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Topological quantum transitions in a two-band Chern insulator with n = 2

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 045601 (http://iopscience.iop.org/0953-8984/27/4/045601) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 045601 (10pp)

doi:10.1088/0953-8984/27/4/045601

Topological quantum transitions in a two-band Chern insulator with n = 2 Juntao Song1 , Yan-Yang Zhang2 , Yuxian Li1 and Qing-feng Sun3,4 1

Department of Physics and Hebei Advanced Thin Film Laboratory, Hebei Normal University, Hebei 050024, People’s Republic of China 2 SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, PO Box 912, Beijing 100083, People’s Republic of China 3 International Center for Quantum Materials, School of Physics, Peking University, Beijing 10087, People’s Republic of China 4 Collaborative Innovation Center of Quantum Matter, Beijing 100871, People’s Republic of China E-mail: [email protected] Received 28 October 2014, revised 29 November 2014 Accepted for publication 5 December 2014 Published 8 January 2015 Abstract

Based on a two-band Chern insulator with Chern number n = 2, we study the transport properties and the topological phase transition induced by either an external magnetic field or disorder. In this paper, a characteristic topological phase transition from n = 2 to n = 0, which is in sharp contrast to the plateau–plateau transition in the integer quantum Hall effect, is observed. This unique feature of the phase transition should be ascribed to the minimal two-band feature of this high Chern insulator. We prove this result by studying the transport properties of many different geometrical structures and the evolution of the Chern number in the presence of magnetic fields and strong disorder. Keywords: Chern insulator, disorder effect, topological phase transition, Anderson localization (Some figures may appear in colour only in the online journal)

as one brand of Chern insulator, was extensively studied in theory [15–19]. More inspiringly, QAHE was successfully observed in magnetic topological insulator thin films of Crdoped (Bi,Sb)2 Te3 by Chang et al [20]. In contrast to the IQHE, the Chern insulator and QSHE can exist in the absence of an external magnetic field. It is therefore natural to ask what would happen when they are exposed to an external magnetic field. Several theoretical and experimental investigations have observed the existence of a TI phase in the presence of a magnetic field below a certain threshold [21, 22] or when the time reversal symmetry (TRS) is broken [23]. In addition, topological phase transitions induced by magnetic fields has been discussed in [24]. In addition to the effect of a magnetic field, the study of the effects of disorder is also a significant subject. It is generally believed that a Chern insulator or topological insulator is robust against weak disorder. In particular, a disorder-induced topological phase (the so-called topological Anderson insulator) was found in 2009 [25]. Many research groups have focused on the underlying basis of this topological phase from different perspectives [26–33]. In particular,

1. Introduction

The integer quantum Hall effect (IQHE) can be observed only in the presence of an externally applied magnetic field. The counterpart in the absence of an externally applied magnetic field is related to a Chern insulator. Although the idea of a Chern insulator was first proposed theoretically by Haldane in 1988 [1], almost no progress was made on finding a real Chern insulator experimentally over the following period of nearly two decades. Significant insight was obtained when the idea of a Chern insulator was theoretically related to the quantum spin Hall effect (QSHE) in graphene by Kane and Mere in 2005 [2]. Almost at the same time, Bernevig et al reported the potential realization of the QSHE in HgTe/CdTe quantum wells [3], and of much note is the experimental observation of QSHE in two dimensional HgTe/CdTe quantum wells [4, 5] in 2007. In addition, with remarkable breakthroughs in three dimensional topological materials [6, 7], a new research field is beginning to thrive and much research attention has been given to the field of topological insulators (TIs) [8–14]. In recent times, the quantum anomalous Hall effect (QAHE), 0953-8984/15/045601+10$33.00

1

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the dependence of the topological Anderson insulator on different disorder types was studied by several authors of the present paper [29]. It was found that in the presence of strong disorder [34, 35], all extended states in the topological phases were localized because of Anderson localization. In recent years, Anderson localization phenomena in topological insulators have caused renewed interest, in particular, in the comparison of the differences and similarities [35, 36] between TIs and the IQHE. For the Haldane model, [1] and for most two-band Chern insulators, the conduction or valence band possesses a simple topological number n = 1. How to construct a n > 1 Chern insulator has become a goal for researchers in recent years. In 2012, a possible realization of the QAHE with a tunable Chern number n 1 was proposed theoretically in magnetic topological insulator film by Jiang et al [37]. In 2013, Fang et al [38] also reported that the large Chern number QAHE can be realized by tuning the Zeeman field, the structural distortion and the thickness of the thin film. Using firstprinciples calculations, the QAHE with a high Chern number was further uncovered in the thin film magnetic topological insulator of the Cr-doped Bi2 (Se,Te)3 by Wang et al [39]. More intriguingly, a surface Chern insulator with a high Chern number was announced on the mirror symmetric surfaces of SnTe and in particular topological transitions between different surface insulating states with different Chern numbers can be observed by tuning the direction of Zeeman field [40]. Besides, the high Chern insulator with arbitrary Chern numbers can be similarly constructed in a multi-band model [41, 42]. By considering electron–electron interaction, fractional Chern insulators with n > 1 were also reported extensively in multilayer films or multi-band models [43–48]. Even though a high Chern number can be easily realized in a multi-layer film or a multi-band model, in some sense these systems can be regarded as a simple superposition of two or more n = 1 Chern insulators. Roughly speaking, the topological behaviors of these systems are similar to those of an n = 1 Chern insulator. Recently, a minimal two-band n = 2 Chern insulator was proposed theoretically [49]. Comparing with the n = 1 Chern insulator, the two-band n = 2 Chern insulator inverts the valence and conduction bands twice in the whole Brillouin zone and thus possesses four distinct Dirac points at the corresponding inverted points. One immediate result from the features of the n = 2 Chern insulator is the emergence of two pairs of topologically protected gapless edge states propagating along the same direction when an open edge is created. Based on the discussion above, the n = 2 Chern insulator actually differs from the multi-layer film or multiband model with n = 2, though it is impossible to distinguish the difference between them from the edge states. Analogous to the localization and delocalization in the IQHE, some questions arise naturally. What will happen to the n = 2 Chern insulator when an external magnetic field or disorder is introduced? Is the quantum topological phase with n = 2 localized to n = 0 directly, or first to n = 1 and then to n = 0? Inspired by these questions, we study the topological transition of this n = 2 Chern insulator in the presence of a magnetic field and disorder. The rest of this paper

is organized as follows: In section 2, we present the topological Hamiltonian with a Chern number n = 2, the formulas used to compute the Chern number in the clean lime and in the presence of disorder, and also the conductance formula. In section 3, we first study the transport properties of both a ribbon and a cylinder and the evolutions of the topological phase diagrams in the presence of an external magnetic field and disorder. In addition, two-dimensional topological phase diagrams are given in the presence of magnetic fields and disorders, respectively, by performing the calculations on a six-terminal Hall bar and a periodic boundary torus. Finally, based on these results, a schematic phase diagram showing how this topological phase gradually degenerates to a trivial phase under the influence of a magnetic field and disorder is given. 2. Theoretical models and methods

In [49], a simple method for constructing a theoretical Chern insulator with an arbitrary topological index [50] n is presented. Here we straightforwardly adopt the Hamiltonian with a possible topological phase n = 2: Hk = 2t1 cos(kx )σ1 + 2t1 cos(ky )σ2 + m + 2t2 cos(kx + ky ) +2t3 (sin(kx ) + sin(ky )) σ3 , (1) where t1 , t2 , t3 and m are the system-correlated parameters and σα ’s represent the Pauli matrices. Following the symmetry conserving operations: T H (k)T −1 = H (−k), P H (k)P −1 = −H (−k), C H (k)C −1 = −H (k), and I H (k)I −1 = H (−k) with defining the time reversal symmetry operator T = K (K the complex conjugate), the particle-hole symmetry operator P = σ3 K, the chiral symmetry operator C = σ3 , and the inversion symmetry operator I = σ1 , it is easy to verify the breaking of all symmetries for the Hamiltonian (1), of which the topological phase, being the same to those in the Haldane model [1] and the BHZ model [3], therefore belongs to the unitary class A in the periodic classification table of topological systems [51]. The more detailed introduction about this Hamiltonian can be found in [49]. In the real space, the corresponding tight-binding Hamiltonian for the square lattice has the form: † m + Wj,s 0 H = ϕj ϕj 0 −m + Wj,p j t1 it3 † ϕj + h.c. + eiφj+ˆx,j ϕj+ˆ x t1 −it3 i it3 −it1 † + ϕj + h.c. eiφj+ˆy,j ϕj+ˆ y it1 −it3 j 0 t2 † + ϕj + h.c. eiφj+ˆx+ˆy,j ϕj+ˆ (2) x+ˆy 0 −t2 j

Here j = (jx , jy ) is the position index, xˆ and yˆ are unit vectors along the x and y directions. ϕj = (ϕj,s , ϕj,p )T represent two annihilation operators of the electron on the site j with the band indices s and p, respectively. Wj,s and Wj,p represent random onsite disorders, which is uniformly distributed in the range [− W2 , W2 ] with the disorder strength W . The effect of an 2

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J Song et al

(b)

L

N

L

N N

(c) N

2

N

(d)

3

1

4 6

5

Figure 1. The schematic diagrams of (a) a ribbon, (b) a cylinder, (c) a six-terminal Hall bar and (d) a torus in real space. Note that the red region represents central region or sample, where disorder is considered in the following calculations, and the green regions represent the external infinite leads. r − retarded/advanced Green’s function, and L/R = i( L/R r/a a L/R ) with L/R being the retarded/advanced self energy due to leads. In addition, a standard structure of a six-terminal Hall bar, as shown in figure 1(c), is adopted to obtain the transverse resistance Rxy = V23 /I14 , where V23 represents the voltage difference between the 2nd and 3rd terminals and I14 denotes the current from the 1st terminal to the 4th terminal. The detailed numerical method can be found in [21] and [55]. When the disorder effect is included, the translation symmetry is broken and thus it becomes impossible to compute the Chern number directly by the formula (3) derived in the clean limit. In recent years, a non-commutative theory [56, 57] of the Chern number is developed in aperiodic systems and has been proved to be very powerful in extensive investigations [35, 36, 57–60]. In the thermodynamic limit, the real-space formula of the non-commutative Chern number can be written as [58]:

(5) n = 2πi dρω Tr Pω − i[x, Pω ], −i[y, Pω ] .

external magnetic field is also included in the hopping terms through the flux-induced phase φj,j = (jx − j x )(jy + j y ) = j e/¯h j A · dl with A being the magnetic vector potential. In general, the Chern number of insulator can a Chern 1 2 d k F where F = be represented as [52]: n = 2π z BZM × A is the Berry curvature, A = i m=1 µm (k)|k |µm (k) is the Berry connection, |µm (k) is the eigenfunctions of Hamiltonian, and M represents the total number of occupied bands |µm (k). Because the Berry connection A is dependent on the gauge chosen, sometimes it makes the numerical implementation of the above formula difficult. A more convenient formula for the Chern number is:

i d2 k Tr Pˆk [∂kx Pˆk , ∂ky Pˆk ] , (3) n= 2π BZ

M where Pˆk = m=1 |µm (k)µm (k)| denotes the projector onto the occupied bands. Due to mutual cancellation of the uncertain phase factors between the eigenfunctions, the projector Pˆk is independent of the gauge. As such, the formula (3) has significant advantages in numerical calculations. Furthermore, from the real space expression of Hamiltonian (2), it is easy to describe a ribbon or cylindrical structure as shown in figure 1. Using the Landauer–B¨uttiker formula, the linear conductance of these two structures at zero temperature and low bias voltage can be represented as [53, 54]: GLR =

e2 e2 T = Tr[L Gr R Ga ], h h

Note that dρω represents the probability measure of a definite disorder configuration ω, Pω denotes the projector onto the occupied states of the Hamiltonian (2), and x and y are the position operators. To perform the numerical computation conveniently, a finite size must be chosen. Given that an N × N lattice with periodic boundary conditions, forming a real-space torus as shown in figure 1(d), is adopted, the formula of the non-commutative Chern formula can be expressed as follows [58, 59]: 2π i j, α Pω − i[x, Pω ], −i[y, Pω ] j, α , (6) n= 2 N

(4)

where T = Tr[L Gr R Ga ] is the transmission coefficient from the left lead (source) to the right lead (drain), Gr/a is the

j,α

3

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4

(a)

2

(c)

0

A

C

B

D

A B C D

0.6

|φ(y)|2

E

0.8

−2

0.4 0.2 0

−4

0

(b)

20

40

60

80

y

E

2 0

−2 −4 −1

−0.5

0

0.5

1

kx (π) Figure 2. (a) and (b) correspond to the energy spectra of the ribbon and cylindrical structures, respectively. (c) is the space distribution of the edge states for the ribbon structure, which corresponds to the blue lines in figure 2(a). The parameters are set to m = −1.4, t1 = 1, t2 = 1, t3 = 0.5 and the ribbon or cylindrical width N = 80.

where |j, α represents the α-th orbital’s projection on the j -th lattice site, and j,α sums over N × N lattice sites j and all orbitals α, respectively. It should be pointed out explicitly that although x or y in equation (6) denotes the position operator and in the strict sense should have the same physical meaning as well as that in the thermodynamic limit, the periodic boundary condition in the numerical implementation makes the definition of the position operator not straightforward. The detailed solution has been given in [59], and also proved to be very efficient by a number of investigations [35, 36, 57–60].

corresponding quantum conductance value |n|e2 / h can be observed inside the energy gap of ribbon structure, as shown figure 3(a). However, for the cylindrical case, all the quantum conductances are zero, as shown in figure 3(b), which is in good consistent with the changes in the band spectrums shown in figure 2. To distinguish more clearly the different topological phases, in figure 4 the conductance is also shown as a function of the parameter m for four definite different Fermi energies. As demonstrated in figure 4(a), four distinct m ranges, namely (−4, −2), (−2, 0), (0, 4) and (4, 5), corresponding to different conductance features, can be recognized. The quantum conductance value at E = 0.1 states clearly that these four distinct m ranges represent the four topological phases, n = 0, n = ±2, n = ±1, and n = 0, respectively. The increase in the conductance for other energies is due to the shift of the Fermi energy into the bulk states. As illustrated in figure 3(b), all transport contributions of gapless states likewise disappear for the cylindrical structure in figure 4(b).

3. Results and discussions 3.1. Band spectra and transport properties for ribbon and cylindrical structures

At first, let us study the spectra for the ribbon and cylindrical structures, which are infinite along the horizonal direction, as shown in figures 1(a) and (b). In figure 2, it can be seen that when the chosen parameters cause the system to lie within the n = −2 topological phase, two pairs of gapless edge states appear on the edges of the ribbon as shown in figure 2(a). Note, however, that these gapless states are absent for the cylindrical geometry. This characteristic is a hallmark of a topologically nontrivial phase. In particular, it should be pointed out that two right-going states are located on one edge of the ribbon while the other two left-going states lie at the other edge of the ribbon, as shown in figure 2(c). The spatial separation of left-going and right-going states makes backscattering between them very difficult, which implies that the n = 2 topological phase and its edge states are robust against disorder. In figure 3, the conductance is plotted for both ribbon and cylindrical structures. When the system is located within different phases with n = −2, n = −1, and n = 0, a

3.2. Effect of an external magnetic field

One of the main focuses in this paper is how the topological phases, especially the n = −2 phase, evolve in the presence of an external magnetic field. In figure 5, we compare the behaviors of the band spectra of a ribbon in the absence or presence of an external magnetic field. Obviously, in figure 5(b) two pairs of gapless states that are present in the absence of a magnetic field in figure 5(a) are destroyed completely and a finite energy gap emerges nearby E 0.3 in the presence of = 0.002π . When the parameters m = −1.4 and t3 = 0.82 are used, as in figures 5(c) and (d), the behavior is distinctly different. Clearly, the external magnetic field destroys only one pair of gapless edge states and keeps the other 4

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14 (a)

(b)

12

m=−3 m=−1 m=3 m=5

G (e2/h)

10 8 6 4 2 0 −1.5 −1.0 −0.5

0 E

0.5

1

−1.5 −1.0 −0.5

0 E

0.5

1

1.5

Figure 3. The conductance for (a) a ribbon and (b) a cylinder. The parameters are set to t1 = 1, t2 = 1, t3 = 0.5 and the ribbon or cylindrical width N = 80.

12

(a)

10

(b)

E=0.8 E=0.5 E=0.1 E=−0.2

G (e2/h)

8 6 4 2 0 −4 −3 −2 −1

0

m

1

2

3

4

5 −3 −2 −1

0

m

1

2

3

4

5

Figure 4. The conductance for (a) a ribbon and (b) a cylinder. Other parameters are set to t1 = 1, t2 = 1, t3 = 0.5 and the ribbon or cylindrical width N = 80. 2

2 (a)

E

1

(b)

0

0

−1

−1

−2

−2

(d)

(c)

1

E

(f)

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kx (π)

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(g)

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0

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(h)

−0.5

0

kx (π)

kx (π)

0.5

−1

−0.5

0

0.5

1

kx (π)

Figure 5. The energy spectra for a ribbon for (a) m = −1.8, t3 = 0.5, = 0; (b) m = −1.8, t3 = 0.5, = 0.002π; (c) m = −1.4, t3 = 0.82, = 0; (d) m = −1.4, t3 = 0.82, = 0.001π; (e) m = −1.4, t3 = 0.5, = 0; (f ) m = −1.4, t3 = 0.5, = 0.02π; (g) m = −0.6, t3 = 0.5, = 0; (h) m = −0.6, t3 = 0.5, = 0.02π. Other parameters are set to t1 = 1, t2 = 1, and the ribbon width N = 80.

5

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Figure 6. Contour plot of the reciprocal of Rxy for (a) m = −1.4 and (b) m = −0.6. Other parameters are set to t1 = 1, t2 = 1, t3 = 0.5 and the ribbon width N = 60.

EF =1.1

4 (a)

(b)

m=−1.6 m=−1 m=−0.2 m=0.2

EF =0.8 EF =0.5

G (e2/h)

3

EF =0.1 2 1 0 0

2

4

6

8

0

W

2

4

6

8

10

W

Figure 7. The effect of disorder on a ribbon’s conductance (a) m = −1 and (b) E = 0.01. Other parameters are set to t1 = 1, t2 = 1, t3 = 0.5, the ribbon width N = 60 and the length of the disordered central region L = 100. Note that the result is averaged over 100 random disorder configurations.

n = 0 phase represented by 1/Rxy = 0. Similarly, when the parameter m increases to m = −0.6 as in figure 6(b), a distinct topological phase transition from n = −2 (1/Rxy = −2) to n = −1 (1/Rxy = −1) can be also seen. To sum up, an external magnetic field can induce distinct topological phase transitions but which kind of topological phase transitions actually appear depends on the system parameters.

pair intact as shown in figure 5(d), which is in sharp contrast with the case of figure 5(b). These observations demonstrate that an external magnetic field not only can induce a topological phase transition from n = −2 to n = 0 but also can trigger a topological phase transition from n = −2 to n = −1. The same conclusion can be obtained by analyzing figures 5(e)–(h). In order to illustrate unambiguously the effect of magnetic fields, we plot a two dimensional phase diagram of 1/Rxy with respect to the magnetic flux and Fermi energy E, where Rxy is numerically calculated by using a six-terminal Hall bar structure [21, 55] as illustrated in figure 1(c). As shown in figure 6(a), when the external magnetic field is increased, an n = −2 phase, which could be simply regarded as the region with 1/Rxy = −2(e2 / h), is transformed to the

3.3. Effect of disorder on conductance and Chern topological phase diagram

When disorder is included in the system, topological phase transitions are also observed. In figure 7(a), when the Fermi energy E = 1.1 is located at the conductance band, the 6

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Figure 8. The phase diagram of the Chern number n in the two dimensional parameter space (m,t1 ). The disorder strength is set to (a) W = 0, (b) W = 6, (c) W = 8, (d) W = 9 and (e) W = 12. Other parameters are set to t1 = t2 = 1 and the lattice sites N 2 = 60 × 60. Note that the result is averaged over 10 different random disorder configurations.

bulk states are gradually localized and the conductance finally decreases to zero as the disorder strength W increases. For other cases where the Fermi energies lie inside the bulk gap with gapless edge states, the conductance is also reduced from the quantized value to zero as the disorder strength become larger than a threshold. More importantly, the topological phase with n = −2 falls directly to the n = 0 phase, and no intermediate n = −1 phase appears between them. This observation regarding the effect of disorder is very different from that in the case of an external magnetic field.

This conclusion is also confirmed in figure 7(b) for four different m values but with a fixed Fermi energy E = 0.01. No obvious n = −1 topological phase is observed in the phase transition from n = −2 to n = 0. In these cases, it is reasonable to expect that the energy gaps between valence and conduction bands host different sizes at different Dirac points (e.g. see figure 5(c)). Given that the mobility gap at one Dirac point first closes and then reopens with increasing disorder strength, while the mobility gap at other Dirac points always remains unclosed, a intermediate n = −1 topological 7

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(a)

Insulating phase

xx

C =0 C =1 C=2 C =4

C =3

1/4 1/3

1

1/2 2

(2h/e )

xy

(b)

Insulating phases

(c)

Insulating phases

C =0 B

W

C =0

C =-1

C =-2

C =-2

C =-1

C =-1 System parameter

C =-1 System parameter

Figure 9. (a) is the phase diagram for the IQHE. ρxx and ρxy can be simply regarded as microscopic measures of the strengths of the disorder and the magnetic field, respectively. (b) and (c) are schematic phase diagrams for the high Chern insulator considered in this paper.

is that no n = −1 topological phase is inserted between the n = −2 and n = 0 topological phases during this topological phase transition. All topologically nontrivial phases fall into the n = 0 phase under the influence of disorder. This result is in good consistent with the observations in figure 7. Because the disorder can not distinguish the topological characteristics of the valence or conductance band and tune them separately at different Dirac points, the topological property would be completely destroyed once the band gap is closed at any one of these Dirac points. Note that this behavior is very different from what occurs with an external magnetic field. However, whether there is a topological phase transition from n = ±2 to n = ±1 in other Chern insulators is still worth studying. Based on the observations and results given above, we can obtain a schematic phase diagram for the topological phase transitions of a high Chern insulator, as shown in figures 9(b) and (c). For comparison, the schematic phase diagram [61] for the IQHE is also shown in figure 9(a). In the plateau– plateau transitions in the IQHE, topological phase with C = n is always adjacent to those phases with C = n ± 1. Therefore, no matter whether an external magnetic field (being simply regarded as ρxy ) or disorder (being simply regarded as ρxx ) is introduced, a topological phase transition from C = n to C = n ± 2 never happens. However, this conclusion is overturned for the topological phase transitions of a high Chern insulator, as studied in this paper. In figure 9(b), an external magnetic field can cause not only a topological phase transition between two adjacent Chern numbers which case is similar to that of the IQHE, but also one from n = −2 to n = 0, which generally is repelled in the IQHE. Much more intriguingly, the first kind of topological phase transition disappears completely

phase will appear during the topological phase transitions from n = −2 to n = 0. However, this intuition is not justified in figure 7(b). In other words, for the two-band Chern insulator considered in this paper, once the mobility gap is closed at one of the Dirac points due to disorder, then the system becomes topologically trivial. It is worth noting that this observation is very different from that of plateau–plateau transitions in the IQHE. For the sake of completeness, a phase diagram in the two dimensional parameter space (m,t3 ) is provided in figure 8. Using the formula (3) of the Chern number, we first compute the phase diagram in the absence of disorders in figure 8(a). However, in the presence of disorders, the following computation of the Chern number could be only performed by the non-commutative Chern formula (6). When the disorder strength, W = 6, is relatively small in figure 8(b), different topological phases remain almost unchanged except for minor disorder-induced fluctuations on the boundaries of the topological phases. When the disorder strength increases further to W = 8 in figure 8(c), the n = −2 topological phase decays to a very small region, the phase boundary become very vague. For the case of W = 9 in figure 8(d), the stable topological phase with the perfect quantized value n = −2 is destroyed completely and replaced by a disorder-induced unstable phase. What is more important is that the topological phase with a quantized value n = −1 also begins fading away from the n = −2 phase. For a very strong disorder W = 12 in figure 8(e), the sign of the n = −2 topological phase could not be seen any more and similarly the n = −1 phase is greatly reduced. Undoubtedly, all topological phases will be localized at sufficiently strong disorder [34]. What fascinates us greatly 8

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and only the second kind of topological phase transition is seen in the presence of disorder alone, as shown in figure 9(c). This characteristic topological phase transition of high Chern insulators is probably rooted in the characteristics of the minimal two-band spectrum. Whether disorder-induced phase transitions between adjacent Chern numbers such as that in the IQHE also exist in high Chern insulators needs to be further studied in other topological systems, e.g. thin films of three dimensional topological insulators.

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4. Conclusions

In summary, we have studied transport properties and quantum topological transition of a two-band Chern insulator with n = 2. The characteristic phase transitions of high Chern insulators are seen in the presence of either an external magnetic field or disorder, which is in sharp contrast to the case of the integer quantum Hall effect. In particular, a high Chern insulator with topological number n = −2 could decrease directly to the n = 0 phase in the cases of both an external magnetic field and disorder. Because most information of topological phase transitions is mainly based on the studies on the integer quantum Hall effect in the past decades, the result in this paper is very helpful to further deepen the understanding of topological phase transitions. It should be also pointed out that the important question as to whether a disorderinduced topological phase transition between two adjacent Chern numbers can actually occur is still unsolved and requires further study. Acknowledgments

We gratefully acknowledge the financial support from NSFChina under Grant Nos. 11204065(JTS), 11474085(JTS), 11374294(YYZ), 10974043(YXL), 11274364(QFS) and NBRP of China under Grant No. 2012CB921303(QFS). JTS is also supported by NSF-Hebei Province under Grant No. A2013205168, BJ2014038, and SRFDP-China under Grant No. 20101303120005. References [1] Haldane F D M 1988 Phys. Rev. Lett. 61 2015 [2] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 146802 [3] Bernevig B A, Hughes T L and Zhang S C 2006 Science 314 1757 [4] K¨onig M, Wiedmann S, Br¨une C, Roth A, Buhmann H, Molenkamp L W, Qi X-L and Zhang S-C 2007 Science 318 766 K¨onig M, Buhmann H, Molenkamp L W, Hughes T L, Liu C-X, Qi X-L and Zhang S-C 2008 J. Phys. Soc. Japan 77 031007 [5] Day C 2008 Phys. Today 61 19 Nagaosa N 2007 Science 318 758 [6] Hsieh D, Qian D, Wray L, Xia Y, Hor Y S, Cava R J and Hasan M Z 2008 Nature 452 970 Fu L, Kane C L and Mele E J 2007 Phys. Rev. Lett. 98 106803 [7] Zhang H, Liu C, Qi X, Dai X, Fang Z and Zhang S C 2009 Nat. Phys. 5 438 [8] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 9

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