Quantum Anomalous Hall Effect and Related Topological Electronic States
Abstract
Over a long period of exploration, the successful observation of quantized version of anomalous Hall effect (AHE) in thin film of magneticallydoped topological insulator completed a quantum Hall trio—quantum Hall effect (QHE), quantum spin Hall effect (QSHE), and quantum anomalous Hall effect (QAHE). On the theoretical front, it was understood that intrinsic AHE is related to Berry curvature and U(1) gauge field in momentum space. This understanding established connection between the QAHE and the topological properties of electronic structures characterized by the Chern number. With the time reversal symmetry broken by magnetization, a QAHE system carries dissipationless charge current at edges, similar to the QHE where an external magnetic field is necessary. The QAHE and corresponding Chern insulators are also closely related to other topological electronic states, such as topological insulators and topological semimetals, which have been extensively studied recently and have been known to exist in various compounds. Firstprinciples electronic structure calculations play important roles not only for the understanding of fundamental physics in this field, but also towards the prediction and realization of realistic compounds. In this article, a theoretical review on the Berry phase mechanism and related topological electronic states in terms of various topological invariants will be given with focus on the QAHE and Chern insulators. We will introduce the Wilson loop method and the band inversion mechanism for the selection and design of topological materials, and discuss the predictive power of firstprinciples calculations. Finally, remaining issues, challenges and possible applications for future investigations in the field will be addressed.
Keywords: anomalous Hall effect; quantum anomalous Hall effect; topological invariants; Chern number and Chern insulators; topological electronic states; firstprinciples calculation
List of Contents
I. Introduction
I.A. Hall Effect and Anomalous Hall Effect (AHE)
I.B. Berry Phase Mechanism for Intrinsic AHE
II. Quantum Hall Family and Related Topological Electronic States
II.A. Chern Number, Chern Insulator, and Quantum AHE (QAHE)
II.B. Invariant, Topological Insulator, and
Quantum Spin Hall Effect (QSHE)
III.C. Magnetic Monopole, Weyl Node and Topological Semimetal
III. FirstPrinciples Calculations for Topological Electronic States
III.A. Calculations of Berry Connection and Berry
Curvature
III.B. Wilson Loop Method for Evaluation of
Topological Invariants
III.C. Boundary States Calculations
IV. Material Predictions and Realizations of QAHE
IV.A. Band Inversion Mechanism
IV.B. QAHE in Magnetic Topological Insulators
IV.C. QAHE in Thin Film of Weyl semimetals
IV.D. QAHE on Honeycomb Lattice
V. Discussion and Future Prospects
VI. Acknowledgements
VII. References
I Introduction
I.A. Hall Effect and Anomalous Hall Effect (AHE)
Edwin H. Hall discovered, in 1879, that when a conductor carrying longitudinal current was placed in a vertical magnetic field, the carriers would be pressed towards the transverse side of the conductor, which led to observed transverse voltage. This is called Hall effect (HE) Hall (1880), and it was a remarkable discovery, although it was difficult to understand at that time since the electron was not to be discovered until 18 years later. We now know that the HE is due to the Lorentz force experienced by the moving electrons in the magnetic field, which is balanced by a transverse voltage for a steady current in the longitudinal direction. The Hall resistivity under perpendicular external magnetic field (along the direction) can be written as , where is the Hall coefficient, which can be related to the carrier density as (in the free electron gas approximation). The HE is a fundamental phenomenon in condensed matter physics, and it has been widely used as as experimental tool to identify the type of carrier and to measure the carrier density or the strength of magnetic fields.
In 1880, Edwin H. Hall further found that this “pressing electricity effect” in ferromagnetic (FM) conductors was larger than in nonmagnetic (NM) conductors. This enhanced Hall effect was then called as the anomalous Hall effect (AHE) Hall (1881), in order to distinguish it from the ordinary HE. Later experiments on Fe, Co and Ni Kundt (1893) suggested that the AHE was related to the sample magnetization (along ), and an empirical relation for the total Hall effect in FM conductors was established as Pugh (1930); Pugh and Lippert (1932)
(1) 
where the second term is the anomalous Hall resistivity, and its coefficient is materialdependent (in contrast to , which depends only on carrier density ).
As schematically shown in Fig. 1, if we plot the Hall resistivity versus external magnetic field , we will generally expect a straight line (which crosses the origin of the coordinates) for NM conductors (see Fig. 1(a)); however, nonlinear behavior will be expected for FM conductors, wherein increases sharply at low field and crosses over into a linear region under high field (see Fig. 1(b)). The initial sharp increase of is due to the saturation of magnetization of the sample under external field. After the saturation of magnetization, will depend on linearly (for high field region), which is dominated by the ordinary Hall contribution. Therefore, the slope of the linear part under high field gives us . If we extrapolate this linear part to the zero field limit (=0), it will not go through the origin of coordinates, and its intercept on axis gives us as can be learned from Eq. (1). This is a surprising result, which immediately implies two important facts: (1) A kind of Hall effect can be observed even in the absence of external magnetic field (i.e., no Lorentz force); (2) The anomalous Hall resistivity is sensitive to magnetic moment , and it has been suggested that this property would be useful for the detection of magnetization of conducting carriers, particularly for the cases of weak itinerant magnetism, such as surface/interface magnetization Bergmann (1978), dilute magnetic semiconductors Ohno et al. (1992, 1996), etc. Experimentally, measurements are usually done in magnetization loops by scanning the magnetic field from positive to negative, and a hysteresis loop in the vs. relationship is to be expected. This is very similar to the familiar hysteresis loop observed in the vs. curve, as usually found in FM conductors. The empirical relation Eq. (1) is very simple and widely used. Unfortunately, its correctness is not well justified. As will be pointed out in a latter part of this article, the anomalous Hall resistivity may have a very complicated form, which is usually nonlinear in .
Although the HE and AHE look quite similar phenomenologically, their underlying physics are completely different. The HE is due to the Lorentz force’s influence on the moving electrons under magnetic field; however the AHE exists in the absence of external magnetic field, where there is no orbital effect of moving electrons. In other words, why do the electrons move towards the transverse direction in the absence of Lorentz force? The mechanism of AHE has been an enigmatic problem since its discovery, a problem that lasted almost a century. This problem involves concepts deeply related to topology and geometry that have been formulated only in recent years Jungwirth et al. (2002); Onoda and Nagaosa (2002); Fang et al. (2003); Yao et al. (2004); Sinova et al. (2004); Nagaosa et al. (2010); Xiao et al. (2010) after the Berry phase was recognized in 1984 Berry (1984).
Karplus and Luttinger provided a crucial step in unraveling this problem as early as 1954 Karplus and Luttinger (1954). They showed that moving electrons under external electric field can acquire an “anomalous velocity”, which is perpendicular to the electric field and contributes to the transverse motion of electrons and therefore to the AHE. This “anomalous velocity” comes from the occupied electronic states in FM conductors with spinorbit coupling (SOC). They suggested that the mechanism leads to an anomalous Hall resistivity proportional to the square of the longitudinal resistivity: . Since this contribution depends only on the electronic band structures of perfect periodic crystal and is completely independent of scattering from impurities or defects, it is called intrinsic AHE, and it was not widely accepted until the concept of Berry phase was well established.
For a long time, two other extrinsic contributions had been considered to be the dominant mechanisms that give rise to the AHE. Smit Smit (1955, 1958) suggested that there always exist defects or impurities in real materials, which will scatter the moving electrons. In the presence of SOC and ferromagnetism, this scattering is asymmetric and should lead to unbalanced transverse motion of electrons. This is called skew scattering, and Smit argued that it is the main source of the AHE. This mechanism predicted , in contrast with the intrinsic contribution. Berger Berger (1970), on the other hand, argued that electrons should experience a difference in electric field when approaching and leaving an impurity, and this leads to another asymmetric scattering process called side jump, which also contributes to the AHE. This mechanism marvelously predicts , the same as the intrinsic contribution.
The debates about the origin of AHE lasted for a long time, and no conclusion could be drawn unambiguously. From the experimental point of view, defects or impurities in samples are unavoidable and are usually complicated with rich varieties. The contributions from both intrinsic and extrinsic mechanisms should generally coexist Onoda et al. (2006). Experimentalists have tried hard to distinguish them Miyasato et al. (2007); Tian et al. (2009); Ye et al. (2012); Zeng et al. (2006). The controversy also arises because of a lack of quantitative calculations that could be compared with experiments.
Still, the discovery of the quantum Hall effect in 1980s Klitzing et al. (1980) and the later studies on the geometric phase Wilczek and Shapere (1989) and the topological properties of quantum Hall states Thouless et al. (1982); Prange and Girvin (1987); Stone (1992), promoted the fruitful study of AHE significantly. Around the early years of this century, the intrinsic AHE mechanism proposed by Karplus and Luttinger was completely reformulated in the language of Berry phase and topology in electronic structures Jungwirth et al. (2002); Onoda and Nagaosa (2002); Fang et al. (2003); Yao et al. (2004); Sinova et al. (2004). It was recognized for the first time that the socalled “anomalous velocity” originates from the Berry curvature of occupied eigen wave functions, which can be understood as effective magnetic field in momentum space. This effective magnetic field modifies the equation of motion of electrons and leads to the intrinsic AHE Jungwirth et al. (2002); Onoda and Nagaosa (2002); Fang et al. (2003); Yao et al. (2004); Sinova et al. (2004); Xiao et al. (2010); Nagaosa et al. (2010). From this understanding, we now know the intrinsic AHE can be evaluated quantitatively from the band structure calculations, thanks to the rapid development in the field of firstprinciples calculations. Detailed calculations for various compounds portray the contributions from intrinsic AHE in a way that convincingly confirms the existing experimental results Fang et al. (2003); Yao et al. (2004, 2007); Zeng et al. (2006); Tian et al. (2009); Lee et al. (2004), establishing the dominant role of the intrinsic contribution to the AHE. In the meantime, the extrinsic AHE was also analyzed carefully Nagaosa et al. (2010), and it was understood that its contribution dominates for the clean limit case (rather than the dirty limit), where the deviation of distribution of electronic states from the equilibrium distribution is significant. In this article, we will not discuss the extrinsic AHE. For readers who want to learn more details, please refer to the review article by Nagaosa et al. Nagaosa et al. (2010) Introduction of the Berry phase mechanism for the understanding of intrinsic AHE was a big step forward in the field, and it is also the fundamental base for understanding the quantum anomalous Hall effect (QAHE), which is the main subject of this article.
I.B. Berry Phase Mechanism for Intrinsic AHE
In quantum mechanics, the Berry phase is the quantal phase acquired by the adiabatic evolution of wave function associated with the adiabatic change of the Hamiltonian in a parameter space , with being the set of parameters (a vector). Let be the th eigenstate of the Hamiltonian . The overlap of two wave functions infinitesimally separated by in the space can be evaluated as
(2) 
where is called the Berry connection. Here is an important quantity, because it can be viewed as a vector potential and its curl , called the Berry curvature, gives an effective magnetic field in the parameter space (as will be addressed below). The Berry phase can then be defined as the integral of the Berry connection along a closed loop in the parameter space, or according to Stoke’s theorem, equivalently as the integral of Berry curvature on the surface enclosed by the adiabatic loop (i.e., the surface with the loop as boundary):
(3) 
where the second equality suggests that the Berry phase can also be regarded as the effective magnetic flux passing through a surface .
Although the concept of the Berry phase has broad applications in physics, its relevance to the band structure in solids has been recognized only in limited situations, such as the quantum Hall effect under a strong magnetic field Thouless et al. (1982) and the calculation of electronic polarization in ferroelectrics KingSmith and Vanderbilt (1993); Resta (1994). Here we will show that the Berry phase concept is also important for understanding intrinsic AHE. In this case, we treat the crystal momentum () space as the parameter space. We consider a crystalline solid with discrete translational symmetry. The eigen equation of the system is given as , where is the Hamiltonian, and are the eigen energy and the eigen wave function, and is the band index. Because of the translational symmetry, the eigen wave function of the system is dependent, where is the momentum defined in the first Brillouin zone (BZ) of momentum space. According to Bloch’s theorem, the eigen wave function should take the form of Bloch state , with being the periodic part of the wave function. Then the eigen equation of the system can be recast as , where is the dependent Hamiltonian. Following the above discussion, we can now define the Berry connection and Berry curvature in the parameter (momentum k) space as
(4)  
(5) 
These two quantities are crucial in understanding the Berry phase mechanism for intrinsic AHE. Before we can go further, we have to clarify several important properties of the Berry connection and Berry curvature.

Berry connection is gauge dependent: As we have learned from the textbooks of solid state physics, there exists an arbitrary phase factor for the eigen wave function that is not uniquely determined by the eigen equation of the system. Under a U(1) gauge transformation, the eigen wave function is transformed into as
(6) where is a real and smooth scalar function of . It is easy to see that is still the eigen wave function of the system for the same eigen state, i.e., is satisfied. However, the corresponding Berry connection will be changed by such a gauge transformation, becoming
(7) If we regard the scalar function as a kind of scalar potential in the space, this form of transformation is the same as that of a vector potential of magnetic field in real space, and therefore the Berry connection can be viewed as an effective vector potential in momentum space. The gauge dependence of the Berry connection suggests that it is not physically observable. However, it becomes physical after integrating around a closed path (i.e., the Berry phase defined in Eq. (3)). This is because the integration of the second term of Eq. (7) around a closed path will only contribute an integer multiple of , and the Berry phase is therefore invariant modulo .

Berry curvature is gauge invariant: This conclusion can be drawn directly from the factor that ; therefore is unchanged under the U(1) gauge transformation of Eq. (6). This vector form of Berry curvature suggests that it can be viewed as an effective magnetic field in momentum space. It is a gauge invariant local manifestation of the geometric properties of the wave function in the parameter () space, and has been proven to be an important physical ingredient for the understanding of a variety of electronic properties Wilczek and Shapere (1989); Xiao et al. (2010); KingSmith and Vanderbilt (1993); Resta (1994); Fang et al. (2003).

Symmetry consideration: Here we consider two important symmetries in solid state physics, namely time reversal symmetry (TRS) and inversion symmetry (IS). Following the above discussion, it is easy to learn that the Berry curvature has the following symmetry properties:
(8) (9) which suggests that for a system with both TRS and IS. They are strong symmetry constraints, which means that in order to study the possible physical effects related to the Berry curvature, a system with either broken TRS or broken IS is generally required. For example, the intrinsic AHE is observed in a ferromagnetic (FM) system with SOC, where the TRS is broken. For a system with TRS only, in general , however its integration over the whole Brillouin zone should be zero as implied by Eq. (9). In such a case, the Berry curvature may take effect only through selective or band sensitive probes.

The choice of periodic gauge: For the convenience of studies on solid states with discrete translational symmetry, we usually use the periodic gauge in practice. That is, we require the Bloch wave function to be periodic in momentum space, and use the condition (where is a translational vector for the reciprocal lattice) to partly fix the phases of wave functions. This periodic gauge is used in most of the existing firstprinciples electronic structure calculation codes. However, we have to note that it is a very weak gaugefixing condition, and abundant gauge degrees of freedom still remain.

Gauge field: Spreading out each component of the vector form discussed above (Eq. (5)), the Berry curvature can be written explicitly as an antisymmetric secondrank tensor ,
(10) where is the LeviCivita antisymmetric tensor, and can be simply treated as the direction indices (i.e., ) for our purposes. For example, for many of our following discussions on the transport properties in a two dimensional (2D) system, we will use the equality,
(11) Writing it in such a way also helps us understand that Berry curvature is simply the U(1) gauge field () in the language of gauge theory. In the presence of gauge freedom, in order to preserve the gauge invariance of Lagrangian under the gauge transformation, we need to use the gauge covariant derivative instead of the original derivative. For example, for the position operator , its gauge covariant form is written as , where is the vector potential (i.e., the Berry connection). In such a case, the gauge covariant position operators and are no longer commutable, and their commutation relation is given as , which gives the gauge field .
Keeping in mind all these properties discussed above for the Berry connection and Berry curvature, we are now ready to understand the intrinsic AHE from the viewpoint of gauge field and geometric Berry phase. For simplicity, let us first consider a 2D system with Hamiltonian defined in the plane. Under an external electric field , the whole Hamiltonian of the system becomes . By evaluating the velocity operator as , we find that the presence of electric field in the system leads to an additional velocity term, , which is along the direction (being transverse to the external electric field ). This result comes from the nonvanishing commutation relation between the gauge covariant position operators. This additional velocity term is nothing but the anomalous velocity term proposed by Karplus and Luttinger Karplus and Luttinger (1954), and is the origin of the intrinsic anomalous Hall effect.
Now let us generalize the above discussion to three dimensional (3D) space and considering both the electric and magnetic external fields, we find that the equation of motion of electrons, in the presence of gauge field, should be recast as Xiao et al. (2010)
(12)  
(13) 
where and are the external electric and magnetic fields, respectively. It is apparent that the group velocity Eq. (12) is different from the textbook one by the inclusion of an additional contribution (i.e., the second term on the righthand side). This term, , is now known as the anomalous velocity term, and it is proportional to the space Berry curvature . This equation of motion can also be obtained from the semiclassical theory of Bloch electron dynamics. To do this, we have to consider a narrow wavepacket made out of the superposition of the Bloch state of a band. Here we will not discuss the detailed derivation of this approach; readers please refer to the review article by Xiao et al. for more details Xiao et al. (2010).
For a ferromagnetic system in the presence of SOC, the time reversal symmetry is broken; therefore the nonvanishing Berry curvature contributes to the anomalous velocity term. Consider a system under external electric field and without magnetic field (i.e., ). The current density can be then obtained from the equation of motion shown above as
(14)  
where is the Fermi distribution function. The dc Hall conductivity is then given as
(15)  
(16) 
which is simply the Brillouin zone (BZ) integral of the Berry curvature weighted by the occupation factor of each state. For the convenience of our later discussions, in the last part of the equation, we have explicitly written the 3D integral as a 2D integral in the plane, supplemented with a line integral along .
On the other hand, we know from textbooks that the DC Hall conductivity can be also derived from the Kubo formula as
(17) 
which is basically equivalent to Eq. (15), by expanding each term of the vector form of the Berry curvature explicitly with some algebra. However, we have to emphasize that the form of Eq. (15) for Hall conductivity is not only compact, but also fundamentally important for the underlying physics. It demonstrates the Berry phase mechanism of the intrinsic AHE, and it is the key equation for our following discussions on the geometric meaning and topological nature of QAHE. For example, for a two dimensional system, the BZ integral of Berry curvature for a fully occupied band must give rise to an integer multiple of . Therefore, the Hall conductivity for a 2D insulator can be finite and be an integer multiple of . This is the underlying physics for the quantization of the AHE.
In addition to the understanding of the Berry phase mechanism for the intrinsic AHE discussed above, another important step forward in this field is the quantitative and accurate evaluation of the anomalous Hall conductivity (AHC) for realistic materials, yielding results that can be compared with experimental observations. Due to the rapid increase of computational power and the development of firstprinciples calculation methods, such quantitative comparisons (though still difficult) become possible and contribute greatly to the development of the field. In a series of papers reporting quantitative firstprinciples calculations for SrRuO Fang et al. (2003), Fe Yao et al. (2004) and CuCrSeBr Yao et al. (2007), it was demonstrated that the calculated AHC can be reasonably compared with experiments, suggesting the importance of the intrinsic AHE. In more recent years, accurate calculations of AHC have been achieved by using the Wannier function interpolation scheme Wang et al. (2006, 2007). The presence of SOC and the breaking of TRS are crucially important for the intrinsic AHE in those systems.
In summary of this part, we have discussed the Berry phase mechanism for intrinsic AHE. Conceptually, this is an important increment in our understanding because the intrinsic AHE is now directly linked to the geometric and topological properties of the Bloch states in momentum space. This understanding has also led to a great number of research projects on the exotic properties of electronic systems with SOC, such as QAHE and the topological insulators.
Ii Quantum Hall Family and Related Topological Electronic States
One of the main subjects of condensed matter physics is the study and classification of different phases and various phase transitions of materials that possess rich varieties of physical properties. Landau developed a general symmetrybreaking theory to understand the phases and phase transitions of materials. He pointed out that different phases really correspond to different symmetries of compounds. An ordered phase, in general, can be described by a local order parameter, and the symmetry of the system changes as a material changes from one phase to another. Landau’s symmetrybreaking theory is very successful in explaining a great many kinds of phases (or states) in materials, but it does not work for the topological quantum states. A topological quantum state cannot be described by a local order parameter, and it can change from one state to another without any symmetrybreaking. Topology, a word mostly used in mathematics, is now used to describe and classify the electronic structures of materials. “Topological electronic state” means an electronic state which carries certain topological properties (in momentum space usually), such as the states in topological insulators Haldane (1988); Kane and Mele (2005a, b); Bernevig and Zhang (2006); Bernevig et al. (2006); König et al. (2007); Zhang et al. (2009a); Roushan et al. (2009); Zhang et al. (2009b); Xia et al. (2009); Chen et al. (2009); Alpichshev et al. (2010); Hasan and Kane (2010); Qi and Zhang (2011); Knez et al. (2011); Beidenkopf et al. (2011); Ando (2013); Schnyder et al. (2008), topological semimetals Wan et al. (2011); Xu et al. (2011); Burkov and Balents (2011); Zyuzin et al. (2012); Wang et al. (2012); Young et al. (2012); Turner et al. (2012); Wang et al. (2013a); Liu et al. (2014a, b); Yi et al. (2014); Neupane et al. (2014); Yang and Nagaosa (2014); Jeon et al. (2014); Xu et al. (2015); Burkov et al. (2011); Lu et al. (2013); Weng et al. (2015a), topological superconductors Fu and Kane (2008); Hasan and Kane (2010); Qi and Zhang (2011); Qi et al. (2008, 2010); Fu and Berg (2010); Hor et al. (2010); Zhang et al. (2011a); Weng et al. (2011), etc. One of the most important characteristics of topology is the robustness against local deformations, or in the language of physics, the insensitivity to environmental perturbations, which makes topological electronic states promising for future applications. To characterize the order of topological states, new parameters, namely the topological invariants, are needed. In this section, we will review some of the topological electronic states and their quantum physics, including the integer quantum Hall (IQH) state, the quantum anomalous Hall (QAH) state, the quantum spin Hall (QSH) state, and the topological semimetal (TSM) state. We will use the concept of Berry phase in momentum space to discuss the topological nature of topological states and the related topological invariants.
II.A. Chern Number, Chern Insulator, and Quantum AHE
In 1980, Klaus von Klitzing discovered Klitzing et al. (1980) that the Hall conductance, viewed as a function of strength of the magnetic field applied normal to the twodimensional electron gas plane, at very low temperature, was quantized and exhibited a staircase sequence of wide plateaus. The values of Hall conductance were integer multiples of a fundamental constant of nature: , with totally unanticipated precision, and independent of the geometry and microscopic details of the experiment. This is an important and fundamental effect in condensed matter physics, and it is called the integer quantum Hall effect (IQHE) — the quantum version of the Hall effect.
The IQHE is now well understood in terms of single particle orbitals of an electron in a magnetic filed (i.e., the Landau levels), and the phenomenon of “exact quantization” has been shown to be a manifestation of gauge invariance. The robustness and the remarkable precision of Hall quantization can also be understood from the topological nature of the electronic states of 2D electron gas under magnetic field. The integer number, originally known as the TKNN number Thouless et al. (1982) in the Hall conductance derived from the Kubo formula, is now characterized as a topological invariant called “Chern number”. This topological understanding of the IQHE is a remarkable leap of progress, opening up the field of topological electronic states in condensed matter physics. IQHE is therefore regarded as the first example of topologically nontrivial electronic states to be identified and understood.
This conceptual breakthrough with regard to the topological nature of IQHE, though important, was not widely generalized, and the relationship of IQHE to the rich variety of condensed materials was not revealed for a long time. This is because the IQHE is observed only in a very particular system, i.e. 2D electron gas under strong external magnetic field, and the formation of Landau levels (usually at very low temperature) is required. Under such extreme conditions, the material’s details and its electronic band structure become irrelevant to the physics. In this sense, the lattice model proposed by Haldane in 1988 Haldane (1988) is very stimulating for the study of topological electronic states. He proposed that a spinless fermion model on a periodic 2D honeycomb lattice without net magnetic flux can in principle support a similar IQHE. Although Haldane’s model is very abstract and unrealistic (at least at that time), his result suggested that a kind of quantized Hall effect can exist even in the absence of magnetic field and the corresponding Landau levels. In other words, certain materials, other than the 2D electron gas under magnetic field, can have topologically nontrivial electronic band structures of their own, which can be characterized by a nonzero Chern number. Such materials were called Chern insulators later. In this way, the concept of topological electronic state was generalized and was connected to the electronic band structure of materials.
The progress in the study of IQHE was very inspiring in the 1980s; on the other hand, not much of it was related to the field of AHE. Around that time, the development of the AHE field was rather independent, with almost no intersection with the study of IQHE. But the understanding of Berry phase mechanism for intrinsic AHE Jungwirth et al. (2002); Onoda and Nagaosa (2002); Fang et al. (2003); Yao et al. (2004); Sinova et al. (2004), reformulated around the early years of the 21st century, changed the situation significantly. This understanding established the connection between the AHE and the topological electronic states. By this connection, it is now understood Onoda and Nagaosa (2002); Fang et al. (2003); Onoda and Nagaosa (2003) that intrinsic AHE can have a quantum version — the quantum anomalous Hall effect (QAHE), which is in principle similar to IQHE but without external magnetic field and the corresponding Landau levels. It turns out that this is exactly the same effect discussed by Haldane for a Chern insulator with a nonzero Chern number. Currently, the fields of IQHE and AHE are closely combined, and the realization of a Chern insulator becomes possible by finding the QAHE in suitable magnetic compounds with strong SOC.
The IQHE and the QAHE have their own characteristics; however, their underlying physics, in terms of the topological properties of their electronic structures, are basically the same. They are all related to the Berry connection, Berry curvature and Berry phase in momentum space, as discussed in the previous section. In this part, we will not discuss the particular details associated with each effect, but rather we will concentrate on their common features, namely the topological property and topological invariant (Chern number). To begin with, we will discuss several important concepts:

Hall Conductivity as Integral of Berry Curvature: Considering a 2D insulating system with broken TRS, the Hall conductivity of the system at low enough temperature can be written as
(18) where the second equality follows because of the existence of an energy gap and (=0) for the occupied (unoccupied) state (see Eqs. (14)(16) for a 3D version). This form of Hall conductivity can be derived either from the Berry phase formalism or from the Kubo formula, as discussed in the previous section. It is also important to note that this formula can be equally applied to the anomalous Hall conductivity of a 2D FM insulator without external field and to the Hall conductivity of a 2D electron gas under magnetic field. For the former it is straightforward; for the latter case, however, special considerations have to be taken. First, we assume that the external magnetic field is strong enough for the system to form the Landau levels. Second, the system must be insulating with the chemical potential located within the gap (i.e., between two neighboring Landau levels). Third, since the presence of external magnetic field will break the translational symmetry, we have to use the concept of magnetic translational symmetry to recover the periodicity of the system. As a result, magnetic Bloch states must be used for the evaluation of Berry curvature, and the magnetic Brillouin zone is used for the integration, correspondingly.

2D Brillouin Zone as a Torus: By adopting the periodic gauge, the wave function is periodic in momentum space. Under such a periodic boundary condition, the 2D BZ becomes a torus as schematically shown in Fig. 2. Therefore, the integral of the Berry curvature over the 2D BZ can be recast as a vector integral over the surface of the torus as
(19) Here we have used an effective Berry curvature defined over the torus surface, which satisfies the condition , where is the unit vector along the normal of the surface .

Quantization and Chern Number: The torus is a closed manifold without boundary, so according to Chern’s theorem Chern (1945) in differential geometry, the vector integral of Berry curvature over a torus surface must be an integer multiple of ,
(20) where is an integer number called Chern number. Having this property, it is now easy to see that the Hall conductivity for a 2D insulator must be quantized as
(21) 
Absence of Smooth Gauge: Based on Stokes’ theorem, the surface integral of Berry curvature (Eq. (20)) can also be evaluated as an loop integral of the Berry connection along the boundary of the BZ, i.e., . Since the BZ is a torus, which has no boundary, the integral must be vanishing if is smoothly defined in the whole BZ. Therefore, a nonzero Chern number indicates that we cannot choose a smooth gauge transformation such that is continuous and single valued over the whole BZ. Thus, nonzero Chern number can be viewed as obstruction to continously define the phase of the occupied wave functions on a 2D BZ, which is a torus. Thouless (1984); Thonhauser and Vanderbilt (2006); Brouder et al. (2007).
For a system with multiple bands, the Berry curvature should be understood as the summation of contributions coming from all occupied bands. Having the properties discussed above, we can now define a Chern insulator as a 2D insulator whose electronic structure gives a nonzero Chern number. The above discussions about the Chern number and the topology of electronic states in momentum space may be still too abstract, in the following we will give a more explicit explanation, in which the none zero Chern number manifests itself as the winding number of the Wannier center evolution for the effective 1D systems with constant . Here we consider the simplest case: a 2D lattice with only one single occupied band (the band index can therefore be neglected). Although it is impossible to choose a single smooth gauge over the whole BZ for a Chern insulator according to our above discussions, it is possible to choose a special gauge that is smooth and periodic along one direction (say ), but not necessarily along the other (say ) Thouless (1984); Thonhauser and Vanderbilt (2006); Brouder et al. (2007); Soluyanov and Vanderbilt (2011). Thus, we can do the integration in Eq. (19) explicitly for the direction with fixed and obtain (we set the lattice parameter =1)
(22)  
Here, is an angle (i.e., Berry phase) calculated from the 1D integration of along the axis (a closed loop) for each fixed . We can then plot as a function of . As shown in Fig. 3, over a cylinder surface (in the cylinder coordinates), is along the longitudinal direction, and the azimuth is the angle for each fixed . Moving from = to =, we can see the difference between the trivial insulator (=0) and the Chern insulator (). The winding number of over the cylinder surface is zero for the former (Fig. 3(a)), and nonzero for the latter (Fig. 3(b), (c)). In this way, we relate the Chern number to some kind of winding number defined by the eigen functions, and see that the Chern insulator has a “twisted” energy band.
It is also interesting to note that the dependent Berry phase calculated as
(23) 
can be related to the Wannier center of a 1D system (where we have recovered the band index dependence). To see this, we consider a generic 2D system and denote the Wannier function in cell associated with band , given in terms of the Bloch states, as
(24) 
with
(25) 
where is considered, and the periodic part of the Bloch function is defined as
(26) 
Then the matrix elements of the position operator between Wannier functions take the form
(27) 
from which we get the center of the Wannier function in the 0th cell as
(28) 
Now let us consider the and directions separately, and define the 1D Wannier function for each fixed as
(29) 
Then the 1D Wannier center (along direction) as function of can be defined as the average value of position operator as
(30) 
from which we see that is nothing but the 1D Wannier center of band . The Chern number can therefore be understood as the winding number of 1D Wannier center when it evolves as a function of (see Fig. 3). We can also see that when the Chern number is not zero, we cannot choose a smooth gauge transformation such that is continuous and single valued over the whole BZ. Although the above discussion is only for the simplest systems with single occupied band, generalization to the multiband situation is quite straightforward. For a general band insulator with multiple occupied bands, the Berry connection contains band index , and becomes nonabelian Then the Berry curvature that defines the Chern number is obtained from the trace of the Berry connection .
The Hall conductivity of a Chern insulator with a nonzero Chern number must be quantized as an integer multiple of . Different Chern numbers give different states whose Hall conductivities are different, but their local symmetry can be the same. Therefore, to distinguish the states, we cannot use local order parameters (in the language of Landau’s symmetry breaking theory); instead we need a topological invariant, the Chern number, as a global order parameter of the system. By rewriting Hall conductivity in terms of the Berry curvature and Berry phase, we can now unify QHE and QAHE. Readers will recall that the QAHE is nothing but the quantum version of the AHE realized in a Chern insulator without the presence of external magnetic field.
Similar to QHE, where the TKNN number is related to the number of edge states in a real 2D sample with boundary Thouless et al. (1982), the Chern number can also be physically related to the number of edge states for a 2D Chern insulator Haldane (1988). The existence of edge states is a direct result of the topological property of the bulk electronic structure, and is due to the phenomenon discussed in the literature as the bulkboundarycorrespondence Hatsugai (1993); Essin and Gurarie (2011). In this case, due to the broken TRS, the edge state must be chiral (i.e, the electrons of the edge state can move only in one direction surrounding the sample boundary, either left or righthanded, as shown in Fig. 4(b)). As discussed with regard to IQHE, the charge transport of the edge state is in principle dissipationless, and backscattering is absent due to the lack of an edge state with opposite velocity MacDonald and Středa (1984); Halperin (1982).
Although the topological properties of Chern insulators and the related QAHE are fundamentally the same as that discussed with regard to IQHE, they are conceptually broadly generalized to a wide field and to a rich variety of materials. This generalization is an important step forward, providing the building blocks for subsequent discussions of many possible topological electronic states. From the application point of view, the Chern insulator (or the related QAHE) is also important because the quantized Hall conductivity can be realized in the absence of magnetic field, greatly simplifying measurement conditions.
II.B. Invariant, Topological Insulator, and Quantum Spin Hall Effect (QSHE)
Although the Chern insulator (or the related QAHE) is the simplest topological electronic state, its realization occurred rather later and was much stimulated by the rapid development in the field of topological insulators—another interesting topological electronic state protected by TRS. Considering symmetry, it is easy to prove that the Chern number in Eq. (20) must be vanishing for an insulator with TRS. However, this does not mean that the electronic state carries no topological property in this case. Kane and Mele Kane and Mele (2005a) introduced a new topological invariant, the number, to classify an insulating system with TRS. They proposed that a time reversal invariant insulator can be further classified as a trivial insulator with or a nontrivial topological insulator with , which is a typical example of a symmetry protected topological state Gu and Wen (2009). A topological insulator in 2D is also called a quantum spin Hall insulator (QSHI) because it can support the quantum spin Hall effect (QSHE) Kane and Mele (2005b); Bernevig et al. (2006); Bernevig and Zhang (2006); König et al. (2007), which shares certain features with the IQHE and QAHE and can be understood from the viewpoint of “bandtwisting” or winding number of Berry phase in momentum space. A QSHI is different from a trivial insulator in the sense that it has gapped insulating states in the bulk but gapless states on the edge due to its topological property. It is also distinguished from the Chern insulator in the sense that QSHI has even number of edge states, composed of pairs of counterpropagating chiral edge states, as shown in Fig. 4(c)). The QSHE has been explicitly discussed for graphene lattice Kane and Mele (2005b) and HgTe quantum well structures Bernevig et al. (2006); Bernevig and Zhang (2006); König et al. (2007). In this part, however, we will take a simple task and discuss the invariant and the topological state with TRS from the viewpoint used above for the Chern insulator.
For a 2D insulator with TRS, the total number of occupied electronic states must be even. Suppose we can divide them into two subspaces, I and II , which are related by TRS and smoothly defined on the whole BZ. Evaluating the Chern number for each subspace independently, if one has Chern number Z, the other one must have Chern number Z (due to the TRS) and the total Chern number of the whole system is always zero. It seems that we can use Z as the topological index to classify the band insulators with TRS. While unfortunately, the smooth partition of the occupied states into two subspaces with one being the time reversal of another is only possible with extra good quantum numbers, such as , where the Chern number obtained within the spin up subspace can be used to describe the topology of the system and is called “spin Chern number”. But generically such a smooth partition of the occupied states can only be made for half of the BZ (not the whole BZ) and we need at least two patches (A and B) to fully cover the whole BZ. The winding number of the U(2N) gauge transformation matrix at half of the boundary between two patches can be used to define a new classification of the band insulators with TRS. As further proved by Fu and Kane, only the even and odd feature of the above winding number is unchanged under the U(2N) gauge transformation and all the band insulators with TRS can be classified into topological trivial and nontrivial classes, which is called invariance accordingly. Fu and Kane further derived the expression for invariance in terms of both Berry connection and Berry curvature as Fu and Kane (2006)
(31) 
where integral of Berry curvature is performed over the half BZ (i.e., /2), and the integral of Berry connection is performed along the boundary of the half BZ (i.e., ). To evaluate invariance using the above equation, one needs to find a smooth gauge for the wave functions on half of the BZ, which is very difficult for the band structure calculations for realistic materials, and as a result, this formula is rarely used in practice. On the other hand, the invariance is a gauge invariant quantity, one should be able to compute it without any gauge fixing condition. For that purpose, some of the authors of the present paper developed an alternative expression for the invariance called Wilson loop method, which will be introduced in detail in section 3.2. Here we just sketch its main idea briefly. Similar with the Wilson loop method for Chern number calculation, the purpose of the Wilson loop method is to calculate the “Wannier center” of each band for the effective 1D insulators with fixed and determine the invariance by looking at their evolution with . With the presence of TRS, a generic system contains 2N occupied bands. As mentioned previously, now the Berry connection becomes 2N*2N matrix. The loop integral of such U(2N) Berry connection gives a 2N*2N unitary matrix . The U(1) part of this matrix contributes to the Chern number as discussed before and the invariance can be obtained from the remaining SU(2) part by taking the phases of its eigenvalues , where denotes the band index. As proved in Ref. Yu et al., 2011, the TRS only guarantees the double degeneracy of at two time reversal invariant loops and . The invariance of the system is determined by whether the s switch parters or not as they evolves from to .
To be explicit, let us again consider the simplest example of 2D insulator, with only two occupied bands, and , which are related by the TRS (say, for example, =, where is the time reversal operator). The two states form a Kramers pair and must be degenerate at the timereversalinvariant momentum (TRIM) of the BZ (i.e, with or ). We can evaluate for all occupied bands (as we have done above for the Chern insulator). As mentioned previously, the two Wannier centers must be degenerate at and , which leads to three situations in general (as shown in Fig. 5(a)(c)). First, if the evolution path of the doesn’t enclose the half BZ, this is the trivial situation with invariance =0 (Fig. 5(a)). Second, if the evolution path of the enclose the half BZ once, this is the nontrivial topological insulator with index =1, where the crossing of two curves at and is protected by the TRS (Fig. 5(b)). Third, if the evolution path of the enclose the half BZ twice, the two curves must cross at some other than the TRIM. Such crossings are not protected by the TRS and are removed by small perturbations, which drive the crossings into anticrossings. As the result, the system becomes equivalent to the trivial case with index =0 (Fig. 5(c)).
The edge states of 2D topological insulators must appear in pairs due to the existence of TRS, and each pair of edge states is composed of two counterpropagating chiral edge states, which are related to each other by the TRS. In this case, if one state at the edge has velocity , we can always find another state with opposite velocity . However, backscattering between the two states is again forbidden because the two states must have opposite spins. Existence of such pair of edge states leads to the observable QSHE Kane and Mele (2005b, a); Bernevig et al. (2006), where the spin Hall conductivity is quantized in units of (if is conserved). According to our above discussions of the topology, if there exist multiple pairs of edge states, two pairs can in principle couple together and open up a gap (because scattering between the pairs is not forbidden). As a result, the number is physically related to the number of pairs of edge states mod 2, in contrast to the case of Chern insulators.
Using the 2D topological invariants (such as or ) as building blocks, a 3D crystal can in principle be characterized by a triplet of 2D topological numbers for the three crystal orientations, respectively. In such a way, we can extend the topological classes from 2D to 3D. There are in general two situations by doing this: (1) one is the trivial extension leading the “weak” 3D topological class, which can be viewed as a simple stacking of 2D topological insulating layers along a certain crystal orientation; (2) another is a nontrivial extension leading to a “strong” topological class, which is a new and stable topological state in 3D. For example, by a simple stacking of 2D Chern insulating layers along the direction, if the band dispersion along is weak enough, each 2D layer with fixed will keep its original topological property (characterized by a nonzero Chern number). As a result, a 3D “weak” Chern insulator can be obtained. In a similar way, a 3D “weak” topological insulator can be obtained too. However, it is important to note that, if we concentrate on insulators only, there will be no “strong” Chern insulator in 3D; on the other hand, a new 3D state with TRS, called “strong” topological insulator, can be obtained Fu et al. (2007); Moore and Balents (2007); Roy (2009) by extending the number from 2D to 3D. (As will be discussed in the next section, the missing “strong” Chern class in 3D has to be a metal—the topological semimetal).
For a 3D insulator with TRS, there will be 8 TRIM points in the BZ (different from 2D BZ which has only 4 TRIM points). For any 4 TRIM points that lie in the same plane, we can define a 2D number as discussed above. In total, there will be 16 invariant configurations and can be distinguished by four independent indices, (), as discussed in Ref. Fu et al., 2007. Here is the total number, and the are the 2D numbers along the directions, respectively. The three weak indices can be evaluated for the , , and planes, as a convention, respectively. The total index , however, is a strong index, and it has to be evaluated globally by considering the 3D structure. That is, we have to consider the change of 2D number between two parallel planes (perpendicular to certain reciprocal lattice vector). For example, we can choose the =0 and planes, or =0 and planes, or =0 and planes. If the numbers of two planes are different, we have , otherwise, if they are the same, we have . A realistic example will be given in Section 3. For a 3D strong topological insulator with strong topological index , on its boundary (i.e, surface), we will expect the Dirac cone type surface states, which are protected by the TRS Hasan and Kane (2010); Qi and Zhang (2011).
II.C. Magnetic Monopole, Weyl node and Topological Semimetal
As mentioned above, if we attempt to consider the possible “strong” topological state characterized by Chern numbers in 3D, we will find that it has to be a metal, and this leads to a very interesting new topological state of quantum matters — topological metals or semimetals. We will see that this is a much more general state, and the 2D Chern insulating state can be regarded as a special cut of the 3D topological semimetal state along a certain plane in momentum space. From another point of view, we can also raise a question like the following. Since we know from the above discussions that insulators can be further classified as topologically trivial and nontrivial insulators, can we do the same classification for metals? If possible, what will be the topological invariant for the proper description of the topological metallic state? This is a very interesting question, and we will show in this part that it is related to the magnetic monopoles Fang et al. (2003); Volovik (2009) and Weyl nodes in momentum space.
Berry phase is called quantum geometric phase since it has a very intuitive geometric picture. It is proportional to the solid angle subtended at the magnetic monopole by the adiabatic loop of Hamiltonian in momentum space. The magnetic monopole is the source or drain of the gauge field . Since
(32) 
it diverges at the point where = (see Eqs. (16) and (17)). This means that the magnetic monopole is formed by energy level crossing, and a twoenergylevel system is the simplest case. Consider the generic form of the Hamiltonian for any twolevel system,
(33) 
where is the identity matrix, is a 3D vector depending on momentum , and are Pauli matrices. There are two eigen states with eigen energies . The term is just a shift of zero energy level, and can be neglected. At the energy degeneracy or level crossing point (), = (i.e, =0) is required. Obviously, is not necessarily on the path of a adiabatic loop change of the Hamiltonian, which means that the band gap between and can be well preserved if the Fermi level is away from the bandcrossing point. Around the neighborhood of the point, can be expanded as . Taking the zero point of the parameter space as , we have =. Now for the simplest case, where , we can have , and the corresponding Berry curvature can be obtained as
(34) 
Obviously, such a magnetic field distribution in momentum space is similar to the electric field distribution of a point charge in real space, and can be understood as magnetic field around a “magnetic charge”–magnetic monopole Fang et al. (2003). In other words, the divergence of magnetic field is no longer zero (i.e., ) but is related to the magnetic charge at the source or drain. The degeneracy or level crossing point is the place where the magnetic monopole with strength is located.
Joshua Zak Zak (1989) pointed out that such Berry phase and Berry curvature can also exist in periodic systems where the eigenstates are Bloch wave functions and the parameter space is the crystal momenta which can vary in closed loops, such as Brillouin zone or Fermi surface. Then, the adiabatic evolution loop forms a compact manifold that has no boundary. For example, the Brillouin zone of a 2D system is a torus, and the Fermi surface of a 3D metal (in the simplest case) is an enclosed sphere. Obviously, Gauss’s law ensures that the total flux penetrating the closed surface must be quantized and is equal to the magnetic charge of monopoles enclosed by the surface (either torus or Fermi sphere).
Now we can see that the magnetic monopole in momentum space (or parameter space in general) plays a crucial role in determining the topology of electronic band structures. Besides the insulating case, in the following we concentrate on a 3D metal with a welldefined Fermi surface in momentum space. Taking the above twolevel system as an example and assuming that , and , the magnetic monopole is located at the origin and the lowenergy Hamiltonian can be written as
(35)  
This Hamiltonian was first proposed by Weyl Weyl (1929), who found that, for massless fermions, the Dirac representation is reducible, and is composed of two (irreducible) Weyl fermions with positive (+) and negative () chirality (and opposite magnetic charge). Similar to the definition of Chern number for a torus (in a 2D insulator), the total flux of the gauge field passing through a Fermi surface of 3D metal must be quantized as a multiple of . We can then define the Fermi surface Chern number as
(36) 
and use it as a new topological invariant to describe topological metallic states. The Fermi surface Chern number is nonzero if a Weyl node (or a magnetic monopole) is enclosed by a Fermi surface, and this leads to a nontrivial topological metallic state (called Weyl metal). If the Fermi level happens to be exactly at the Weyl node, we will get a topological semimetal state (i.e, Weyl semimetal). Unfortunately, according to the “nogo theorem” Nielsen and Ninomiya (1981a, b), for any lattice model, the Weyl nodes with opposite chirality have to appear in pairs (although they might be separated in momentum space), and the summation of for all pieces of Fermi surfaces must be vanishing Haldane (2014). This makes it difficult to give a proper definition for Weyl metal. Nevertheless, we should note that Weyl nodes (and magnetic monopoles) are stable topological objects, which can be well defined in 3D momentum space Balents (2011); Murakami (2007). Two Weyl nodes with opposite signs may be separated in momentum space and lead to two pieces of Fermi surfaces, each of which has a nonzero Fermi surface Chern number . The two Weyl nodes may annihilate each other if and only if they overlap in space Wan et al. (2011); Burkov and Balents (2011); Balents (2011); Zyuzin et al. (2012); Wang et al. (2012).
Special attentions must be paid to the systems with both TRS and IS, where Kramers degeneracy exists for every momentum . In such cases, each pair of Weyl nodes, if any, must overlap exactly in the space. In other words, the minimum effective Hamiltonian to describe such system must be at least 44 and contain two Weyl nodes (with opposite signs) simultaneously, as shown below,
(37) 
This is called a 3D Dirac node, and is a straightforward extension of the 2D graphene to 3D space Zhang et al. (2010a). As we have discussed above, in such a case, a perturbative mass term can be introduced in principle, opening up a gap and leading to an insulating state. However, if we consider additional symmetries in the system, such as the crystalline symmetry, the mass term may again be forbidden, and this would stabilize the 3D Dirac cone and lead to a 3D Dirac metal or semimetal state Young et al. (2012); Wang et al. (2012). Therefore, the 3D Dirac node is not as stable as the Weyl node, however, it is a good starting point for us to reach the true Weyl semimetal state by breaking either TRS or IS in the 3D Dirac semimetals Zyuzin et al. (2012).
In general, the energy level crossing may happen in many materials and at any energy and momentum; however, it is quite rare to have bandcrossing exactly at Fermi level, particularly for the cases with broken TRS or IS. Up to now, only a few materials have been theoretically proposed to host such Weyl semimetal Wan et al. (2011); Xu et al. (2011); Burkov and Balents (2011); Liu and Vanderbilt (2014) or Dirac semimetal states Young et al. (2012); Wang et al. (2012, 2013a). Among them, only NaBi Wang et al. (2012); Liu et al. (2014a) and CdAs Wang et al. (2013a); Liu et al. (2014b) have been confirmed experimentally to be Dirac semimetals. The proposed Weyl semimetals breaking TRS, such as pyrochlore Iridate Wan et al. (2011) and HgCrSe Xu et al. (2011), have not been confirmed yet due to experimental difficulties. The proposals for Weyl semimetals keeping TRS but breaking IS are thought to be a way to overcome those difficulties. Presently the following are typical proposals: One is a superlattice system formed by alternately stacking normal and topological insulators Halász and Balents (2012). The second involves Tellurium, Selenium crystals or BiTeI under pressure Hirayama et al. (2014); Liu and Vanderbilt (2014). The third one is the solid solutions LaBiSbTe and LuBiSbTe Liu and Vanderbilt (2014) tuned around the topological transition points Murakami (2007). The fourth is TaAsfamily compounds, including TaAs, TaP, NbAs and NbP, which are natural Weyl semimetals and each of them possesses a total of 12 pairs of Weyl points Weng et al. (2015b); Huang et al. (2015).
The Weyl semimetal is a new state of quantum matters. It is of particular interest here that the Weyl semimetal with broken TRS is closely related to the Chern insulators and provides a unique way to realize QAHE. Let us first consider a Weyl semimetal with a single Weyl node in a continuous model defined as , which as mentioned previously generates a monopole in Berry curvature right at the origin. The Gauss’s theorem then requires that the total flux flowing through out any closed surface enclosing the Weyl point must be , i.e, equal to the chirality of the Weyl point (Eq. (36)). For a lattice system, there is an important “nogo theorem” indicating that Weyl points must appear in pairs with opposite chirality. Originally this theorem has been proved by field theory. Nielsen and Ninomiya (1981a, b) Here we provide a much intuitive way to understand it in terms of the concept of Chern number. Given a Weyl semimetal in 3D lattice (assuming all trivial states are far away from the Fermi level), first let’s consider a 2D plane with fixed , where band structures within the plane should be fully gapped unless the plane cut through a Weyl point exactly. The integral of Berry curvature over such a 2D insulating plane must be quantized and gives rise to a welldefined integer Chern number. Moving the plane (i.e, ) from  to , we will then get the Chern number as a function of , as shown in Fig. 6. Now, it is important to note that the Chern number as a function of should jump by +1 or 1 (depending on the chirality of the Weyl point), whenever the movingplane goes across a Weyl point. This is because there exists a topological phase transition at the Weyl point, and the band gap of 2D plane is closed and reopened when the plane goes across the Weyl point. This jump can also be understood by the following consideration. Selecting two planes at the opposite sides of a Weyl point, we can construct a closed manifold surrounding the Weyl point, i.e, the cube formed by the two parallel planes and the four sidesurfaces, shown in Fig. 6 as the shaded area. The flux flowing through the four sidesurfaces should exactly cancel each other, being net zero. Then, the total flux flowing through the closed manifold (i.e, the cube), which is now equivalent to the difference of Chern numbers for the two parallel planes, must be equal to the chirality of the Weyl point enclosed by the manifold (namely ) as discussed above. Having understood the jump, let’s look at the Chern number evolution as the function of from  to . In order to satisfy the periodic boundary condition in the BZ of lattice, the positive jump (+1) and the negative jump (1) must appear in pairs, if there is any. This leads to the important conclusion that Weyl points with opposite chirality must appear in pairs.
Finally, the total Hall conductivity of the system is given by the integral of as
(38) 
In such a way, we see the relationship between 2D Chern insulators and 3D Weyl semimetals. Given a 3D Weyl semimetal as defined above, we can form a 2D thin film (or quantum well structure) such that is quantized by the sample thickness, and only particular values of are allowed. If it happens that, for some particular thicknesses, the 2D Chern number are nonzero, we will expect quantized total Hall conductivity, and this leads to a 2D Chern insulator and the QAHE.
To explicitly show the Hall conductivity of a FM Weyl semimetal, we introduce a simple Hamiltonian based on a cubic lattice BHZ model Qi et al. (2006),
(39) 
It has two Weyl nodes at (0,0, ) with = when . The energy dispersion along axis is shown in Fig. 7 for =. The Hall conductivity is calculated by using Eqs. (18) and (32) for each plane with fixed . When chemical potential is 0 and passes exactly through the Weyl nodes, is quantized to be 1 for and 0 for , consistent with the above discussions based on the continuum model. Integrating over , the total Hall conductivity is given as Chen et al. (2013a); Burkov (2014a, b), which depends only on the separation of Weyl nodes in momentum space (i.e, ). Shifting the chemical potential away from 0 leads to partially occupied bands, and the corresponding Fermi surfaces change from two isolated spheres () to two touched ones () and finally they merge into a peanut shape (), as shown in Fig. 7. The total Hall conductivity decreases with rising chemical potential , because the Berry curvature from the upper band tends to cancel out the contribution from the lower band. Finally, when both bands are fully occupied the sum of Berry curvature over all bands must vanish Bernevig and Hughes (2013). It is important to note that a fully occupied band (without a corresponding Fermi surface) may still contribute to the Hall conductivity, and this should be carefully considered in practical calculations. In such a case, the Hall conductivity is usually evaluated by the volume integrals of the Berry curvature of occupied bands over the whole BZ, as is done for the calculation of intrinsic AHE Fang et al. (2003); Yao et al. (2004); Wang et al. (2006); this is also discussed by Chen et al. Chen et al. (2013a) and Vanderbilt et al. Vanderbilt et al. (2014); Chen et al. (2013b); Haldane (2004); Wang et al. (2007).
Iii FirstPrinciples Calculations for Topological Electronic States
Theoretical predictions, particularly firstprinciples electronic structure calculations based on density functional theory Martin (2004), have played important roles in the exploration of topological electronic states and materials. This is not an accidental success, but rather due to several deep reasons: (1) To describe complicated electronic band structures of real materials, particularly with the spinorbit coupling (SOC) included, firstprinciples calculations are necessary; (2) Firstprinciples calculations nowadays can reach the accuracy even up to 90% for most of the physical properties of “simple” materials (i.e., weakly correlated electronic materials), which makes prediction possible; (3) The topological electronic properties of materials are robust, nonperturbative, and not sensitive to small error bars, if any. Given those advantages of firstprinciples calculations and its great success in this field, however, we have to be aware that the numerical determination of topological invariants (such as integer or Z numbers) is still technically demanding, because those numbers are related to the phases of eigen wave functions, which are gauge dependent and randomized in most of the calculations. In addition to that, we also have to note that: (1) We still have a bandgap problem in either the local density approximation (LDA) or the generalized gradient approximation (GGA) for the exchangecorrelation potential; (2) The present firstprinciples calculations based on LDA or GGA can not treat strongly correlated systems properly; (3) The evaluation of Berry phase Wilczek and Shapere (1989) and topological numbers may require some additional complicities, such as fine points meshes, gaugefixing condition, etc. In this section, we will discuss some important issues and techniques related to the first principles studies of topological electronic states.
III.A. Calculations of Berry Connection and Berry Curvature
From the computational point of view, a selfconsistent firstprinciples electronic structure calculation for a real material is typically performed based on the LDA or GGA for the exchangecorrelation potential. Such a calculation will generate a set of single particle eigen states with wave functions , eigen values , and occupation ) for the ground state of the system. Having obtained those quantities, the remaining task is to evaluate the Berry connection and the Berry curvature . The Hall conductivity can be obtained either from the summation of Berry curvature over the BZ (Eq. (16)) or from the Kubo formula directly (Eq. (17)). In the latter case, the matrix element of velocity operator is given as
(40) 
The strategy is simple, but the computational task is hard, because although the Berry curvature does not depend on the gauge choice, the Berry connection does. It is also true, according to our above discussions, that choosing a smooth gauge for a topologically nontrivial state is generally difficult or even impossible. In most calculations, the Berry connection is a rapidly varying function of momentum , which makes the convergence of the calculation difficult. To overcome this problem, a straightforward way is to increase the number of points in momentum space. Another possible way is to use the Wannier representation, such as the maximally localized Wannier functions Marzari et al. (2012).
The firstprinciples calculations of Berry curvature and its integration over the BZ to get the intrinsic anomalous Hall conductivity (AHC) have been performed for SrRuO and bodycentered cubic Fe by Fang et al. Fang et al. (2003) and Yao et al. Yao et al. (2004), respectively. These were the first quantitative demonstrations of strong and rapid variation of Berry curvature over the BZ. The sharp peaks and valleys make the calculation of AHC challenging. Millions of sampling points are required to ensure convergence. Wang et al. Wang et al. (2006) proposed an efficient approach employing the Wannier interpolation technique. They first perform the normal selfconsistent calculation by using relatively coarse grid — good enough to ensure the total energy convergence. Then, Maximally Localized Wannier Functions (MLWF) are constructed for the states around the Fermi level. The number of Wannier functions should be carefully chosen so that the isolated group of bands around the Fermi level can be well reproduced, especially those bands passing through the Fermi level (because the small energy splitting due to spinorbit coupling or band anticrossing can produce sharp peaks or valleys of Berry curvature). Once MLWFs are obtained, the interpolation technique is applied to obtain the necessary quantities, such as eigen energies, eigen states, and velocity operator, on a much finer grid. Since the number of MLWFs is typically much smaller than the number of basis functions used in calculations, this approach saves a lot of computation cost, both in time and in storage.
In additional to the direct methods discussed above, some advanced techniques with wellcontrolled convergency with respect to the random phases can be developed. This is because Berry connection is not the physical quantity that we can measure, and our ultimate goal is to evaluate the Berry curvature which is gaugeinvariant. For example, it was proposed that for calculation of electrical polarization within the modern theory of polarization KingSmith and Vanderbilt (1993); Resta (1994) the change of polarization is directly proportional to the sum of Berry phase terms of occupied states. Since Berry phase is the integral of the Berry connection along a closed loop in the parameter space (see Eq. (3)), it is well defined and gaugeindependent. To calculate the change of polarization along the direction of , a series of loops (called Wilson loops Wilson (1974); Giles (1981)) in momentum space are constructed. Each loop has fixed =(), but is periodic along . We can then discretize as = (=0, 1, …, J1), where is the reciprocal lattice vector along . Taking the periodic gauge , the Berry phase along each loop can be calculated by
(41) 
with band indices and for occupied states. By doing this, the random phase problem associated with eigen wave functions can be largely eliminated, and the main computational task is to evaluate the inner product of two neighboring wave functions along the loop.
For a metallic system, the intrinsic AHC can also be calculated through this method, as has been implemented by Wang et al. Wang et al. (2007). It was justified by Haldane Haldane (2004) that the nonquantized part of intrinsic AHC in a FM metal can be recast as a Fermi surface integration of the Berry curvature, so we can construct the Wilson loops over the Fermi surface rather than the Brillouin zone. Considering a FM metal with magnetization along the direction, a 2D plane (in momentum space) perpendicular to will cut through the Fermi surface. The intersection between the plane and the Fermi surface defines loops, which are the Wilson loops used for calculating the AHC. Following our above discussions, we can discretized each loop and evaluate the Berry phase along the loop efficiently by using Eq. (41). To do this, several things should be borne in mind: (1) The periodic boundary condition has to be adapted as if the loop does not cross the Brillouin zone boundary; (2) The direction of Wilson loops should be defined consistently; (3) For the case with multiple branches, a continuity condition must be used to choose each branch carefully. Finally, in additional to the nonquantized part of AHC, the quantized contribution to the AHC (coming from the deep occupied states) can be determined by the Fermisea integration of the Berry curvature, which is gauge invariant and has no ambiguity.
This Wilson loop method is a convenient tool, and it can also be applied to investigate the topological invariant for systems with or without TRS as will be discussed below.
III.B. Wilson Loop Method for Evaluation of Topological Invariants
In real materials, the band structures are usually very complicated, and the determination of topological invariants becomes essential and computationally demanding. The band degeneracy, either accidentally or due to symmetry, makes the numerical determination of phases of wave functions a tough task. Here we will present the Wilson loop method for determining topological indices efficiently Yu et al. (2011); Taherinejad et al. (2014); Soluyanov and Vanderbilt (2011); Ringel and Kraus (2011). This method is computationally easy and is equally applicable for Chern insulators, topological insulators (2D and 3D), and the topological crystalline insulators Fu (2011); Hsieh et al. (2012), which have attracted lots of interest recently.
For band insulators with inversion symmetry, the indices can easily be computed as the product of parity eigen values for half of the occupied states (Kramers pairs have identical parities) at the timereversalinvariantmomentum (TRIM) points Fu and Kane (2007). In such a case, parity is a good quantum number for the TRIM points and can be computed from the eigen wave functions obtained from the firstprinciples calculations. This method has been used frequently and is very efficient. Unfortunately, for general cases where inversion symmetry is absent, the evaluation of the number becomes difficult. Three different methods have been considered for such a case: (i) Directly compute the numbers from the integration of the Berry curvature over half of the Brillouin zone Fukui and Hatsugai (2007). In order to do so, one has to set up a fine mesh in the space and calculate the corresponding quantity for each point. Since the calculation involves the Berry connection , one has to numerically fix the gauge on the half BZ, which is not easy for the realistic wave functions obtained by firstprinciples calculations. (ii) Start from an artificial system with inversion symmetry, and then adiabatically deform the Hamiltonian towards the realistic one. If the energy gap never closes at any point in the BZ during the deformation process, the realistic system must share the same topological nature with the initial reference system, whose number can easily be counted by the parity eigenvalue formula Fu and Kane (2007). Unfortunately, making sure that the energy gap remains open on the whole BZ is very difficult numerically, especially in 3D. (iii) Calculate the boundary (edge or surface) states. Due to the open boundary condition, the firstprinciples calculation for the boundary states is numerically heavy. The Wilson loop method, which we will present here, has the following advantages: firstly, it uses only the periodic bulk system; second, it does not require a gaugefixing condition – thereby greatly simplifying the calculation; third, it can easily be applied to a general system with or without inversion symmetry.
As discussed in the previous section for 2D insulators, to determine the topological number (either or ), the key is the angle and its 1D evolution. In the single band model, the angle is given simply as the 1D integration of along the axis. However, for real compounds with multiple bands, its computation requires the Wilson loop method Yu et al. (2011). Once the angle is computed, all the topological numbers can be determined easily, following our previous discussions. For the mathematic details of this method, please refer to our paper Yu et al. (2011). Here we will focus on the practical steps for real calculations.
Considering an insulator with occupied states, after the selfconsistent electronic structure calculations, we get the converged charge density, the total energy, the band structure, etc. Then, without loss of generality, we consider a periodic plane of (, ) with and , and consider the axis as a closed loop. For each fixed , we discretize the line into an mesh with meshindex ranging from 0 to . Due to the periodic condition, the = point is equivalent to the =0 point. The Wilson loop method consists of the following steps:
(1) We need to calculate the eigen wave functions of occupied states for all mesh points. This can be done by continuing from the converged charge density and with no need for charge selfconsistency furthermore.
(2) Using the obtained wave functions, we can calculate the inner products among them and evaluate the matrix , where and are band indices of occupied states ranging from 1 to . The matrix Yu et al. (2011) is nothing but the discretized version of Berry connection .
(3) Now we can construct the matrix as the product of ,