Zhi-Wen Chang, Wei-Chang Hao and Xin Liu,?

1 Institute of Theoretical Physics, Faculty of Sciences, Beijing University of Technology, Beijing 100124,China

2 School of Physics, Beihang University, Beijing 100191, China

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 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.

Keywords: Chern insulator, Chern number, monopole, meron

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 [1–3] based on the quantum spin Hall effect [4] and quantum anomalous Hall effect [5].

A TI has a quantized Hall conductance,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 σxythe performance of the TI interconnect devices;a larger C leads to a larger σxyand 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.

Figure 1.Topological map from the base manifold (k-space,Brillouin zone) to the group space, T 2 →S 2.

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

h as an SU(2) Lie algebraic vector can be expanded onto the Clifford algebraic basis{I,σ}, with I the identity andσ= (σx,σy,σz) the Pauli matrices:

The Euler–Lagrangian equation is given by

Regarding it as an eigen equation, one has the eigenvalues giving the energy levels,E=?±‖h‖ ;usually one takes?=0, thusE= ±‖h‖.The eigen-functions give the Block wave functions:

? hx=hy=hz=0, yielding

corresponding to the center O of the sphere S2in 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‖h‖ ≠0and

where anis introduced for convenience,Equations (7) and (8) correspond respectively to the north-pole N and the south-pole S of the sphere S2in 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.

3.Gauge theory

For the purpose of studying the topological defects let us start with the Weyl spinor Ψ,which has a field distribution over the base manifold T2.Ψ induces anSU(2) covariant derivative DμΨ acting as a parallel field on the manifold:

fμνis understood as aU(1)sub-field of theSU(2)tensor Fμν.

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

withdenoting a pull-back operation.The significance of C lies in its relationship to the Hall conductance σxy=Ce2/h.

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-formcannot be pulled back to a triple integral, because T2lacks the third coordinate beyond kxand 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 Kμνtoo.In this regard we need to change the angle of view from three dimensions to two dimensions, by rewriting Kμνas a U(1) field tensor:

Wμis the so-called Wu–Yang potential[15],constructed with two unit vectorsandon the surface of theformed S2,as shown in figure 3:

Figure 3.Orthonormal frame {}: the unit vector forms a 2-sphere S2 in theSU (2) group space, whileand 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.

According to equations (7) and (8), the meron defects occur at the north pole N, where hx=hy=0 andE=hz=‖h‖≠ 0(corresponding to the conductance band),and at the south-pole S, where hx=hy=0 andE=hz= ?‖h‖≠ 0(corresponding to the valence band).Figure 4 illustrates the h-distribution in the neighborhood of N.The winding number of the vector(hx,hy) in two dimensions gives the topological charge of the meron, as detailed below.The situation for the south-pole S is similar.

The winding number is computed as follows.Introducing a two-dimensional field

the orthonormal vectorsandof equation (17) can be evaluated as

Figure 4.Neighborhood of the pole N,where the vector h is parallel to the axis, with hx=hy=0.The winding number of the vector(h x , hy) in the two-dimensional vicinity of N gives the topological charge of the meron.

The Chern number on the north hemisphere becomes

where d2k=dkx∧dky, and

is a so-called two-dimensional topological current [16, 17].CScan be similarly obtained.

The δ-function does not vanish only at the zeroes of the vector field φ, which requires solving the zero equations:

According to the implicit function theorem,under the regular conditionequation (27) has a number LNcopies of isolated solutions: kμ=kμ,l, μ=x, y; l=1, 2, …, LN,corresponding to the LNmerons in two dimensions.

The δ-function can be further expanded onto the merons as

Wl,Ndenotes the winding number of the lth defect, with βl,Nas 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,

Similarly, for the south hemispherewe have

whereWl,Sis the winding number, βl,Sthe Hopf index and ηl,S=±1 the Brouwer degree.

It should be addressed that CNand CSare not additive,becauseandare 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]

where CSis 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,

wherem?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) Monopoles: They occur at the Dirac points withE=‖h‖= 0.The solutions of the zero point equations

give the locations of the monopole defects:

?(kx,ky) =(0 , 0), with m=?2, due to equation (32);

?(kx,ky) = (0 ,π) and (π, 0), with m=0;

?(kx,ky) = (π,π), with m=2.

(0, 0), (0,π), (π, 0) and (π,π) 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) Merons: They occur at the two-dimensional singular points where

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 = ±≠ 0.

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 = ±≠ 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(hx,hy) in the neighborhoods of the poles N and S.

The asymptotic behavior of hxand hyaround kx,ky=0,π is observed via a Taylor expansion:

Thus,

Table 2.Winding numbers at the meron defects of the |C|=1 model, and their total contribution to the north pole N =+1).

Table 2.Winding numbers at the meron defects of the |C|=1 model, and their total contribution to the north pole N =+1).

?

Table 3.Winding numbers at the meron defects of the |C|=1 model, and their total contribution to the south pole S =?1).

Table 3.Winding numbers at the meron defects of the |C|=1 model, and their total contribution to the south pole S =?1).

?

The above winding numbers are illustrated by a plot for(hx,hy) 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.

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.

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:

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

In conclusion, when 0

? When ?2

In conclusion, when ?2

? When m

In conclusion, when m

In summary,the above monopole and meron topological charges together present the evaluations for the Chern numbers CNand CS:

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.

On north hemisphere (hz= +1):

It is stressed that the CSof equation (36), which has=?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 aμis given byaμ=Aμ·where Aμis an SU(2)gauge potential.In order to study the topology arising from aμlet 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 Aμin terms of the Weyl spinor Bloch wave function [19],

and Bμνbecomes a Berry curvature,

According to equations (24)–(30), from the δ-function arise the meron defects, with topological charges

As far as the typical example of square lattice is concerned,hx= sinkx,hy= sinkyandhz=m+ coskx+ cosky, 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.

Table 4.Locations of the monopole defects of the high-C model,and the corresponding m values, where hx=hy=hz=0.

The Hamiltonian reads

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

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.

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

The solutions of equation (45) are given in table 5, due to equation (43).

hz(m) kx ky 0 π/2 π 3π/m+3 (m ≠?3) m+1 (m ≠?1) m+3 (m ≠?3) m+1 (m 2 0≠?1)2π/3 m (m ≠0) m ?2 (m ≠2) m (m ≠0) m ?2 (m ≠2)π m+1 (m ≠?1) m ?1 (m ≠1) m+1 (m ≠?1) m ?1 (m ≠1)4π/3 m (m ≠0) m ?2 (m ≠2) m (m ≠0) m ?2 (m ≠2)T ana db l Ee 5=.L ho z c=a t i±o ns of th≠e m 0.eron defects of the high-C model, and the corresponding m values, where hx=hy=0

The topological charge of every meron can be found by recognizing its winding number, from the configurations of the two-dimensional vector(hx,hy) 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(hx,hy) 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.

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

? When 0

? When ?1

? When ?3

? When m

In summary,the above monopole and meron topological charges together present the evaluations for the Chern numbers CNand CS:

It is addressed that the CS, which has=?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.

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.

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 subfield fμνis 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.

Acknowledgments

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.

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