Welcome to visit Communications in Theoretical Physics,
Condensed Matter Theory

A gauge theory for two-band model of Chern insulators and induced topological defects

  • Zhi-Wen Chang 1 ,
  • Wei-Chang Hao 2 ,
  • Xin Liu , 1, *
Expand
  • 1Institute of Theoretical Physics, Faculty of Sciences, Beijing University of Technology, Beijing 100124, China
  • 2School of Physics, Beihang University, Beijing 100191, China

*Author to whom any correspondence should be addressed.

Received date: 2021-08-20

  Revised date: 2021-11-06

  Accepted date: 2021-11-10

  Online published: 2022-02-09

Copyright

© 2021 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this paper a gauge theory is proposed for the two-band model of Chern insulators. Based on the so-called 't Hooft monopole model, a U(1) Maxwell electromagnetic sub-field is constructed from an SU(2) gauge field, from which arise two types of topological defects, monopoles and merons. We focus on the topological number in the Hall conductance ${\sigma }_{{xy}}=\tfrac{{e}^{2}}{h}C$, where C is the Chern number. It is discovered that in the monopole case C is indeterminate, while in the meron case C takes different values, due to a varying on-site energy m. As a typical example, we apply this method to the square lattice and compute the winding numbers (topological charges) of the defects; the C-evaluations we obtain reproduce the results of the usual literature. Furthermore, based on the gauge theory we propose a new model to obtain the high Chern numbers ∣C∣ = 2, 4.

Cite this article

Zhi-Wen Chang , Wei-Chang Hao , Xin Liu . A gauge theory for two-band model of Chern insulators and induced topological defects[J]. Communications in Theoretical Physics, 2022 , 74(1) : 015701 . DOI: 10.1088/1572-9494/ac381e

1. Introduction

Topological insulators (TIs) are one of the most important scientific frontiers in condensed matter physics, which have a gapped bulk band but gapless edge states, protected by topology and symmetry. The edge states are insensitive to disorder and electron–electron interactions thanks to the forbidden backscattering states. The most promising applications of TIs are spintronic devices and dissipationless devices for quantum computers [13] based on the quantum spin Hall effect [4] and quantum anomalous Hall effect [5].
A TI has a quantized Hall conductance, ${\sigma }_{{xy}}=\tfrac{{e}^{2}}{h}C$, together with a vanishing longitudinal conductance that leads to dissipationless edge channels. Here C is an integer-valued topological number—the so-called Chern number. The evaluation of C determines through σxy the performance of the TI interconnect devices; a larger C leads to a larger σxy and thus a significantly improved performance of the device [6]. Physically, as pointed out in [7], when the meron defects are specially concerned, the topological charge C computes the TKNN integers of the total curvature for the (abelian) Berry connections arising from the spinors over the Brillouin zone (BZ). See details below.
To investigate the non-trivial topology of a two-dimensional quantum Hall system, Zhang et al proposed a gauge theory through a Chern–Simons–Ginzburg–Landau model [8] by employing a so-called statistical gauge potential to construct a Chern–Simons type action in (2 + 1) dimensions. That gauge theory has been extended to (4 + 1) or even higher dimensions, which gives rise to observable effects in the real (2 + 1)-dimensional physical system through an auxiliary technique of dimensional reduction [9].
Usually, to establish an effective topological gauge field theory needs to introduce a gauge potential as a principal bundle connection, to construct a topological invariant such as a topological characteristic class (Chern, Euler, etc) or a secondary characteristic class (Chern–Simons) [10]. In this paper we alternatively propose a new gauge theory, by employing a Hamiltonian vector and a Bloch wave function to act as the basic fields, to re-express the inner structure of the induced gauge potential (called a gauge potential decomposition). As a result the topological information of the base manifold is input to the constructed topological characteristic classes through the connection and curvature, and the Chern number C is delivered by the topological charges of the monopole and meron defects.
The paper is arranged as follows. In section 2, a brief introduction is presented for the two-band model—the minimal model to demonstrate nontrivial topology, with the emphasis placed on the singularities of the wave functions where the topological defects arise. In section 3, a gauge theory based on the 't Hooft monopole model is proposed, which gives rise to different types of topological defects. In section 4, we focus on the discussions on monopoles and merons, and provide a typical example of a square lattice. The results we obtain agree to the data of literature. In section 5, the Berry connection-induced meron excitations are studied, the result being identical to that of section 4.2. In section 6, a model with high Chern number ∣C∣ = 2, 4 is proposed, i.e. through studying the structures of the monopole and meron defects. The paper is summarized in section 7.

2. Two-band model

Consider a tight-binding model for an N-band topological insulator [7, 9]: $H={\sum }_{{\boldsymbol{k}};\alpha ,\beta }{\psi }_{\alpha {\boldsymbol{k}}}^{* }{h}_{{\boldsymbol{k}}}^{\alpha \beta }{\psi }_{\beta {\boldsymbol{k}}}$, where ${\boldsymbol{k}}=\left({k}_{x},{k}_{y}\right)$ is the two-dimensional momentum (for convenience the subscript k being ignored below). kx and ky satisfy the periodic condition, ${k}_{x},{k}_{y}\in \left[0,2\pi \right)$, i.e. kx, kyS1; hence the first BZ becomes a torus, T2 = S1 × S1. The α, β = 1, 2, ⋯, N are the band indices; if specially N= 2, it is a two-band structure, the minimal model to have nontrivial topology. In this paper we focus on the N = 2 case. ${\psi }_{\alpha }\in {\mathbb{C}}$ is a Bloch wave function, and hαβ a one-particle Hamiltonian. Introducing a Weyl spinor ${\rm{\Psi }}={\left(\begin{array}{cc}{\psi }_{1}, & {\psi }_{2}\end{array}\right)}^{{\rm{T}}}$, the Hamiltonian becomes
$\begin{eqnarray}H={{\rm{\Psi }}}^{\dagger }h{\rm{\Psi }}.\end{eqnarray}$
h as an SU(2) Lie algebraic vector can be expanded onto the Clifford algebraic basis $\left\{I,{\boldsymbol{\sigma }}\right\}$, with I the identity and ${\boldsymbol{\sigma }}=\left({\sigma }^{x},{\sigma }^{y},{\sigma }^{z}\right)$ the Pauli matrices:
$\begin{eqnarray}h={h}_{0}I+{\boldsymbol{h}}\cdot {\boldsymbol{\sigma }}=\epsilon I+\sum _{a=x,y,z}{h}_{a}{\sigma }^{a}.\end{eqnarray}$
The 3-vector ${\boldsymbol{h}}=\left({h}_{x},{h}_{y},{h}_{z}\right)$ induces a unit vector, $\hat{{\boldsymbol{h}}}=\tfrac{{\boldsymbol{h}}}{\parallel {\boldsymbol{h}}\parallel }$, with the norm $\parallel {\boldsymbol{h}}\parallel =\sqrt{{h}_{x}^{2}+{h}_{y}^{2}+{h}_{z}^{2}}$. $\hat{{\boldsymbol{h}}}$ forms a unit 2-sphere S2 in the SU(2) group space, and thus defines a topological map $\hat{{\boldsymbol{h}}}({\boldsymbol{k}}):{T}^{2}\to {S}^{2}$, as illustrated in figure 1.
Figure 1. Topological map from the base manifold (k-space, Brillouin zone) to the group space, $\hat{{\boldsymbol{h}}}:{T}^{2}\to {S}^{2}$.
The Euler–Lagrangian equation is given by
$\begin{eqnarray}\begin{array}{l}h{\rm{\Psi }}=E{\rm{\Psi }},\qquad {\rm{i}}.{\rm{e}}.\\ ({\boldsymbol{h}}\cdot {\boldsymbol{\sigma }}){\rm{\Psi }}=\left(\begin{array}{cc}{h}_{z} & {h}_{x}-{\rm{i}}{h}_{y}\\ {h}_{x}+{\rm{i}}{h}_{y} & -{h}_{z}\end{array}\right)\\ {\rm{\Psi }}=\left(E-\epsilon \right){\rm{\Psi }}.\end{array}\end{eqnarray}$
Regarding it as an eigen equation, one has the eigenvalues giving the energy levels, $E=\epsilon \pm \parallel {\boldsymbol{h}}\parallel ;$ usually one takes ε = 0, thus $E=\pm \parallel {\boldsymbol{h}}\parallel $. The eigen-functions give the Block wave functions:
$\begin{eqnarray}\begin{array}{l}\mathrm{conductance}:\quad E=+\parallel {\boldsymbol{h}}\parallel ,\\ {\rm{\Psi }}=\left(\begin{array}{c}{h}_{x}-{\rm{i}}{h}_{y},\\ -{h}_{z}+\parallel {\boldsymbol{h}}\parallel \end{array}\right),\quad \mathrm{with}\\ \parallel {\rm{\Psi }}\parallel =\sqrt{2\parallel {\boldsymbol{h}}\parallel \left(\parallel {\boldsymbol{h}}\parallel -{h}_{z}\right)};\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{valence}:\quad E=-\parallel {\boldsymbol{h}}\parallel ,\\ {\rm{\Psi }}=\left(\begin{array}{c}{h}_{x}-{\rm{i}}{h}_{y},\\ -{h}_{z}-\parallel {\boldsymbol{h}}\parallel \end{array}\right),\quad \mathrm{with}\\ \parallel {\rm{\Psi }}\parallel =\sqrt{2\parallel {\boldsymbol{h}}\parallel \left(\parallel {\boldsymbol{h}}\parallel +{h}_{z}\right)}.\end{array}\end{eqnarray}$
The singular points of $\hat{{\rm{\Psi }}}=\tfrac{{\rm{\Psi }}}{\parallel {\rm{\Psi }}\parallel }$ occur at the zeroes of $\Psi$. According to equations (4) and (5) there are two cases to reach $\parallel {\rm{\Psi }}\parallel =0$:

hx = hy = hz = 0, yielding

$\begin{eqnarray}\begin{array}{l}E=\parallel {\boldsymbol{h}}\parallel =0\qquad \mathrm{and}\\ \parallel {\boldsymbol{h}}\parallel \mp {h}_{z}=0,\end{array}\end{eqnarray}$
corresponding to the center O of the sphere S2 in the group space, as shown in figure 2. In the forthcoming sections it will be pointed out that equation (6) gives the monopole-type defects. For this case, since O is not on the surface of the sphere, topological transition will take place and the corresponding topological charges will be indeterminately singular.

hx = hy = 0 and hz ≠ 0, yielding $\parallel {\boldsymbol{h}}\parallel \ne 0$ and

$\begin{eqnarray}\begin{array}{l}\mathrm{when}\ E=+\parallel {\boldsymbol{h}}\parallel :\parallel {\boldsymbol{h}}\parallel -{h}_{z}=0,\\ {\rm{i}}.{\rm{e}}.\quad {\hat{h}}_{z}=+1;\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{when}\ E=-\parallel {\boldsymbol{h}}\parallel :\parallel {\boldsymbol{h}}\parallel +{h}_{z}=0,\\ {\rm{i}}.{\rm{e}}.\quad {\hat{h}}_{z}=-1,\end{array}\end{eqnarray}$
where an ${\hat{h}}_{z}$ is introduced for convenience, ${\hat{h}}_{z}=\tfrac{{h}_{z}}{\parallel {\boldsymbol{h}}\parallel }$. Equations (7) and (8) correspond respectively to the north-pole N and the south-pole S of the sphere S2 in the group space, as shown in figure 2. It will be pointed out in the forthcoming sections that this case corresponds to the meron excitations, i.e. the point defects in two dimensions.

Figure 2. Sphere S2 in the group space formed by the three-dimensional unit vector $\hat{{\boldsymbol{h}}}$. $\left\{{\hat{{\boldsymbol{e}}}}_{1},{\hat{{\boldsymbol{e}}}}_{2},{\hat{{\boldsymbol{e}}}}_{3}\right\}$ is an orthonormal frame. The center is O, where monopole defects take place. The north-pole is N, where $E={h}_{z}=\parallel {\boldsymbol{h}}\parallel \ne 0;$ the south-pole is S, where $E={h}_{z}=-\parallel {\boldsymbol{h}}\parallel \ne 0$. At N and S there is hx = hy = 0, hence meron defects take place, with the topological charges given by the winding numbers of the vector fields in the local Euclidean charts. See section 4 below.

3. Gauge theory

For the purpose of studying the topological defects let us start with the Weyl spinor $\Psi$, which has a field distribution over the base manifold T2. $\Psi$ induces an ${SU}\left(2\right)$ covariant derivative Dμ$\Psi$ acting as a parallel field on the manifold:
$\begin{eqnarray}{D}_{\mu }{\rm{\Psi }}={\partial }_{\mu }{\rm{\Psi }}-{{gA}}_{\mu }{\rm{\Psi }}=0,\end{eqnarray}$
where ${\partial }_{\mu }=\tfrac{\partial }{\partial {k}_{\mu }},\,\mu =x,y$. The g is a coupling constant; for simplicity we take g = 1. The Aμ is an ${SU}\left(2\right)$ gauge potential induced by $\Psi$ through equation (9), which can be expanded onto the ${SU}\left(2\right)$ generator basis as ${A}_{\mu }=\tfrac{1}{2{\rm{i}}}{\sum }_{a=x,y,z}{A}_{\mu a}{\sigma }^{a}=\tfrac{1}{2{\rm{i}}}\ {{\boldsymbol{A}}}_{\mu }\cdot {\boldsymbol{\sigma }}$. From Aμ arises an ${SU}\left(2\right)$ gauge field tensor
$\begin{eqnarray}{F}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }-\left[{A}_{\mu },{A}_{\nu }\right],\end{eqnarray}$
which can be expanded onto the ${SU}\left(2\right)$ generator basis as ${F}_{\mu \nu }=\tfrac{1}{2{\rm{i}}}{\sum }_{a=x,y,z}{F}_{\mu \nu ,a}{\sigma }^{a}=\tfrac{1}{2{\rm{i}}}{{\boldsymbol{F}}}_{\mu \nu }\cdot {\boldsymbol{\sigma }}$.
It is noted that the parallel field condition (9) arises from an analogue of the Christoffel connection in Riemannian geometry. For the metric gμν a requirement of Riemannian geometry is ${{\rm{\nabla }}}_{\mu }{g}_{\nu \lambda }={\partial }_{\mu }{g}_{\nu \lambda }-{{\rm{\Gamma }}}_{\mu \nu }^{\rho }{g}_{\rho \lambda }-{{\rm{\Gamma }}}_{\mu \lambda }^{\rho }{g}_{\nu \rho }=0$, from which the Christoffel connection is solved out, ${{\rm{\Gamma }}}_{\nu \lambda }^{\mu }={{\rm{\Gamma }}}_{\nu \lambda }^{\mu }\left({g}_{\alpha \beta }\right)$. Through that inner structure, the distribution information of the basic field gμν on the base manifold is input to the Riemannian curvature: ${R}_{\nu \lambda \rho }^{\mu }={R}_{\nu \lambda \rho }^{\mu }\left({g}_{\alpha \beta }\right)$. For details see Faddeev, Cho, et al [11, 12].
From the ${SU}\left(2\right)$ tensor Fμν and the covariant derivative Dμ one can construct a $U\left(1\right)$ Maxwell electromagnetic field tensor in terms of the 't Hooft monopole model [13, 14]:
$\begin{eqnarray}{f}_{\mu \nu }={{\boldsymbol{F}}}_{\mu \nu }\cdot \hat{{\boldsymbol{h}}}-\hat{{\boldsymbol{h}}}\cdot \left({D}_{\mu }\hat{{\boldsymbol{h}}}\times {D}_{\nu }\hat{{\boldsymbol{h}}}\right).\end{eqnarray}$
fμν is understood as a $U\left(1\right)$ sub-field of the ${SU}\left(2\right)$ tensor Fμν. The ${{\boldsymbol{F}}}_{\mu \nu }\cdot \hat{{\boldsymbol{h}}}$ is the projection of the ${SU}\left(2\right)$ vector Fμν onto the unit direction $\hat{{\boldsymbol{h}}}$, with $\hat{{\boldsymbol{h}}}\cdot \left({D}_{\mu }\hat{{\boldsymbol{h}}}\times {D}_{\nu }\hat{{\boldsymbol{h}}}\right)=\tfrac{1}{3!}{\epsilon }_{{abc}}{\hat{h}}_{a}{D}_{\mu }{\hat{h}}_{b}{D}_{\nu }{\hat{h}}_{c}$. Here εabc is the Levi-Civita full anti-symmetric tensor, and the Einstein summation convention applies. The covariant derivative ${D}_{\mu }\hat{{\boldsymbol{h}}}$ is defined as ${D}_{\mu }\hat{{\boldsymbol{h}}}={\partial }_{\mu }\hat{{\boldsymbol{h}}}-{{\boldsymbol{A}}}_{\mu }\times \hat{{\boldsymbol{h}}}$. The Maxwell field tensor fμν is ${SU}\left(2\right)$-covariant thanks to the covariance of ${\hat{h}}_{a}$, Fμν,a and ${D}_{\mu }{\hat{h}}_{a}$. Substituting ${D}_{\mu }\hat{{\boldsymbol{h}}}$ into equation (11) we arrive at
$\begin{eqnarray}{f}_{\mu \nu }=\left({\partial }_{\mu }{a}_{\nu }-{\partial }_{\nu }{a}_{\mu }\right)-\hat{{\boldsymbol{h}}}\cdot \left({\partial }_{\mu }\hat{{\boldsymbol{h}}}\times {\partial }_{\nu }\hat{{\boldsymbol{h}}}\right),\end{eqnarray}$
where ${a}_{\mu }={{\boldsymbol{A}}}_{\mu }\cdot \hat{{\boldsymbol{h}}}$ is a $U\left(1\right)$ electromagnetic potential obtained from the projection of the ${SU}\left(2\right)$ potential Aμ onto the chosen direction $\hat{{\boldsymbol{h}}}$. The second term $\hat{{\boldsymbol{h}}}\cdot \left({\partial }_{\mu }\hat{{\boldsymbol{h}}}\times {\partial }_{\nu }\hat{{\boldsymbol{h}}}\right)=\tfrac{1}{3!}{\epsilon }_{{abc}}{\hat{h}}_{a}{\partial }_{\mu }{\hat{h}}_{b}{\partial }_{\nu }{\hat{h}}_{c}$ is a topological term.
The significance of equation (12) lies in that it satisfies both the U(1) and SU(2) gauge covariance. It serves as the starting point of our study in the following sections on various topological defects.

4. Monopoles and merons: with typical example

4.1. Monopoles and merons

Let us begin with the second term of equation (12); for convenience one uses the notation ${K}_{\mu \nu }=\hat{{\boldsymbol{h}}}\cdot \left({\partial }_{\mu }\hat{{\boldsymbol{h}}}\times {\partial }_{\nu }\hat{{\boldsymbol{h}}}\right)$.
Monopoles. A 2-form can be constructed from Kμν as $K=\tfrac{1}{2}{K}_{\mu \nu }{{dk}}^{\mu }\wedge {{dk}}^{\nu }=\hat{{\boldsymbol{h}}}\cdot \left({\rm{d}}\hat{{\boldsymbol{h}}}\times {\rm{d}}\hat{{\boldsymbol{h}}}\right)$, which carries the geometric meaning of a solid angle, i.e. a surface element on the sphere S2 in the SU(2) group space. Its integral is the area of S2, ${\int }_{{S}^{2}}K=4\pi ;$ the pull-back of the integral, known as the Chern number C, gives the topological degree of the map $\hat{{\boldsymbol{h}}}$ [7]:
$\begin{eqnarray}\begin{array}{l}C={\hat{{\boldsymbol{h}}}}^{* }\left(\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{S}^{2}}K\right)\\ =\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{\hat{{\boldsymbol{h}}}}^{* }\left({S}^{2}\right)}{\hat{{\boldsymbol{h}}}}^{* }\left[\hat{{\boldsymbol{h}}}\cdot \left({\rm{d}}\hat{{\boldsymbol{h}}}\times {\rm{d}}\hat{{\boldsymbol{h}}}\right)\right]\\ =\displaystyle \frac{1}{8\pi }{\displaystyle \int }_{{T}^{2}}\hat{{\boldsymbol{h}}}\cdot \left({\partial }_{\mu }\hat{{\boldsymbol{h}}}\times {\partial }_{\nu }\hat{{\boldsymbol{h}}}\right){\rm{d}}{k}^{\mu }\wedge {\rm{d}}{k}^{\nu },\end{array}\end{eqnarray}$
with ${\hat{{\boldsymbol{h}}}}^{* }\left(\bullet \right)$ denoting a pull-back operation. The significance of C lies in its relationship to the Hall conductance σxy = Ce2/h.

In the group space, the double integral on the closed surface S2, ${\int }_{{S}^{2}}K$, could be turned into a triple integral over a body B3 via the Stokes theorem:

$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{S}^{2}}K=\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{S}^{2}}\hat{{\boldsymbol{h}}}\cdot \left({\rm{d}}\hat{{\boldsymbol{h}}}\times {\rm{d}}\hat{{\boldsymbol{h}}}\right)\\ =\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{B}^{3}}{\rm{d}}\hat{{\boldsymbol{h}}}\cdot \left({\rm{d}}\hat{{\boldsymbol{h}}}\times {\rm{d}}\hat{{\boldsymbol{h}}}\right)\\ =\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{B}^{3}}\left(-\displaystyle \frac{2}{{\parallel {\boldsymbol{h}}\parallel }^{3}}\right){\rm{d}}{\boldsymbol{h}}\cdot \left({\rm{d}}{\boldsymbol{h}}\times {\rm{d}}{\boldsymbol{h}}\right)\\ =\displaystyle \frac{1}{4\pi }{\displaystyle \int }_{{B}^{3}}{{\rm{\nabla }}}^{2}\displaystyle \frac{1}{\parallel {\boldsymbol{h}}\parallel }{\rm{d}}V,\end{array}\end{eqnarray}$
where B3 is the unit ball in three dimensions, with boundary ∂B3 = S2. The volume element reads ${\rm{d}}V={\rm{d}}{\boldsymbol{h}}\cdot \left({\rm{d}}{\boldsymbol{h}}\times {\rm{d}}{\boldsymbol{h}}\right)=\tfrac{1}{3!}{\epsilon }_{{abc}}{\rm{d}}{h}_{a}\wedge {\rm{d}}{h}_{b}\wedge {\rm{d}}{h}_{c}$. The integrand ${{\rm{\nabla }}}^{2}\tfrac{1}{\parallel {\boldsymbol{h}}\parallel }$ is the Laplacian of a point-source potential, ${{\rm{\nabla }}}^{2}\tfrac{1}{\parallel {\boldsymbol{h}}\parallel }=4\pi {\delta }^{3}\left({\boldsymbol{h}}\right)$. Thus,
$\begin{eqnarray}\displaystyle \frac{1}{4\pi }{\int }_{{S}^{2}}K={\int }_{{B}^{3}}{\delta }^{3}\left({\boldsymbol{h}}\right){\rm{d}}V=1.\end{eqnarray}$
In equation (15) the three-dimensional δ-function does not vanish only at h = 0, which implies the existence of monopoles (i.e. three-dimensional point defects) as mentioned in equation (6).

However, from the angle of view of the base manifold T2, the 3-form ${\rm{d}}\hat{{\boldsymbol{h}}}\cdot \left({\rm{d}}\hat{{\boldsymbol{h}}}\times {\rm{d}}\hat{{\boldsymbol{h}}}\right)$ cannot be pulled back to a triple integral, because T2 lacks the third coordinate beyond kx and ky. This corresponds to an indeterminate evaluation for the Chern number C. That is, when the monopole excitations occur at h = 0, the insulator experiences a topological transition with the Chern number C taking an indeterminate value.

Merons. As mentioned in equations (7) and (8), there exist another type of defects, the merons, that can be derived from Kμν too. In this regard we need to change the angle of view from three dimensions to two dimensions, by rewriting Kμν as a U(1) field tensor:
$\begin{eqnarray}{K}_{\mu \nu }=\hat{{\boldsymbol{h}}}\cdot \left({\partial }_{\mu }\hat{{\boldsymbol{h}}}\times {\partial }_{\nu }\hat{{\boldsymbol{h}}}\right)={\partial }_{\mu }{W}_{\nu }-{\partial }_{\nu }{W}_{\mu }.\end{eqnarray}$
Wμ is the so-called Wu–Yang potential [15], constructed with two unit vectors $\hat{{\boldsymbol{e}}}$ and $\hat{{\boldsymbol{f}}}$ on the surface of the $\hat{{\boldsymbol{h}}}$-formed S2, as shown in figure 3:
$\begin{eqnarray}{W}_{\mu }=\hat{{\boldsymbol{e}}}\cdot {\partial }_{\mu }\hat{{\boldsymbol{f}}}.\end{eqnarray}$
$\hat{{\boldsymbol{e}}}$ and $\hat{{\boldsymbol{f}}}$ are perpendicular to each other, and $\left\{\hat{{\boldsymbol{h}}},\hat{{\boldsymbol{e}}},\hat{{\boldsymbol{f}}}\right\}$ forms an orthonormal frame. Wμ is independent of the choice of $\left\{\hat{{\boldsymbol{e}}},\hat{{\boldsymbol{f}}}\right\};$ indeed, if introducing another pair $\left\{{\hat{{\boldsymbol{e}}}}^{{\prime} },{\hat{{\boldsymbol{f}}}}^{{\prime} }\right\}$ by rotating $\left\{\hat{{\boldsymbol{e}}},\hat{{\boldsymbol{f}}}\right\}$ with an angle θ, we will have another potential ${W}_{\mu }^{{\prime} }$. It can be proved that ${W}_{\mu }^{{\prime} }={W}_{\mu }+{\partial }_{\mu }\theta $, hence ${W}_{\mu }^{{\prime} }\sim {W}_{\mu }$, as the ∂μθ has no contribution to Kμν.
Figure 3. Orthonormal frame $\left\{\hat{{\boldsymbol{h}}},\hat{{\boldsymbol{e}}},\hat{{\boldsymbol{f}}}\right\}$: the unit vector $\hat{{\boldsymbol{h}}}$ forms a 2-sphere S2 in the ${SU}\left(2\right)$ group space, while $\hat{{\boldsymbol{e}}}$ and $\hat{{\boldsymbol{f}}}$ are two perpendicular unit vectors on the S2. The base manifold chosen to perform this technique of Wu–Yang potential is a hemisphere, identical to a local Euclidean chart.
The integral of equation (16) is able to give a Chern number C. It should be addressed that, the pre-condition of obtaining a Chern number is to perform the integral on a local Euclidean chart diffeomorphic to a two-dimensional disk D2, rather than a whole S2 in the group space. This local chart could be, say, the north-hemisphere (denoted as ${S}_{N-\mathrm{hemi}}^{2}$) or the south-hemisphere (denoted as ${S}_{S-\mathrm{hemi}}^{2}$). Thus, the Chern number C turns to have two copies:
$\begin{eqnarray}\begin{array}{l}{C}_{N}={\hat{{\boldsymbol{h}}}}^{* }\left(\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{S}_{N-\mathrm{hemi}}^{2}}K\right)\\ \,=\ \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{T}_{N}^{2}}\displaystyle \frac{1}{2}{K}_{\mu \nu }{\rm{d}}{k}^{\mu }\wedge {\rm{d}}{k}^{\nu }\\ \,=\ \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{T}_{N}^{2}}\displaystyle \frac{1}{2}\left({\partial }_{\mu }{W}_{\nu }-{\partial }_{\nu }{W}_{\mu }\right){\rm{d}}{k}^{\mu }\wedge {\rm{d}}{k}^{\nu },\\ \mathrm{with}{\,T}_{N}^{2}={\hat{{\boldsymbol{h}}}}^{* }\left({S}_{N-\mathrm{hemi}}^{2}\right);\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{C}_{S}={\hat{{\boldsymbol{h}}}}^{* }\left(\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{S}_{S-\mathrm{hemi}}^{2}}K\right)\\ \,=\ \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{T}_{S}^{2}}\displaystyle \frac{1}{2}{K}_{\mu \nu }{\rm{d}}{k}^{\mu }\wedge {\rm{d}}{k}^{\nu }\\ \,=\ \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{T}_{S}^{2}}\displaystyle \frac{1}{2}\left({\partial }_{\mu }{W}_{\nu }-{\partial }_{\nu }{W}_{\mu }\right){\rm{d}}{k}^{\mu }\wedge {\rm{d}}{k}^{\nu },\\ \mathrm{with}\ {T}_{S}^{2}={\hat{{\boldsymbol{h}}}}^{* }\left({S}_{S-\mathrm{hemi}}^{2}\right).\end{array}\end{eqnarray}$
According to equations (7) and (8), the meron defects occur at the north pole N, where hx = hy = 0 and $E={h}_{z}=\parallel {\boldsymbol{h}}\parallel \ne 0$ (corresponding to the conductance band), and at the south-pole S, where hx = hy = 0 and $E={h}_{z}=-\parallel {\boldsymbol{h}}\parallel \ne 0$ (corresponding to the valence band). Figure 4 illustrates the h-distribution in the neighborhood of N. The winding number of the vector $\left({h}_{x},{h}_{y}\right)$ in two dimensions gives the topological charge of the meron, as detailed below. The situation for the south-pole S is similar.
Figure 4. Neighborhood of the pole N, where the vector h is parallel to the ${\hat{{\boldsymbol{e}}}}_{3}$-axis, with hx = hy = 0. The winding number of the vector $\left({h}_{x},{h}_{y}\right)$ in the two-dimensional vicinity of N gives the topological charge of the meron.
The winding number is computed as follows. Introducing a two-dimensional field
$\begin{eqnarray}\begin{array}{l}{\boldsymbol{\phi }}=\left({\phi }_{1},{\phi }_{2}\right),\qquad \mathrm{where}\quad {\phi }_{1}={h}_{x},\\ {\phi }_{2}=-{h}_{y},\quad \mathrm{with\; norm}\parallel {\boldsymbol{\phi }}\parallel =\sqrt{{\phi }_{1}^{2}+{\phi }_{2}^{2}},\end{array}\end{eqnarray}$
the orthonormal vectors $\hat{{\boldsymbol{e}}}$ and $\hat{{\boldsymbol{f}}}$ of equation (17) can be evaluated as
$\begin{eqnarray}{\hat{e}}_{A}=\displaystyle \frac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel },\qquad {\hat{f}}_{A}={\epsilon }_{{AB}}\displaystyle \frac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel },\end{eqnarray}$
where $\hat{{\boldsymbol{e}}}=\left({\hat{e}}_{1},{\hat{e}}_{2}\right)$ and $\hat{{\boldsymbol{f}}}=\left({\hat{f}}_{1},{\hat{f}}_{2}\right)$, and A, B = 1, 2. Substituting equation (21) into (17) one immediately obtains
$\begin{eqnarray}\begin{array}{l}{W}_{\mu }={\epsilon }_{{AB}}\displaystyle \frac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel }{\partial }_{\mu }\displaystyle \frac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel },\qquad \mathrm{and}\ \mathrm{thus}\\ {K}_{\mu \nu }=2{\epsilon }_{{AB}}{\partial }_{\mu }\displaystyle \frac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel }{\partial }_{\nu }\displaystyle \frac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel }.\end{array}\end{eqnarray}$
The Chern number on the north hemisphere becomes
$\begin{eqnarray}\begin{array}{l}{C}_{N}={c}_{1,N}=\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{T}_{N}^{2}}{\epsilon }^{\mu \nu }{\epsilon }_{{AB}}{\partial }_{\mu }\displaystyle \frac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel }{\partial }_{\nu }\displaystyle \frac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel }\\ {{\rm{d}}}^{2}{\boldsymbol{k}}={\displaystyle \int }_{{T}_{N}^{2}}j\left({\boldsymbol{k}}\right){{\rm{d}}}^{2}{\boldsymbol{k}},\end{array}\end{eqnarray}$
where d2k = dkx ∧ dky, and
$\begin{eqnarray}j\left({\boldsymbol{k}}\right)=\displaystyle \frac{1}{2\pi }\displaystyle \frac{1}{2}{\epsilon }^{\mu \nu }{\epsilon }_{{AB}}{\partial }_{\mu }\displaystyle \frac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel }{\partial }_{\nu }\displaystyle \frac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel }\end{eqnarray}$
is a so-called two-dimensional topological current [16, 17]. CS can be similarly obtained.
In terms of the formula $\tfrac{\partial }{\partial {\phi }_{A}}\tfrac{\partial }{\partial {\phi }_{A}}\mathrm{ln}\parallel {\boldsymbol{\phi }}\parallel =2\pi {\delta }^{2}\left({\boldsymbol{\phi }}\right)$, with $\tfrac{\partial }{\partial {\phi }_{A}}\tfrac{\partial }{\partial {\phi }_{A}}$ as a Laplacian operator and ${\delta }^{2}\left({\boldsymbol{\phi }}\right)$ a two-dimensional δ-function, one can prove
$\begin{eqnarray}j\left({\boldsymbol{k}}\right)={\delta }^{2}\left({\boldsymbol{\phi }}\right)D\left(\displaystyle \frac{\phi }{k}\right),\end{eqnarray}$
where $D\left(\tfrac{\phi }{k}\right)$ is a Jacobian, ${\epsilon }_{{AB}}D\left(\tfrac{\phi }{k}\right)={\epsilon }^{\mu \nu }{\partial }_{\mu }{\phi }_{A}{\partial }_{\nu }{\phi }_{B}$. Hence
$\begin{eqnarray}{C}_{N}={\int }_{{T}_{N}^{2}}{\delta }^{2}\left({\boldsymbol{\phi }}\right)D\left(\displaystyle \frac{\phi }{k}\right){{\rm{d}}}^{2}{\boldsymbol{k}}.\end{eqnarray}$
The δ-function does not vanish only at the zeroes of the vector field φ, which requires solving the zero equations:
$\begin{eqnarray}{\phi }_{1}({k}_{x},{k}_{y})=0,\qquad {\phi }_{2}({k}_{x},{k}_{y})=0.\end{eqnarray}$
According to the implicit function theorem, under the regular condition $D\left(\tfrac{\phi }{k}\right)\ne 0$, equation (27) has a number LN copies of isolated solutions: kμ = kμ,l, μ = x, y; l = 1, 2, …, LN, corresponding to the LN merons in two dimensions.
The δ-function can be further expanded onto the merons as
$\begin{eqnarray}\begin{array}{l}{\left.{\delta }^{2}\left({\boldsymbol{\phi }}\right)\right|}_{{T}_{N}^{2}}=\displaystyle \sum _{l=1}^{{L}_{N}}{{ \mathcal W }}_{l,N}{\delta }^{2}\left({k}_{\mu }-{k}_{\mu ,l}\right),\\ \mathrm{where}\,{{ \mathcal W }}_{l,N}={\beta }_{l,N}{\eta }_{l,N}.\end{array}\end{eqnarray}$
${{ \mathcal W }}_{l,N}$ denotes the winding number of the lth defect, with βl,N as the Hopf index and ηl,N = ±1 the degree of the Brouwer mapping. Thus, the Chern number is given by an algebraic sum of the topological charges of the merons,
$\begin{eqnarray}{C}_{N}=\sum _{l=1}^{{L}_{N}}{{ \mathcal W }}_{l,N}=\sum _{l=1}^{{L}_{N}}{\beta }_{l,N}{\eta }_{l,N}.\end{eqnarray}$
Similarly, for the south hemisphere ${S}_{N-\mathrm{hemi}}^{2}$ we have
$\begin{eqnarray}{C}_{S}={\int }_{{T}_{S}^{2}}j\left({\boldsymbol{k}}\right){{\rm{d}}}^{2}{\boldsymbol{k}}=\sum _{l=1}^{{L}_{S}}{{ \mathcal W }}_{l,S}=\sum _{l=1}^{{L}_{S}}{\beta }_{l,S}{\eta }_{l,S},\end{eqnarray}$
where ${{ \mathcal W }}_{l,S}$ is the winding number, βl,S the Hopf index and ηl,S = ±1 the Brouwer degree.
It should be addressed that CN and CS are not additive, because ${S}_{N-\mathrm{hemi}}^{2}$ and ${S}_{S-\mathrm{hemi}}^{2}$ are not additive. They are two separate covers of the S2, cannot be topological-trivially stuck to each other, i.e. their overlap is topologically non-trivial. Physically, it means each energy band has independent topological numbers. The existence of meron defects indicates the two bands open an energy gap in the whole BZ. In this case, if the lower band is fully filled, the Hall conductance is given by [18]
$\begin{eqnarray}{\sigma }_{{xy}}=-\displaystyle \frac{{e}^{2}}{h}{C}_{S},\end{eqnarray}$
where CS is the Chern number of the valence band.

4.2. Typical example

Let us use the above theory to examine a typical example in literature [9]. Consider a two-band model of a square lattice,
$\begin{eqnarray}\begin{array}{l}{h}_{x}=\sin {k}_{x},\qquad {h}_{y}=\sin {k}_{y},\\ {h}_{z}=m+\cos {k}_{x}+\cos {k}_{y},\end{array}\end{eqnarray}$
where $m\in {\mathbb{R}}$ is an on-site energy to open up an energy gap. As per equations (6)–(8) one needs to find the monopoles at h = 0, and the merons at hx = hy = 0 and hz ≠ 0.
i

(i)Monopoles: They occur at the Dirac points with $E=\parallel {\boldsymbol{h}}\parallel =0$. The solutions of the zero point equations

$\begin{eqnarray}{h}_{x}={h}_{y}={h}_{z}=0\end{eqnarray}$
give the locations of the monopole defects:

$\left({k}_{x},{k}_{y}\right)=\left(0,0\right)$, with m = −2, due to equation (32);

$\left({k}_{x},{k}_{y}\right)=\left(0,\pi \right)$ and $\left(\pi ,0\right)$, with m = 0;

$\left({k}_{x},{k}_{y}\right)=\left(\pi ,\pi \right)$, with m = 2.

$\left(0,0\right)$, $\left(0,\pi \right)$, $\left(\pi ,0\right)$ and $\left(\pi ,\pi \right)$ are four Dirac points on T2, hence they inevitably get involved in the integral (13), which forces C to take an indeterminate value when m = 0, ±2.

ii

(ii)Merons: They occur at the two-dimensional singular points where

$\begin{eqnarray}{h}_{x}={h}_{y}=0,\qquad E={h}_{z}=\pm \parallel {\boldsymbol{h}}\parallel \ne 0.\end{eqnarray}$

The solutions of equation (34) are given in table 1, due to equation (32).
Table 1. Locations of the meron defects of the ∣C∣ = 1 model, and the corresponding m values, where hx = hy = 0 and $E={h}_{z}=\pm \sqrt{\parallel {\boldsymbol{h}}\parallel }\ne 0$.
Moment kx Moment ky On-site energy m hz i.e. E
0 0 m ≠ −2 m + 2
0 π m ≠ 0 m
π 0 m ≠ 0 m
π π m ≠ 2 m − 2
The desired topological charges of the merons certainly could be computed by means of equations (29) and (30) though, they fortunately can be found out alternatively in an easier way: direct recognition from the configurations of the two-dimensional vector $\left({h}_{x},{h}_{y}\right)$ in the neighborhoods of the poles N and S.
The asymptotic behavior of hx and hy around kx, ky = 0, π is observed via a Taylor expansion:

Near kx = 0: ${h}_{x}=\sin {k}_{x}\approx {\rm{\Delta }}{k}_{x}$, where Δkx = kx − 0. Similarly, near ky = 0: hy ≈ Δky, where Δky = ky − 0.

Near kx = π: ${h}_{x}=\sin {k}_{x}\approx \pi -{\rm{\Delta }}{k}_{x}$, where Δkx = kxπ. Similarly, near ky = π: hyπ − Δky, where Δky = kyπ.

Thus,

Near $\left({k}_{x},{k}_{y}\right)=(0,0)$, the asymptotic behavior of $\left({h}_{x},{h}_{y}\right)$ is $\left({\rm{\Delta }}{k}_{x},{\rm{\Delta }}{k}_{y}\right)$. Such a vector field distribution represents a source-point, so the winding number is +1.

Near $\left({k}_{x},{k}_{y}\right)=(0,\pi )$, that of $\left({h}_{x},{h}_{y}\right)$ is $\left({\rm{\Delta }}{k}_{x},-{\rm{\Delta }}{k}_{y}\right)$. This is a saddle-point, with winding number −1.

Near $\left({k}_{x},{k}_{y}\right)=(\pi ,0)$, that of $\left({h}_{x},{h}_{y}\right)$ is $\left(-{\rm{\Delta }}{k}_{x},{\rm{\Delta }}{k}_{y}\right)$. This is a saddle-point, with winding number −1.

Near $\left({k}_{x},{k}_{y}\right)=(\pi ,\pi )$, that of $\left({h}_{x},{h}_{y}\right)$ is $\left(-{\rm{\Delta }}{k}_{x},-{\rm{\Delta }}{k}_{y}\right)$. This is a congruence-point, with winding number +1.

The above winding numbers are illustrated by a plot for $\left({h}_{x},{h}_{y}\right)$ in two dimensions as in figure 5.
Figure 5. Merons, i.e. two dimensional point defects, of the ∣C∣ = 1 model, marked as red spots: (0, 0) and (π, π) are source/congruence points, so they have a winding number +1; (0, π) and (π, 0) are saddle points, so they have a winding number −1. The pink-boxed region indicates the first Brillouin zone.
Figure 6. The different evaluations of the Chern number C of the ∣C∣ = 1 model due to varying on-site energy m, as described by [9]. This figure is produced by directly substituting equation (32) into (13) and plotting the result of the integral.
Thus, with respect to table 1, we obtain tables 2 and 3 below to list out the topological charges of the meron defects as well as their respective contributions to the north pole N and the south pole S.
Table 2. Winding numbers at the meron defects of the ∣C∣ = 1 model, and their total contribution to the north pole N (${\hat{h}}_{z}$ = +1).
On-site energy m $\left(0,0\right)$ $\left(0,\pi \right)$ $\left(\pi ,0\right)$ $\left(\pi ,\pi \right)$ Total contribution to N, i.e. CN
m > 2 +1 −1 −1 +1 0
0 < m < 2 +1 −1 −1 / −1
−2 < m < 0 +1 / / / +1
m < −2 / / / / 0
Table 3. Winding numbers at the meron defects of the ∣C∣ = 1 model, and their total contribution to the south pole S (${\hat{h}}_{z}$ = −1).
On-site energy m $\left(0,0\right)$ $\left(0,\pi \right)$ $\left(\pi ,0\right)$ $\left(\pi ,\pi \right)$ Total contribution to S, i.e. CS
m > 2 / / / / 0
0 < m < 2 / / / +1 +1
−2 < m < 0 / −1 −1 +1 −1
m < −2 +1 −1 −1 +1 0
The detailed computations for the last columns of tables 2 and 3—the total contributions to the north-pole N, i.e. CN, and that to the south-pole S, i.e. CS—are presented as follows:

When m > 2:

1. At (kx, ky) = (0, 0): according to equation (32), hz = m + 2 > 0, hence hz = +∥h∥ and ${\hat{h}}_{z}=\tfrac{{h}_{z}}{\parallel {\boldsymbol{h}}\parallel }$ = +1, corresponding to the north-pole. This means the topological charge +1 of (kx, ky) = (0, 0) has contribution to the north-pole N.

2. At (kx, ky) = (0, π) and (π, 0): according to equation (32), hz = m > 0, hence hz = +∥h∥ and ${\hat{h}}_{z}=\tfrac{{h}_{z}}{\parallel {\boldsymbol{h}}\parallel }$ = +1, the north-pole. This means both the topological charges of (0, π) and (π, 0), −1 −1 = −2, have contributions to N.

3. At (kx, ky) = (π, π): hz = m − 2 > 0, hence hz = +∥h∥ and ${\hat{h}}_{z}=\tfrac{{h}_{z}}{\parallel {\boldsymbol{h}}\parallel }$ = +1, the north-pole. This means the topological charge +1 of (kx, ky) = (π, π) has contribution to N.

In conclusion, when m > 2 the total contributions to the north-pole N is +1 − 1 − 1 + 1 = 0, while there are no contributions to the south-pole S.

When 0 < m < 2:

1. At (kx, ky) = (0, 0): hz = m + 2 > 0, hence ${\hat{h}}_{z}$ = +1, the north-pole. This means the topological charge +1 of (kx, ky) = (0, 0) has contribution to N.

2. At (kx, ky) = (0, π) and (π, 0): hz = m > 0, hence ${\hat{h}}_{z}$ = +1, the north-pole. This means both the topological charges of (0, π) and (π, 0), −1 − 1 = −2, have contributions to N.

3. At (kx, ky) = (π, π): hz = m − 2 < 0, hence hz = − ∥h∥ and ${\hat{h}}_{z}=\tfrac{{h}_{z}}{\parallel {\boldsymbol{h}}\parallel }$ = −1, the south-pole. This means the topological charge +1 of (kx, ky) = (π, π) has contribution to the south-pole S.

In conclusion, when 0 < m < 2 the total contributions to N and S are +1 − 1 − 1 = −1 and +1, respectively.

When −2 < m < 0:

1. At (kx, ky) = (0, 0): hz = m + 2 > 0, hence ${\hat{h}}_{z}$ = +1, the north-pole. This means the topological +1 charge of (kx, ky) = (0, 0) has contribution to N.

2. At (kx, ky) = (0, π) and (π, 0): hz = m < 0, hence ${\hat{h}}_{z}$ = −1, the south-pole. This means both the topological charges of (0, π) and (π, 0), −1 − 1 = −2, have contributions to S.

3. At (kx, ky) = (π, π): hz = m − 2 < 0, hence ${\hat{h}}_{z}$ = −1, the south-pole. This means the topological charge +1 of (kx, ky) = (π, π) has contribution to S.

In conclusion, when −2 < m < 0 the total contributions to N and S are +1 and −1 − 1 + 1 = −1, respectively.

When m < −2:

1. At (kx, ky) = (0, 0): hz = m + 2 < 0, hence ${\hat{h}}_{z}$ = −1, the south-pole. This means the topological charge +1 of (kx, ky) = (0, 0) has contribution to S.

2. At (kx, ky) = (0, π) and (π, 0): hz = m < 0, hence ${\hat{h}}_{z}$ = −1, the south-pole. This means both the topological charges of (0, π) and (π, 0), −1 − 1 = −2, have contributions to S.

3. At (kx, ky) = (π, π): hz = m − 2 < 0, hence ${\hat{h}}_{z}$ = −1, the south-pole. This means the topological charge +1 of (kx, ky) = (π, π) has contribution to S.

In conclusion, when m < −2 the total contributions to S is +1 − 1 − 1 + 1 = 0, while there are no contributions to N.
In summary, the above monopole and meron topological charges together present the evaluations for the Chern numbers CN and CS:
$\begin{eqnarray}\begin{array}{l}\mathrm{On}\,\mathrm{north}\,\mathrm{hemisphere}({\hat{h}}_{z}=+1):\\ {C}_{N}=\left\{\begin{array}{lll}0, & \mathrm{when} & m\lt -2,\,m\gt 2;\\ -1, & \mathrm{when} & 0\lt m\lt 2;\\ +1, & \mathrm{when} & -2\lt m\lt 0;\\ \mathrm{indeterminate}, & \mathrm{when} & m=0,\pm 2.\end{array}\right.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{On}\,\mathrm{south}\,\mathrm{hemisphere}({\hat{h}}_{z}=-1):\\ {C}_{S}=\left\{\begin{array}{lll}0, & \mathrm{when} & m\lt -2,\,m\gt 2;\\ +1, & \mathrm{when} & 0\lt m\lt 2;\\ -1, & \mathrm{when} & -2\lt m\lt 0;\\ \mathrm{indeterminate}, & \mathrm{when} & m=0,\pm 2.\end{array}\right.\end{array}\end{eqnarray}$
It is stressed that the CS of equation (36), which has ${\hat{h}}_{z}$ = −1 and corresponds to the valence band, exactly reproduces the evaluation of the Hall conductance equation (31) (see [9]). The CN-evaluation agrees to the integration result of direct substitution of equation (32) into (13), as shown in figure 6.

5. Berry connection and Chern number of merons: with example

Next let us move to the first term of equation (12); for convenience one uses the notation Bμν = ∂μaν − ∂aμ. The U(1) potential aμ is given by ${a}_{\mu }={{\boldsymbol{A}}}_{\mu }\cdot \hat{{\boldsymbol{h}}}$, where Aμ is an SU(2) gauge potential. In order to study the topology arising from aμ let us first investigate the inner structure of Aμ.
In the light of the parallel field condition (9) and its Hermitian conjugate, one obtains an expression for Aμ in terms of the Weyl spinor Bloch wave function [19],
$\begin{eqnarray}\begin{array}{l}{{\boldsymbol{A}}}_{\mu }={\hat{{\rm{\Psi }}}}^{\dagger }{\boldsymbol{\sigma }}{\partial }_{\mu }\hat{{\rm{\Psi }}}-{\rm{h}}.{\rm{c}}.,\\ \hat{{\rm{\Psi }}}=\displaystyle \frac{{\rm{\Psi }}}{\parallel {\rm{\Psi }}\parallel },\end{array}\end{eqnarray}$
where the Clifford algebraic definition applies, σaσb + σbσa = 2δabI, a, b = x, y, z. This expression satisfies ${A}_{\mu }^{{\prime} }={{SA}}_{\mu }{S}^{\dagger }+{\partial }_{\mu }{{SS}}^{\dagger }$ under an ${SU}\left(2\right)$ gauge transformation S, with Aμ = Aμ · σ. On the other hand, the unit vector $\hat{{\boldsymbol{h}}}$ can be expressed by a projection of the density operator onto the Lie algebraic basis σ: $\hat{{\boldsymbol{h}}}=\mathrm{Tr}\left[\left(\hat{{\rm{\Psi }}}{\hat{{\rm{\Psi }}}}^{\dagger }\right){\boldsymbol{\sigma }}\right]={\hat{{\rm{\Psi }}}}^{\dagger }{\boldsymbol{\sigma }}\hat{{\rm{\Psi }}}$. Thus the $U\left(1\right)$ gauge potential aμ becomes a Berry connection,
$\begin{eqnarray}{a}_{\mu }=\displaystyle \frac{1}{2{\rm{i}}}\left({\hat{{\rm{\Psi }}}}^{\dagger }{\partial }_{\mu }\hat{{\rm{\Psi }}}-{\rm{h}}.{\rm{c}}.\right)=-{\rm{i}}\left\langle \hat{{\rm{\Psi }}}\right|{\overleftrightarrow{\partial }}_{\mu }\left|\hat{{\rm{\Psi }}}\right\rangle ,\end{eqnarray}$
and Bμν becomes a Berry curvature,
$\begin{eqnarray}{B}_{\mu \nu }=-{\rm{i}}\left\langle {\partial }_{\mu }\hat{{\rm{\Psi }}}\right|\left.{\partial }_{\nu }\hat{{\rm{\Psi }}}\right\rangle .\end{eqnarray}$
For ${\rm{\Psi }}={\left(\begin{array}{cc}{\psi }_{1} & {\psi }_{2}\end{array}\right)}^{{\rm{T}}}$ the two components ψ1 and ψ2 separately satisfy a U(1) symmetry. Hence one should consider the ${\rm{\Psi }}={\left(\begin{array}{cc}{\psi }_{1} & 0\end{array}\right)}^{{\rm{T}}}$ and ${\rm{\Psi }}={\left(\begin{array}{cc}0 & {\psi }_{2}\end{array}\right)}^{{\rm{T}}}$ cases individually; without loss of generality the former is adopted (—adopting the latter will lead to the same result). Below one denotes ψ1ψ. Then
$\begin{eqnarray}\begin{array}{l}{B}_{\mu \nu }=-{\rm{i}}\left\langle {\partial }_{\mu }\hat{\psi }\right|\left.{\partial }_{\nu }\hat{\psi }\right\rangle ,\\ \mathrm{with}\quad \hat{\psi }=\displaystyle \frac{\psi }{\parallel \psi \parallel }.\end{array}\end{eqnarray}$
Moreover, as happened in equations (18) and (19), to study the Chern number arising from Bμν one similarly needs to do the two-dimensional integral on the two separate hemispheres ${S}_{N-\mathrm{hemi}}^{2}$ and ${S}_{S-\mathrm{hemi}}^{2}$, to highlight their topologically non-trivial overlap.
Equations (4) and (5) require ψ = φ1 + iφ2 = hx − ihy, implying the φ1 and φ2 have the same definition as in equation (20), φ1 = hx, φ2 = −hy. Thus, correspondingly, the Berry connection aμ and curvature Bμν have the same expressions as in equation (22): ${a}_{\mu }={\epsilon }_{{AB}}\tfrac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel }{\partial }_{\mu }\tfrac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel }$ and ${B}_{\mu \nu }=2{\epsilon }_{{AB}}{\partial }_{\mu }\tfrac{{\phi }_{A}}{\parallel {\boldsymbol{\phi }}\parallel }{\partial }_{\nu }\tfrac{{\phi }_{B}}{\parallel {\boldsymbol{\phi }}\parallel },\,$$ A,B=1,2$. A same two-dimensional topological current can be defined, $j\left({\boldsymbol{k}}\right)=\tfrac{1}{8\pi }{\epsilon }^{\mu \nu }{B}_{\mu \nu }$, which has the inner structure
$\begin{eqnarray}j\left({\boldsymbol{k}}\right)={\delta }^{2}\left({\boldsymbol{\phi }}\right)D\left(\displaystyle \frac{\phi }{k}\right).\end{eqnarray}$
According to equations (24)–(30), from the δ-function arise the meron defects, with topological charges
$\begin{eqnarray}\begin{array}{l}{c}_{1,N}=\displaystyle \sum _{l=1}^{{L}_{N}}{{ \mathcal W }}_{l,N}=\displaystyle \sum _{l=1}^{{L}_{N}}{\beta }_{l,N}{\eta }_{l,N},\\ {c}_{1,S}=\displaystyle \sum _{l=1}^{{L}_{S}}{{ \mathcal W }}_{l,S}=\displaystyle \sum _{l=1}^{{L}_{S}}{\beta }_{l,S}{\eta }_{l,S}.\end{array}\end{eqnarray}$
As far as the typical example of square lattice is concerned, ${h}_{x}=\sin {k}_{x}$, ${h}_{y}=\sin {k}_{y}$ and ${h}_{z}=m+\cos {k}_{x}+\cos {k}_{y}$, it is predictable that the analysis for the merons and their winding numbers will exactly be the same as equations (34)–(36) in section 4.2, so detailed discussion is ignored here.

6. Higher Chern number insulator and corresponding topological defects

The model studied in section 4.2 is a Chern insulator, i.e. a quantum anomalous Hall system, which has been proposed as the basis for designing the interconnects in integrated circuits due to the existence of dissipationless edge channels [6, 20]. The performance of interconnecting devices highly relies on the contact resistance between normal metal electrodes and edge channels, as expressed by h/(Ce2) [21, 22]. A high-C lattice structure is able to lower the contact resistance and significantly improve the performance of the devices. In this section we aim to propose a two-band model to achieve high Chern numbers ∣C∣ > 1.
The Hamiltonian reads
$\begin{eqnarray}\begin{array}{l}{h}_{x}=\sin 2{k}_{x},\\ {h}_{y}=\sin 2{k}_{y}+\sin {k}_{y},\\ {h}_{z}=m+\cos 2{k}_{x}+\cos 2{k}_{y}+\cos {k}_{y}.\end{array}\end{eqnarray}$
Physically, such a model can be realized in a rectangle lattice with nearest-neighbor and next-nearest-neighbor hoppings.
As presented in section 4.2, the Chern number C can be obtained through studying the monopole and meron defects.
i

(i)Monopoles: Monopole defects occur at the Dirac points with E = ∥h∥ = 0. The solutions of the zero point equations

$\begin{eqnarray}{h}_{x}={h}_{y}={h}_{z}=0\end{eqnarray}$

give the locations of the monopole defects. The solutions of equation (44) are given in table 4, with respect to equation (43). These Dirac points inevitably get involved in the integral (13), which forces C to take an indeterminate value when m = 2, 1, 0, −1, −3.
Table 4. Locations of the monopole defects of the high-C model, and the corresponding m values, where hx = hy = hz = 0.
i

(ii)Merons: Meron defects occur at the two-dimensional singular points where

$\begin{eqnarray}{h}_{x}={h}_{y}=0,\qquad E={h}_{z}=\pm \parallel {\boldsymbol{h}}\parallel \ne 0.\end{eqnarray}$

The solutions of equation (45) are given in table 5, due to equation (43).
Table 5. Locations of the meron defects of the high-C model, and the corresponding m values, where hx = hy = 0 and $E={h}_{z}=\pm \sqrt{\parallel {\boldsymbol{h}}\parallel }\ne 0$.
The topological charge of every meron can be found by recognizing its winding number, from the configurations of the two-dimensional vector $\left({h}_{x},{h}_{y}\right)$ in the respective neighborhoods of the poles N and S. The topological charge evaluations are summized in table 6, and the winding numbers related to $\left({h}_{x},{h}_{y}\right)$ are illustrated in figure 7.
Figure 7. Merons in the high-C model, marked as red spots: (0, 0), (0, π), (π, 0) and (π, π) are the source points, each having a winding number +1; (π/2, 2π/3), (π/2, 4π/3), (3π/2, 2π/3) and (3π/2, 4π/3) are the congruence points, each having a winding number +1; (0, 2π/3), (0, 4π/3), (π/2, 0), (π/2, π), (π, 2π/3), (π, 4π/3), (3π/2, 0) and (3π/2, π) are the saddle points, each having a winding number −1.
Figure 8. The different evaluations of the Chern number C of the high-C model due to varying on-site energy m. This figure is produced by directly substituting equation (43) into (13) and plotting the result of the integral.
Table 6. The winding numbers at the meron defects in the high-C model.
With respect to table 5, we list out the topological charges of the meron defects as well as their respective contributions to the north pole N and the south pole S.

When m > 2: all topological charges have contributions to the north-pole N, but no contributions to the south-pole S. Thus, CN = CS = 0.

When 1 < m < 2: the topological charges of (0, 0), (0, 2π/3), (0, π), (0, 4π/3), (π/2, 0), (π/2, π), (π, 0), (π, 2π/3), (π, π), (π, 4π/3), (3π/2, 0), (3π/2, π) have contributions to the north-pole N, while the others have contributions to the south-pole S. Thus, CN = −4, CS = +4.

When 0 < m < 1: the topological charges of (0, 0), (0, 2π/3), (0, π), (0, 4π/3), (π/2, 0), (π, 0), (π, 2π/3), (π, π), (π, 4π/3), (3π/2, 0) have contribution to the north-pole N, while the others have contribution to the south-pole S. Thus, CN = −2, CS = + 2.

When −1 < m < 0: the topological charges of (0, 0), (0, π), (π/2, 0), (π, 0), (π, π), (3π/2, 0) have contributions to the north-pole N, while the others have contributions to the south-pole S. Thus, CN = +2, CS = −2.

When −3 < m < −1: the topological charges of (0, 0), (π, 0) have contributions to the north-pole N, while the others have contributions to the south-pole S. Thus, CN = +2, CS = −2.

When m < −3: all the topological charges have contributions to the south-pole S, but no contributions to the north-pole N. Thus CN = CS = 0.

In summary, the above monopole and meron topological charges together present the evaluations for the Chern numbers CN and CS:
$\begin{eqnarray}\begin{array}{l}\mathrm{On}\,\mathrm{north}\,\mathrm{hemisphere}({\hat{h}}_{z}=+1):\\ {C}_{N}=\left\{\begin{array}{lll}0, & \mathrm{when} & m\lt -3,\,m\gt 2;\\ -4, & \mathrm{when} & 1\lt m\lt 2;\\ -2, & \mathrm{when} & 0\lt m\lt 1;\\ +2, & \mathrm{when} & -3\lt m\lt -1;\\ \mathrm{indeterminate}, & \mathrm{when} & m=2,1,0,-1,-3.\end{array}\right.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{On}\,\mathrm{south}\,\mathrm{hemisphere}({\hat{h}}_{z}=-1):\\ {C}_{S}=\left\{\begin{array}{lll}0, & \mathrm{when} & m\lt -3,\,m\gt 2;\\ +4, & \mathrm{when} & 1\lt m\lt 2;\\ +2, & \mathrm{when} & 0\lt m\lt 1;\\ -2, & \mathrm{when} & -3\lt m\lt -1;\\ \mathrm{indeterminate}, & \mathrm{when} & m=2,1,0,-1,-3.\end{array}\right.\end{array}\end{eqnarray}$
It is addressed that the CS, which has ${\hat{h}}_{z}$ = −1 and corresponds to the valence band, gives the Hall conductance equation (31). The CN-evaluation in this section agrees to the integration result of direct substitution of equation (43) into (13), as shown in figure 8.

7. Conclusion

In this paper the two-band model of Chern insulators is studied. We propose a gauge theory based on the so-called 't Hooft monopole model, where a U(1) Maxwell electromagnetic sub-field fμν is constructed from an SU(2) gauge field, as in equation (12). Using the Hamiltonian vector and Bloch wave function as the basic fields, we re-express the inner structure of the induced gauge potential and then analyze the second term, Kμν, and the first term, Bμν, of fμν. These two terms give rise to the two types of topological defects, monopoles and merons. It is shown that the monopoles cause indeterminate evaluation of Chern number C, while the merons cause different evaluations of C due to the varying on-site energy m. We examine the example of square lattice and achieve the results in perfect agreement to the data of literature [9]. Furthermore, a two-band model with higher Chern number ∣C∣ > 1 is proposed. The topological defects give the same results as the traditional method, implying our method is promising to be extended to other more complicated lattice structures.

The authors XL and ZC acknowledge the financial support from the Natural Science Foundation of Beijing Grant No. Z180007 and the National Science Foundation of China Grant No. 11572005. WH acknowledges the support from the National Science Foundation of China Grant No. 11874003 and Grant No. 51672018.

1
Hasan M Z Kane C L 2010 Colloquium: topological insulators Rev. Mod. Phys. 82 3045 3067

DOI

2
Qi X-L Zhang S-C 2011 Topological insulators and superconductors Rev. Mod. Phys. 8 1057 1110

DOI

3
Wang X L 2008 Proposal for a new class of materials: spin gapless semiconductors Phys. Rev. Lett. 100 156404

DOI

4
könig M 2007 Quantum spin Hall insulator state in HgTe quantum wells Science 31 766 770

DOI

5
Chang C-Z 2013 Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator Science 340 167 170

DOI

6
Wang J 2013 Quantum anomalous Hall effect with higher plateaus Phys. Rev. Lett. 111 136801

DOI

7
Fradkin E 2013 Field Theories of Condensed Matter Physics 2nd edn Cambridge Cambridge University Press

8
Zhang S C Hansson T H Kivelson S 1989 Effective-field-theory model for the fractional quantum Hall effect Phys. Rev. Lett. 62 82 85

DOI

9
Qi X-L Huges T L Zhang S-C 2008 Topological field theory of time-reversal invariant insulators Phys. Rev. B 78 195424

DOI

10
Nakahara M 2018 Geometry, Topology and Physics 2nd edn Boca Raton, FL CRC Press

11
Faddeev L Niemi A J 1999 Partially dual variables in SU(2) Yang-Mills theory Phys. Rev. Lett. 82 1624 1627

DOI

12
Cho Y M Pak D G 2001 Magnetic confinement in QCD arXiv:hep-th/9906198

13
Duan Y-S Liu X Zhang P-M 2003 Mermin-ho vortices and monopoles in three-component spinor BEC J. Phys. A: Math. Gen. 36 563 571

DOI

14
't Hooft G 1974 Magnetic monopoles in unified gauge theories Nucl. Phys. B 79 276 284

DOI

15
Wu T T Yang C N 1975 Concept of nonintegrable phase factors and global formulation of gauge fields Phys. Rev. D 12 3845 3857

DOI

16
Duan Y Liu X Zhang P 2002 Decomposition theory of the U(1) gauge potential and the London assumption in topological quantum mechanics J. Phys.: Condens. Matter 14 7941 7947

DOI

17
Li X Liu X Huang Y-C 2017 Tackling tangledness of cosmic strings by knot polynomial topological invariants Int. J. Mod. Phys. A 32 1750164

DOI

18
Qi X-L Wu Y-S Zhang S-C 2006 Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors Phys. Rev. B 74 085308

DOI

19
Duan Y S Liu X Fu L B 2003 Spinor decomposition of SU(2) gauge potential and spinor structures of Chern–Simons and Chern density Commun. Theor. Phys. 40 447 450

DOI

20
Zhang X Zhang S-C 2012 Chiral interconnects based on topological insulators Proc. SPIE 8373 837309

DOI

21
Zhao Y-F 2020 Tuning the Chern number in quantum anomalous Hall insulators Nature 588 419 423

DOI

22
Chang C-Z 2015 Zero-field dissipationless chiral edge transport and the nature of dissipation in the quantum anomalous Hall state Phys. Rev. Lett. 115 057206

DOI

Outlines

/