Hall measurement cannot characterize their topological nature of topological superconductors. Instead, since the energy is still conserved, thermal transport, especially the thermal Hall conductivity, reflects the topological character of topological superconductors as

κH=sgn(m)π26kB22hT
(100)

for the massive Majorana fermion with mass m.26,29

A spatial gradient in energy is related to a temperature gradient, as one can infer from the thermodynamic equality dU=TdS as follows. Here, U is the internal energy, S is the entropy, and T is the temperature. For simplicity, let us first divide the total system into two subsystems (subsystems 1 and 2). The equilibrium of the total system is achieved when the total entropy is maximized: dS=dS1+dS2=0. Since the energy is conserved, dE2=dE1, and hence dS1/dE1dS2/dE2=0, i.e., T1=T2. Let us now turn on a gradient in the “gravitational potential,” so that the gravitational potential felt by subsystems 1 and 2 differs by δϕg. In this case, we have dE2=dE1(1+δϕg). This suggests the generation of a temperature difference T2=T1(1+δϕg). In other words, we can view the “electric” field Eg associated with the gradient of ϕg, which we call a “gravitoelectric field,” as a temperature gradient,156 

Eg=ϕg=T1T.
(101)

In analogy with electromagnetism, let us next consider the following quantity described in terms of a vector potential Ag, which we call a “gravitomagnetic field”:

Bg=×Ag.
(102)

For example, in a system rotating with the angular velocity Ωz around the z axis, Ag can be expressed as Ag=(1/v)Ωzez×r,45,157 which gives Bg=(2/v)Ωzez. Here, v is the Fermi velocity of the system. Therefore, the gravitomagnetic field Bg can be understood as an angular velocity vector. A gravitomagnetic field Bg can also be introduced as a quantity which is conjugate to the energy magnetization (momentum of energy current) ME in the free energy of a Lorentz-invariant system.29 It follows that ME=(v/2)L with L the angular momentum in Lorentz-invariant systems, which also leads to Bg=(2/v)Ω.29,45

Now, we study the responses of 3D topological superconductors to a temperature gradient Eg and a mechanical rotation Bg. For simplicity, we consider a sample in a cylindrical geometry with height and radius r as illustrated in Fig. 28(b). We assume that magnetic impurities are doped near the surface and the magnetization directions are all perpendicular to the surface so that a uniform mass gap is formed in the surface Majorana state. Let us first introduce a temperature gradient in the z direction, which generates the energy current jE=κHzT on the surface. Since jE/v2 corresponds to the momentum per unit area, total momentum due to the surface energy current is Pφ=(2πr)jE/v2 and thus the induced orbital angular momentum per volume is given by

Lz|Ωz=rPφ(πr2)=2v2κHzT.
(103)

Similarly, upon rotating the cylinder with Ω=Ωzez (without a temperature gradient), we obtain the induced thermal energy density (the induced entropy change) localized on the top and bottom surfaces,29 

ΔQ(z)|T=2TΩzv2[κHtδ(z/2)+κHbδ(z+/2)],
(104)

where κHt (κHb) is the thermal Hall conductivity on the top (bottom) surface given by Eq. (100). Here, κHt=κHb because the magnetization directions on the top and bottom surfaces are opposite to each other, resulting in different signs of m [see Fig. 28(b)].

FIG. 28.

Electromagnetic responses in (a) 3D topological insulators and thermal and mechanical (rotation) responses in (b) 3D topological superconductors. In (a), an electric field E induces the surface Hall current j. In (b), a temperature gradient Eg induces the surface thermal Hall current jE. A uniform mass gap is induced in the surface fermion spectra by doping magnetic impurities near the surface of the 3D topological insulator (a) and topological superconductor (b) such that the magnetization directions are all perpendicular to the surfaces (as indicated by red arrows).

FIG. 28.

Electromagnetic responses in (a) 3D topological insulators and thermal and mechanical (rotation) responses in (b) 3D topological superconductors. In (a), an electric field E induces the surface Hall current j. In (b), a temperature gradient Eg induces the surface thermal Hall current jE. A uniform mass gap is induced in the surface fermion spectra by doping magnetic impurities near the surface of the 3D topological insulator (a) and topological superconductor (b) such that the magnetization directions are all perpendicular to the surfaces (as indicated by red arrows).

Close modal

In terms of the gravitoelectric field Eg=T1T and the momentum of the energy current (i.e., energy magnetization) ME, Eq. (103) can be written as ME=(TκH/v)Eg from the relation ME=(v/2)L. Furthermore, introducing the thermal polarization PE by ΔQ=PE, Eq. (104) can be written similarly as PE=(TκH/v)Bg. Combining these, we find the correspondence between topological insulators and topological superconductors,

TI:MaEb=PaBbTSC:MEaEgb=PEaBgb.
(105)

Since the orbital angular momentum is obtained from the internal energy functional as La=δUθ/δΩa, the coupling energy of the temperature gradient and angular velocity is written as29 

Uθg=d3x2v2κHTΩ=d3xkB2T224vθgEgBg.
(106)

This is analogous to the axion electromagnetic response with e2/c(πkBT)2/6v and θg=π playing the same role as θ in the θ term. Here, note that we have considered the contribution from one Majorana fermion to the internal energy (106). In general, 3D time-reversal invariant (class DIII) topological superconductors with topological invariant N possesses N gapless Majorana fermions localized at the surface.153 When uniform mass gaps (of the same sign) are induced in these Majorana fermions, each Majorana fermion gives rise to the half-integer thermal Hall effect [Eq. (100)].27,30 Therefore, it follows that θg=Nπ in Eq. (106) for this generalized case.

In the case of 2D topological superconductors, the corresponding term is written as

U=d2x(2/v2)TκHϕΩz.
(107)

This is the thermodynamical analog of the Chern–Simons term. A similar term has been derived in the context of 3D 3He A phase with point nodes,158 where the current flows parallel to the Ω vector. A comparison between cross correlations in topological insulators and topological superconductors in two and three spatial dimensions is summarized in Table II.

TABLE II.

Comparison between cross correlations in topological insulators (TIs) and topological superconductors (TSCs) in two and three spatial dimensions. In topological superconductors, the orbital angular momentum L (momentum of energy current ME) and the entropy S (thermal polarization PE in three dimensions) are generated by a temperature gradient Eg=T1T and by a mechanical rotation with angular velocity vector Ω = (v/2)Bg. In analogy with the orbital magnetoelectric polarizability χθab=δabe2/(4πc) in 3D topological insulators, the gravitomagnetoelectric polarizability χθ,gab=δabπkB2T2/(24v) can be introduced in 3D topological superconductors. Note that the relations for topological superconductors applies also to the thermal response of topological insulators.

TITSC
2D σH=ecMzμ=ecNBz κH=v22LzT=v22SΩz 
3D χθab=MaEb=PaBb χθ,gab=MEaEgb=PEaBgb 
TITSC
2D σH=ecMzμ=ecNBz κH=v22LzT=v22SΩz 
3D χθab=MaEb=PaBb χθ,gab=MEaEgb=PEaBgb 

Here, we overview a topological field theory approach to the gravitational (thermal) response of 3D topological superconductors and superfluids.27,28 In Sec. VII A, we have introduced gravitoelectric and gravitomagnetic fields that are written in terms of (fictitious) scalar and vector potentials. Strictly speaking, the presence of a gravitational background should be described as a curved spacetime. Let us consider the Bogoliubov–de Gennes Hamiltonian of the 3He B phase,

HBdG(k)=(Δp/kF)kα+ξkα4,
(108)

where kF is the Fermi wave number, Δp is the p-wave pairing amplitude, ξk=2k2/2mμ with μ the chemical potential is the kinetic energy, and 4×4 matrices αμ satisfy the Clifford algebra {αμ,αν}=2δμν. Clearly, Eq. (108) is a massive Dirac Hamiltonian. When μ>0 (μ<0), the system is topologically nontrivial (trivial).153,159 In the presence of such a gravitational background, the action of a 3D topological superconductor such as the 3He B phase is written as160 

S=d4xgL,L=ψ¯eaμiγa(μi2ωμabΣab)ψmψ¯ψ,
(109)

where μ=0,1,2,3 is a spacetime index, a,b=0,1,2,3 is a flat index, g=det(g) with gμν the metric tensor, eaμ is the vielbein, ωμab is the spin connection, and Σab=[γa,γb]/(4i) is the generator of Lorentz transformation. As in the case of topological insulators (Sec. II E2) and Weyl semimetals (Sec. VI A), we can apply Fujikawa’s method to the action (109), in which the topological term of a system comes from the Jacobian. After a calculation, we arrive at a gravitational effective action27,28

Sg=11536π2d4xθεμνρσRβμναRαρσβ,
(110)

where θ=π and Rβμνα is the Riemannian curvature tensor.

It follows that the coefficient θ in Eq. (110) is θ=0 or π (mod 2π) due to time-reversal symmetry. However, in 3D time-reversal invariant topological insulators with topological number N, topological actions should have θ=Nπ. This is because the Hamiltonian of a noninteracting 3D time-reversal invariant topological insulator with topological number N can be decomposed into N copies of the Hamiltonian of the form (108). The gravitational effective action (110) provides only a Z2 classification of 3D time-reversal invariant topological insulators, which is weaker than the Z classification that they have.

As we have seen in Sec. VII A, the derivation of the internal energy term [Eq. (106)] for 3D topological superconductors and superfluids is not a microscopic derivation but a heuristic one based on the surface thermal Hall effect. It has been suggested that the fluctuation of θg in a p+is-wave superconductor can be written as a function of the relative phase between the two superconducting gaps.161,162 Such a fluctuation of a relative phase is known as the Leggett mode and can depend on time. Also, in analogy with 3D topological insulators, it is expected that the internal energy term can be extended to the form of an action (see Sec. II D for the derivation of the θ term in 3D topological insulators). Therefore, it would be appropriate to consider the action of the form162,163

Sθg=kB2T0224vdtd3rθg(r,t)EgBg
(111)

for non-quantized and dynamical values of θg, instead of the internal energy Ugθ [Eq. (106)].

In order to induce the deviation of θg from the quantized value Nπ (with N the topological number of the system), time-reversal symmetry of the bulk needs to be broken, as in the case of insulators. It has been shown theoretically that the imaginary s-wave pairing in class DIII topological superconductors such as the 3He B phase leads to the deviation of the value of θg from π such that θg=π+tan1(ΔsIm/μ) with ΔsIm the imaginary s-wave pairing amplitude.161 Such an imaginary s-wave pairing term in a Bogoliubov–de