One of the intriguing properties characteristic to three-dimensional topological materials is the topological magnetoelectric phenomena arising from a topological term called the term. Such magnetoelectric phenomena are often termed the axion electrodynamics since the 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 term.
I. INTRODUCTION
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 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 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 takes a quantized value (mod ) for topological insulators, while in trivial insulators. However, in systems with broken time-reversal symmetry, e.g., in magnetically ordered phases, the value of can be arbitrary, i.e., can deviate from the quantized value or , which means that the value of can even depend on space and time as . It should be noted that spatial-inversion symmetry breaking can also lead to the deviation of from the quantized value or .
In the field theory literature, the phenomena described by the term is termed the axion electrodynamics12 because the 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 term, , is a field describing the axion, observing the magnetoelectric responses in materials whose effective action is described by a 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 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 is schematically shown.
Schematic of a classification of 3D insulators in terms of time-reversal symmetry and the orbital magnetoelectric coupling coefficient . 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 (mod ). 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 (mod ). In magnetic insulators, the value of is arbitrary, including . In a class of magnetic insulators termed topological magnetic insulators, is proportional to their magnetic order parameters such as the Néel vector (i.e., antiferromagnetic order parameter), and the fluctuation of the order parameter realizes a dynamical axion field in condensed matter. Here, note that spatial-inversion symmetry must be broken in order for the value of 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.
Schematic of a classification of 3D insulators in terms of time-reversal symmetry and the orbital magnetoelectric coupling coefficient . 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 (mod ). 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 (mod ). In magnetic insulators, the value of is arbitrary, including . In a class of magnetic insulators termed topological magnetic insulators, is proportional to their magnetic order parameters such as the Néel vector (i.e., antiferromagnetic order parameter), and the fluctuation of the order parameter realizes a dynamical axion field in condensed matter. Here, note that spatial-inversion symmetry must be broken in order for the value of 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.
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 reversal . | Inversion . | Value of θ (mod 2π) . |
---|---|---|
✓ | ✓ | 0 or π |
✓ | × | 0 or π |
× | ✓ | 0 or π |
× | × | Arbitrary |
Time reversal . | Inversion . | Value of θ (mod 2π) . |
---|---|---|
✓ | ✓ | 0 or π |
✓ | × | 0 or π |
× | ✓ | 0 or π |
× | × | Arbitrary |
The effective action of the form of the 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 . 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 term with ,21–25 where is the distance between the two Weyl nodes in momentum space and 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 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 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 is quantized due to a combined symmetry (effective time-reversal symmetry), regardless of the breaking of time-reversal symmetry, focusing on MnBi Te family of materials. In Sec. IV, we consider generic expressions for in insulators and extend the derivation of the 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 term. In Sec. VIII, we summarize this Tutorial and outlook future directions of this fascinating research field.
II. QUANTIZED MAGNETOELECTRIC EFFECT IN 3D TOPOLOGICAL INSULATORS
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 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.
A. Overview
Schematic picture of the topological magnetoelectric effect in a 3D topological insulator. (a) Magnetization induced by an external electric field . is the anomalous Hall current on the side surface induced by the electric field. (b) Electric polarization induced by an external magnetic field . Surface states are gapped by magnetic impurities (or a proximitized ferromagnet) whose magnetization direction is perpendicular to the surface, as indicated by green arrows.
Schematic picture of the topological magnetoelectric effect in a 3D topological insulator. (a) Magnetization induced by an external electric field . is the anomalous Hall current on the side surface induced by the electric field. (b) Electric polarization induced by an external magnetic field . Surface states are gapped by magnetic impurities (or a proximitized ferromagnet) whose magnetization direction is perpendicular to the surface, as indicated by green arrows.
B. Symmetry analysis of the magnetoelectric coupling
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 and 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).
C. Surface half-quantized anomalous Hall effect
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 , 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) Te 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 . 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.
(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 dependence of the Hall conductivity and the longitudinal conductivity in a thin film of Cr-doped (Bi,Sb) Te . Reproduced with permission from Chang et al., Science 340, 167 (2013). Copyright 2013 American Association for the Advancement of Science.
(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 dependence of the Hall conductivity and the longitudinal conductivity in a thin film of Cr-doped (Bi,Sb) Te . Reproduced with permission from Chang et al., Science 340, 167 (2013). Copyright 2013 American Association for the Advancement of Science.
D. Phenomenological derivation of the θ term
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.
E. Microscopic derivation of the θ term
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 term microscopically from a low-energy effective model of 3D topological insulators. There are several ways to derive the 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 term based on Fujikawa’s method.
1. Effective Hamiltonian for 3D topological insulators
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 point represents the usual (continuum) massive Dirac fermions with mass , while Eq. (23) around other momentum points, e.g., , represent massive Dirac fermions with the mass .
2. Fujikawa’s method
F. Toward observations of the topological magnetoelectric effect
1. Utilizing topological insulator thin films
(a) Quantum Hall effect in a topological insulator (Bi Sb ) Te 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.
(a) Quantum Hall effect in a topological insulator (Bi Sb ) Te 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.
(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.
(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.
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.
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.
2. Faraday and Kerr rotations
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.
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.
(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 as a function of dc Hall conductance towards the universal relationship . From Okada et al., Nat. Commun. 7, 12245 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
(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 as a function of dc Hall conductance towards the universal relationship . From Okada et al., Nat. Commun. 7, 12245 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
III. AXION INSULATORS
In Sec. II, we have seen that the topological magnetoelectric effect with the quantized coefficient (mod ) occurs in 3D time-reversal invariant topological insulators. In general, the value of 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 (mod ). In this section, we review theoretical and experimental studies on such antiferromagnetic topological insulators, which are also called the axion insulators. Starting from the basics of the antiferromagnetic topological insulators, we focus on the MnBi Te family of materials that are layered van der Waals compounds and have recently been experimentally realized.
A. Quantized magnetoelectric effect in antiferromagnetic topological insulators
Schematic illustration of an antiferromagnetic topological insulator protected by the symmetry.
Schematic illustration of an antiferromagnetic topological insulator protected by the symmetry.
Next, let us consider the resulting surface states. Since on the Brillouin zone plane satisfying , the 2D subsystem on the plane is regarded as a quantum spin-Hall system with time-reversal symmetry. This means that the or dependence of the surface spectra must be gapless because the line of the surface states is the boundary of the 2D subsystem (the plane) in the bulk Brillouin zone. In other words, at the surfaces that are parallel to , which preserve the 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 , which break the 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 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 term, as in the case of time-reversal invariant 3D topological insulators. To see this, recall that the magnetoelectric effect resulting from a term is expressed as , and , where and are the electric polarization and the magnetization, respectively. Under time-reversal , the coefficient changes sign , because and , while and . On the other hand, the lattice translation does not affect .69 Combining these, the operation implies the transformation such that with being an integer. Then, it follows that or modulo .
B. MnBi2Te4
1. Electronic structure of MnBi2Te4 bulk crystals
With the knowledge of antiferromagnetic topological insulators with the symmetry, here we review recent experimental realizations of the antiferromagnetic topological insulator state in MnBi Te .71–81 The crystal structure of MnBi Te 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 direction), which is called A-type AFM- . The Néel temperature is reported to be about .71,74,79,81 The unit cell of the antiferromagnetic insulator state consists of two septuple layers (Fig. 10), where is the half-cell translation vector along the 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 symmetry, indicating that the system is a topological antiferromagnetic insulator, which we have discussed in Sec. III A. Interestingly, the bulk bandgap is estimated to be about 0.2 eV,71,72 which is comparable to that of the time-reversal invariant topological insulator Bi Se .
Crystal and magnetic structure of the antiferromagnetic topological insulator state in MnBi Te . The unit cell consists of two septuple layers. is the half-cell translation vector along the 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.
Crystal and magnetic structure of the antiferromagnetic topological insulator state in MnBi Te . The unit cell consists of two septuple layers. is the half-cell translation vector along the 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.
The surface states of antiferromagnetic MnBi Te are somewhat complicated. Theoretical studies have predicted that the (0001) surface state (i.e., at the surface perpendicular to the axis) which breaks the symmetry of the A-type AFM- state is gapped,71,72 as indicated by the property of antiferromagnetic topological insulators (see Sec. III A). The first experimental study reported that the (0001) surface state is gapped.71 However, subsequent studies reported that it is gapless.77–79,84,85 Figure 11(a) shows an ARPES measurement of the bulk and surface states, in which the surface state is clearly gapless Dirac cone at the (0001) surface. Among possible spin configurations that are allowed by symmetry, Ref. 77 proposed that the gapless surface state is protected by the mirror symmetry , while the symmetry is broken at the surface. (Note that the mirror symmetry is broken in the A-type AFM- state.) In other words, A-type AFM with the magnetic moments along the axis (i.e., the in-plane direction), whose bulk and surface spectra obtained by a first-principles calculation is shown in Fig. 11(b), might be realized in MnBi Te instead of the A-type AFM- shown in Fig. 11(c). These observations of the gapless surface states imply the occurrence of a surface-mediated spin reconstruction.
(a) Bulk and surface spectra of MnBi Te obtained by an ARPES measurement. Bulk and surface spectra of MnBi Te obtained by a first-principles calculation, which assumes (b) A-type AFM with the magnetic moments along the axis and (c) A-type AFM with the magnetic moments along the axis. From Hao et al., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
(a) Bulk and surface spectra of MnBi Te obtained by an ARPES measurement. Bulk and surface spectra of MnBi Te obtained by a first-principles calculation, which assumes (b) A-type AFM with the magnetic moments along the axis and (c) A-type AFM with the magnetic moments along the axis. From Hao et al., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
As pointed in Ref. 72, it should be noted here that the antiferromagnetic order in MnBi Te is essentially different from such an antiferromagnetic order in Fe-doped Bi Se which has been proposed to realize a dynamical axion field.20 In the latter case, time-reversal and inversion symmetries are both broken, allowing the deviation of the value of from . The antiferromagnetic fluctuation contributes to the dynamical axion field at linear order in the Néel field. In contrast, in MnBi Te , an effective time-reversal symmetry and inversion symmetry are both preserved, keeping the quantization and making no contribution to the dynamical axion field at linear order in the Néel field.
2. Transport properties of MnBi2Te4 thin films
Due to the intralayer ferromagnetism and interlayer antiferromagnetism of the Mn layers, the layered van der Waals crystal MnBi Te exhibit interesting properties in its few-layer thin films. In even-septuple-layer films, and symmetries are both broken, but symmetry is preserved.73 As we have seen above, the presence of symmetry leads to doubly degenerate bands. On the other hand, in odd-septuple-layer films, symmetry is preserved, but and 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 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 is realized by the combination of half-quantized anomalous Hall states with opposite conductivities 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 (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 that results from the half-quantized anomalous Hall conductivity of the same sign at the top and bottom surfaces, giving rise to the Chern number as shown in Fig. 12(b).
Experimental observations that are consistent with theoretical predictions have been made. Figure 13 shows the resistivity measurement in a six-septuple-layer MnBi Te 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 in a strong magnetic field were clearly observed. Also, the change in the Chern number between was observed in response to the change in the magnetic field direction. Figure 14 shows the resistivity measurement in a five-septuple-layer MnBi Te film,88 in which a quantum anomalous Hall effect with the quantized Hall resistivity was clearly observed.
Schematic illustration of (a) an axion insulator state realized in an even-septuple-layer MnBi Te film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi Te 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 ( ), or equivalently, the Chern number ( ). From Li et al., Sci. Adv. 5, eaaw5685 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
Schematic illustration of (a) an axion insulator state realized in an even-septuple-layer MnBi Te film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi Te 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 ( ), or equivalently, the Chern number ( ). From Li et al., Sci. Adv. 5, eaaw5685 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
Resistivity measurement in a six-septuple-layer MnBi Te 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 in a magnetic field of . Reproduced with permission from Liu et al., Nat. Mater. 19, 522 (2020). Copyright 2020 Springer Nature.
Resistivity measurement in a six-septuple-layer MnBi Te 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 in a magnetic field of . Reproduced with permission from Liu et al., Nat. Mater. 19, 522 (2020). Copyright 2020 Springer Nature.
Resistivity measurement in a five-septuple-layer MnBi Te film, showing a quantum anomalous Hall effect with the quantized transverse resistivity 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.
Resistivity measurement in a five-septuple-layer MnBi Te film, showing a quantum anomalous Hall effect with the quantized transverse resistivity 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.
C. MnBi2Te4 family of materials
Taking advantage of the nature of van der Waals materials, the layered van der Waals heterostructures of (MnBi Te ) (Bi Te ) can be synthesized. Here, it is well known that Bi Te is a time-reversal invariant topological insulator.37 So far, MnBi Te ( )89–93 and MnBi Te ( and )89,93,94 have been experimentally realized. Figure 15 shows schematic illustrations of MnBi Te and MnBi Te and their STEM images. In MnBi Te , a quintuple layer of Bi Te and a septuple layer of MnBi Te stack alternately. In MnBi Te , two quintuple layers of Bi Te are sandwiched by septuple layers of MnBi Te . As in the case of MnBi Te , interlayer antiferromagnetism (between Mn layers) develops with a Néel temperature in MnBi Te 89,90,93 and in MnBi Te ,93 and this antiferromagnetic insulator state is protected by the symmetry, which indicates that MnBi Te and MnBi Te 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 .89,90 A magnetic phase diagram of MnBi Te is shown in Fig. 16. Also, two distinct types of topological surface states are realized depending on the Bi Te quintuple-layer termination or the MnBi Te septuple-layer termination.91,92 ARPES studies showed that the Bi Te quintuple-layer termination gives rise to gapped surface states, while the MnBi Te septuple-layer termination gives rise to gapless surface states.91,92 Note that these terminations break the symmetry, which implies in principle gapped surface states (see Sec. III A). It is suggested that the gap opening in the Bi Te quintuple-layer termination can be explained by the magnetic proximity effect from the MnBi Te septuple layer beneath and that the gaplessness in MnBi Te septuple-layer termination can be explained by the restoration of time-reversal symmetry at the septuple-layer surface due to disordered spin.92 On the other hand, an ARPES study of MnBi Te observed a gapped Dirac surface state in the MnBi Te septuple-layer termination.94
Since the bulk crystals of MnBi Te and MnBi Te are realized by van der Waals forces, various heterostructures in the 2D limit, which are made from the building blocks of the MnBi Te septuple layer and the Bi Te quintuple layer, can be obtained by exfoliation. A theoretical calculation shows that such 2D heterostructures exhibit the quantum spin-Hall effect without time-reversal symmetry and the quantum anomalous Hall effect.95 Theoretically, it is suggested that (MnBi Te )(Bi Te ) is a higher-order topological insulator hosting surface states with a Möbius twist.96 In contrast to MnBi Te in which the value of is quantized to be , it is suggested that the antiferromagnetic insulator phases of Mn Bi Te (with and )97 and Mn2Bi Te 98 in which the symmetry is absent, break both time-reversal and inversion symmetries, realizing a dynamical axion field.
Schematic illustrations of (a) MnBi Te and (b) MnBi Te . STEM images of (c) MnBi Te and (d) MnBi Te , showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi Te and a septuple layer of MnBi Te , respectively. From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
Schematic illustrations of (a) MnBi Te and (b) MnBi Te . STEM images of (c) MnBi Te and (d) MnBi Te , showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi Te and a septuple layer of MnBi Te , respectively. From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
Magnetic phase diagram of MnBi Te 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.
Magnetic phase diagram of MnBi Te 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.
D. EuIn2As2 and EuSn2As2
EuIn As and EuSn As have also been considered a candidate class of materials for antiferromagnetic topological insulators with inversion symmetry.99 Different from MnBi Te which is a layered van der Waals material, EuIn As has a three-dimensional crystal structure as shown in Fig. 17. EuSn As has a very similar crystal and magnetic structure to EuIn As . Two metastable magnetic structures with the magnetic moments parallel to the axis (AFM ) and the axis (AFM ) have been known in EuIn As and EuSn As .100,101 As in the case of MnBi Te , the antiferromagnetic insulator phases of EuIn As and EuSn As are protected by the symmetry, with the half-unit-cell translation vector connecting four Eu atoms along the axis. Indeed, ARPES measurements in EuIn As 102 and EuSn As 78 suggests that they are antiferromagnetic topological insulators. Theoretically, it is suggested that antiferromagnetic EuIn As (both AFM and AFM ) is at the same time a higher-order topological insulator with gapless chiral hinge states lying within the gapped surface states.99
IV. EXPRESSIONS FOR θ IN INSULATORS
We have seen in Sec. II that time-reversal symmetry and inversion symmetry impose the constraint on the coefficient of the topological magnetoelectric effect such that in 3D topological insulators and in 3D normal insulators. In this section, first, we derive a generic expression for which is given in terms of the Bloch-state wave function. Then, we show explicitly that the value of 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.
Crystal and magnetic structures of EuIn As . There are two metastable magnetic structures where the magnetic moments align parallel to (a) the axis and (b) the axis. Reproduced with permission from Xu et al., Phys. Rev. Lett. 122, 256402 (2019). Copyright 2019 American Physical Society.
Crystal and magnetic structures of EuIn As . There are two metastable magnetic structures where the magnetic moments align parallel to (a) the axis and (b) the axis. Reproduced with permission from Xu et al., Phys. Rev. Lett. 122, 256402 (2019). Copyright 2019 American Physical Society.
A. General expression for θ from the dimensional reduction
Numerically obtained value of in the Fu–Kane–Mele model on a diamond lattice. Here, with being a staggered Zeeman field in the [111] direction of the diamond lattice, and being the hopping strength anisotropy due to the lattice distortion in the [111] direction. When ( ), 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.
Numerically obtained value of in the Fu–Kane–Mele model on a diamond lattice. Here, with being a staggered Zeeman field in the [111] direction of the diamond lattice, and being the hopping strength anisotropy due to the lattice distortion in the [111] direction. When ( ), 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.
B. Expression for θ in topological magnetic insulators
1. Four-band Dirac model
2. Fu–Kane–Mele–Hubbard model on a diamond lattice
(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 points with (represented by green circles), massive Dirac Hamiltonians are derived.
(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 points with (represented by green circles), massive Dirac Hamiltonians are derived.
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 , the value of has a linear dependence on when (i.e., around or ). Thus, the analytical result [Eq. (63)] is in agreement with the numerical result when the deviation from the quantized value ( or ) is small, since in Eq. (63), when .
C. Values of θ in real materials from first principles
In real materials, there are two contributions to the linear magnetoelectric coupling: electronic and ionic (i.e., lattice) contributions. These contributions can be further decomposed in to spin and orbital parts. Among the electronic contribution, Eq. (52) represents on an electronic orbital contribution to the isotropic linear magnetoelectric coupling. Here, note that there exist two additional electronic orbital (but non-topological) contributions to the isotropic linear magnetoelectric coupling.105,106 Cr O 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 and . Figure 20 shows the value of in Cr O obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-space mesh .36 The value of extrapolated in the limit is , which corresponds to ( ). This value is about two orders of magnitude smaller than the experimentally observed value (i.e., full response) of the linear magnetoelectric tensor in Cr O . The values of in other conventional magnetoelectrics have also been evaluated in Ref. 36 as in BiFeO and in GdAlO , which are both very small compared to the quantized value . As a different approach, it has been proposed that the value of may be extracted from experimental observed parameters.107,108
What are the conditions for larger values of in real materials? It was also shown in Ref. 36, the value of in Cr O is approximately proportional to the spin–orbit coupling strength, which implies that materials with strong spin–orbit coupling can have large values of . 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 from the quantized values or . The value of changes continuously from [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 , then it will exhibit a significantly large magnetoelectric effect of .
Value of in Cr O 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 ( ). Reproduced with permission from Coh et al., Phys. Rev. B 83, 085108 (2011). Copyright 2011 American Physical Society.
Value of in Cr O 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 ( ). Reproduced with permission from Coh et al., Phys. Rev. B 83, 085108 (2011). Copyright 2011 American Physical Society.