High Chern number quantum anomalous Hall phases in graphene ribbons with Haldane orbital coupling
Abstract
We investigate possible phase transitions among the different quantum anomalous Hall (QAH) phases in a zigzag graphene ribbon under the influence of the exchange field. The effective tightbinding Hamiltonian for graphene is made up of the hopping term, the KaneMele and Rashba spinorbit couplings as well as the Haldane orbital term. We find that the variation of the exchange field results in bulk gapclosing phenomena and phase transitions occur in the graphene system. If the Haldane orbital coupling is absent, the phase transition between the chiral (antichiral) edge state () and the pseudoquantum spin Hall state () takes place. Surprisingly, when the Haldane orbital coupling is taken into account, an intermediate QSH phase with two additional edge modes appears in between phases and . This intermediate phase is therefore either the hyperchiral edge state of high Chern number or antihyperchiral edge state of when the direction of exchange field is reversed. We present the band structures, edge state wave functions and current distributions of the different QAH phases in the system. We also report the critical exchange field values for the QAH phase transitions.
pacs:
71.70.Ej, 72.25.Dc, 73.43.Nq, 81.05.ueI Introduction
The anomalous integer quantum Hall effect observed in monolayer graphenes subjected to an external magnetic field Novoselov05 ; Zhang05 has recently attracted considerable attention. A theoretical investigation Gusynin05 showed that plateaus located at the halfoddinteger position originate from an additional Landau level at zero energy Gusynin06 ; Peres06 , which is unlike the behavior of the conventional quantum Hall effect observed in twodimensional (2D) heterostructure semiconductors.Klitzing80 The quantum Hall effect has also been experimentally observed in ABstacked bilayer graphenes Novo06 , and this has been studied theoretically as well. Cann06 ; Nakamura08 Furthermore, much experimental evidence is available for the existence of AAstacked bilayer graphenes. Lauffer08 Interestingly, AAstacked bilayer graphenes has been shown to exhibit zero transverse conductivity. Fan10
According to Laughlin’s gauge invariance argument, the sample edges are essential in generating the localized currentcarrying states (edge states) Laughlin81 ; Halperin . The edge states on the sample boundary are protected by the bulk band structure topology which is a manifestation of the Chern number, as elucidated by Thouless et al. (TKNN). Thouless The TKNN integer (or Chern number) relates the topological class of the bulk band structure to the number of chiral edge states on the sample boundary (bulkboundary correspondence) and hence gives rise to the quantized Hall conductivity . The precise quantization of the Hall conductivity arises in the 2D electron system with an integer filling of the Landau levels. The Chern number corresponding to the number of the chiral edge currents equals to the number of Landau levels below the Fermi level. When the system undergoes a phase transition from one chiral edge state to another, the corresponding Chern number varies discontinuously from one integer to .
The Chern number must vanish in a system with time reversal symmetry (TRS). A TRS breaking mechanism is thus required for a 2D system to achieve a nonzero Chern number, either with or without the Landau levels. It has been shown that the chiral edge state in the quantum Hall phase is related to the parity anomaly of 2D Dirac fermions. Jackiw84 ; Fradkin86 Therefore, in a remarkable paper Haldane88 , Haldane constructed a tightbinding Hamiltonian in the 2D honeycomb lattice with a staggered magnetic field that produces zero average magnetic flux per unit cell (i.e., no Landau levels) and showed that the gapped state exhibits the quantum Hall phase with . In this sense, the Haldane model is the prototype for the quantum anomalous Hall (QAH) effect. The relationship between the Chern number and the winding number of the edge state was investigated in Ref. Hatsugai93 .
On the other hand, when the bulk band gap of a system having a spin degree of freedom is opened due to the spinorbit interaction, the system might be in the quantum spin Hall (QSH) state where the gapless edge states appearing on the sample boundary are protected by the TRS. KaneRMP10 The quantization of the spin Hall conductivity has been predicted in a graphene system with the KaneMele spinorbit interaction as well as in a semiconductor superlattice. Kane051 ; Kane052 ; Bern06 The quantization of the spin Hall conductivity, however, may be destroyed by the paritybreaking perturbations via spin nonconserving term or disorder. The associated topological invariant classifying the band structure topology of the timereversal invariant systems is a topological index Kane051 ; Kane052 . The connection between the Chern number and the topological index is explained in Ref. Moore07 . The topological number represents the number of the Kramer pairs of the gapless edge modes. An important result of this classification is that these gapless edge modes with an odd number of the Kramer pairs in the 2D systems Kane051 ; Kane052 ; Bern06 and an odd number of surface Dirac cones in the threedimensional (3D) systems Fu07 ; HZhang09 are robust to impurity scattering; the other systems are just a conventional band insulator.
Recently, another topological invariant (i.e., the spin Chern number) has been proposed by Sheng et al. Sheng05 , and can be evaluated by imposing twisted boundary conditions on a finite sample. In Ref. Fukui07 , it has been shown that the spin Chern number and topological orders would yield the same classification by investigating the bulk gapclosing phenomena in the timereversal invariant systems. The phase diagram of the 3D QSH systems has been investigated systematically. Murakami08 The topological winding number related to the spin edge states of graphene with the KaneMele Hamiltonian has also been studied. Zhang09 Furthermore, the bulkboundary correspondence is generalized to classify topological defects in insulators and superconductors, where the gapless boundary excitations are Majorana fermions. Kane10
In this paper, we first model the bulk graphene and also a zigzag graphene ribbon in the presence of the exchange field Weiss09 by using the KaneMeleRashba Hamiltonian [see Eq. (1)]. Here the spin degeneracy is lifted by the TRS breaking term (i.e., the exchange field) and the mirror symmetry is broken by the Rashba term. We calculate the Chern number of the bulk system as a function of the exchange field strength. Furthermore, we study how the edge current in the corresponding graphene ribbon varies during a phase transition induced by the exchange field.
The quantum anomalous Hall effect in HgMnTe quantum wells Liu08 and tetradymite semiconductors (BiTe, BiSe, and SbTe) Yu10 has been investigated. Graphene with the Rashba spinorbit coupling [, see Eq. (2c)] and the exchange field has also been studied before. Qiao10 However, in Ref. Qiao10 , the KaneMele spinorbit coupling [, see Eq. (2b)] was neglected because it was thought to be weaker than the Rashba spinorbit interaction. In this paper, we find that, in the presence of both the Rashba and KaneMele couplings, a phase transition from either a chiral () or antichiral () edge state () to the pseudoQSH state () would occur in the graphene ribbon, because of the change of the Chern number due to the bulk gapclosing phenomena. This phase transition is different from the transition between the QSH state and the insulator state when the exchange field is absent.
We then add the Haldane orbital coupling term which couple the electron orbital motion to the exchange field Haldane88 , to the KaneMeleRashba Hamiltonian for graphene. Interestingly, we find that this leads to an anomalous change in the Chern number pattern. Note that the Haldane orbital term does not lift the spin degeneracy. Furthermore, we find that the presence of the Haldane orbital coupling would give rise to a new intermediate phase between phases and . This intermediate phase has two new edge modes, and is thus either a hyperchiral edge state with or an antihyper chiral edge state with when the direction of the exchange field is reversed.
The rest of this paper is organized as follows. In Sec. II, we describe the effective tightbinding Hamiltonian for graphene used in this work. In Sec. III, we report the energy bands of a graphene ribbon in the presence of the exchange field. In Sec. IV, we present the phase transition and the variation of the Chern number with the exchange field in the KaneMeleRashba system. In particular, we show that the graphene ribbon undergoes a phase transition from the chiral (or antichiral) state to the pseudoQSH state. In Sec. V, we show that a hyperchiral (or antihyperchiral) state would appear in between the chiral and antichiral states in the HaldaneRashba system. The conclusions are given in Sec. VI.
Ii Effective tightbinding Hamiltonian for graphene
We consider the effective tightbinding model for graphene given by the KaneMeleRashba Hamiltonian Kane051 ; Kane052 :
(1) 
with
(2a)  
(2b)  
(2c) 
The symbols and denote the nearest neighbors and the next nearest neighbors, respectively. The Hamiltonian is the tightbinding energy for the nearestneighbor hopping. The KaneMele Hamiltonian describes the intrinsic spinorbit interaction. The sitedependent Haldane phase factor Haldane88 is defined as
(3) 
where denotes the vector from one carbon atom to one of its nearest neighbors. Two vectors and are required to represent the second neighbor hopping (see Fig. 1). In the twodimensional case, the nonzero component becomes a sign function and we take the values of (i.e., counterclockwise/clockwise). The extrinsic spinorbit interaction is described by the Rashba Hamiltonian , which can be produced by, e.g., applying an electric field perpendicular to the graphene sheet. is proportional to , where and denotes the vector from site to site (see Fig. 1).
Recent ab initio density functional calculations showed that intrinsic ferromagnetism in pure and ontopFedoped graphene monolayers may exist. Qiao10 ; Son06 Furthermore, proximityinduced ferromagnetism in graphene was recently reported. Tom07 Therefore, we consider the interaction of the 2D electrons in graphene with the exchange field produced by the ferromagnetism. Weiss09 The coupling of the orbital motion and also spin of the electrons on graphene to the exchange field would give rise to an additional Hamiltonian:
(4) 
with
(5a)  
(5b) 
where is the (rescaled) exchange field strength. The coupling is proportional to , where is the exchange interaction and is the effective magnetic moment associated with the exchange field. The magnetic field generated by is denoted as . The Hamiltonian describes the response of an electron spin magnetic moment to the exchange field à la Zeeman effect.
In the meantime, the orbital angular momentum of an electron in graphene would be coupled to the exchange field because of its associated orbital magnetic moment. The Haldane phase factor behaves like an effective orbital angular momentum, and hence gives rise to the interaction between the electron orbital motion and the magnetic field , as described by Eq. (5b), where is a function of . Spatial parity symmetry requires to be an odd function of . Unlike the interaction between the spin and exchange field, the energy of the Haldane orbital coupling cannot be linear in the exchange filed . Instead, the response of Haldane orbital motion would be saturated rapidly because the exchange field alters the orbital velocity of electrons and induces an orbital magnetic moment against it. Phenomenologically, we can adopt the simple yet sensible approximation:
(6) 
where we use instead of for simplicity since the sign function is independent of the field strength but its direction. In the present study, the sign of is fixed, and hence the sign change of corresponds to the change in the direction of , which is experimentally possible. Accordingly, we choose the constant to be negative to have a diamagnetic response to the magnetic field .
Iii Chern numbers and edge current chirality
The total Hamiltonian for graphene in the presence of the exchange field is given by . For the bulk graphene, the Hamiltonian which satisfies the periodicity ( stands for a 2D reciprocallattice vector), is given by
(7) 
where and . The state vector is represented by , where and denote the two different sublattice points in the unit cell, respectively, and the arrows represent the spin directions. The matrix elements are given by , , ,
(8) 
The Chern number is then obtained by summing the Berry curvatures for all the occupied states below the Fermi level for each and subsequently integrating over the entire first Brillouin zone:
(9) 
The bulk Hamiltonian Eq. (7) is simplified greatly if we consider the following simple systems:

KaneMele system: ;

Rashba system: ;

Haldane system: .
The sign of Chern number indicates the chirality of the edge current. To verify the occurrence of the edge currents, we compute the energy band structure for a zigzag graphene ribbon. The unit cell of the zigzag graphene ribbon is shown in Fig. 1(a), where the ribbon direction is denoted by the axis and the transverse direction is along the direction. The width of the zigzag ribbon () is , where is the bond length [see Fig. 1(b)], i.e., there are C atoms in the transverse direction [see Fig. 1(a)]. The nearest neighbor hopping integral .
Figure 2 shows the ribbon band structure and the edge state probability distribution in the KaneMele system (a), the Rashba system (b), and the Haldane system (c). The Fermi level is assumed to be above zero, as indicated by the dashed horizontal line, and thus, has four intersections with the conduction bands, denoted as A, B, C, and D, in the left panels in Fig. 2. This gives rise to four edge currents on the ribbon edges, as indicated by the A, B, C, and D arrows in the right panels in Fig. 2. The direction of an edge current, denoted by an arrow, is given by where the electron group velocity is determined using . The and states have the same velocity direction, which is opposite to that of the and states. Hereafter, we use the notation to express the charge current distributions on the lefthand side and righthand side edges, respectively. In terms of the bulkboundary correspondence, for each of the three systems, the pair A and D would form a single handed loop (the turning point is at infinity in the direction), and the pair B and C would constitute the other loop of the opposite handedness, as can be seen from the probability distribution shown in the middle panels in Fig. 2.
In the KaneMele system , the current distribution is , as indicated in the right panel in Fig. 2(a). The two edge states and are on the same edge, and so are the and states. As mentioned above, the handedness of the current loop due to the A and D edge states would produce a Chern number of while that of the pair B and C would give a Chern number of . Therefore, the KaneMele system is composed of two integer quantum Hall subsystems, namely, , Kane051 ; XLQ06 and has . Since this state has the same distribution of the edge currents as that of the quantum spin Hall case with the TRS Kane051 , except that the TRS is broken here, we call this state as the pseudoquantum spin Hall state.
In the Rashba system , the current distribution is , as shown in Fig. 2(b), which constitute a paramagnetic response to the exchange field. Both and are located at the same edge, confirming that the Rashba system has a Chern number of , since the two edge current pairs have the same chirality. Qiao10 It is important to note that both the Chern number and the current distribution in the Rashba system is invariant under the transformation . On the other hand, the current distribution becomes when the direction of the exchange field is reversed. Therefore, the Rashba system is equivalent to two integer quantum Hall subsystems, namely, for or for . Qiao10
In the Haldane system , the current distribution is , as shown in Fig. 2(c). Both and are also located at the same edge, but the chirality of the edge current is opposite to that of the Rashba system, as a result that the Haldane system with exhibits a diamagnetic response to . The Chern number of this system is . Therefore, the Haldane system, being diamagnetic, is equivalent to two integer quantum Hall subsystems, namely, for or for .
In the next two sections, we will consider the following two combinations of the three simple systems discussed in this section:

KaneMeleRashba system: .

HaldaneRashba system: .
We find that, because of the bulk gapclosing phenomena, both systems will undergo a change of the edge current chirality caused by varying the exchange field.
Iv Phase transition in the KaneMeleRashba system
In this section, we will neglect the Haldane orbital coupling term of Eq. (5b). We will find that the phase transition is different from the QSH phase transition in the presence of the exchange field. We consider the interplay between and :
(10) 
In the presence of both KaneMele and Rashba spinorbit couplings, the phase transition between the chiral (or antichiral) state and the pseudoQSH state must occur when the bulk gapclosing phenomena take place. On the other hand, although the locations of four currents become under the transformation , the phase transition between (anti) chiral and pseudoQSH states still applies because both and correspond to .
In order to verify the the phase transition in the finite system (the graphene ribbon), we use the expectation value of position (i.e., ) as a parameter for specifying the angular momentum of the current in the system, and define . When the KaneMele coupling is dominant (, see below), and are on the opposite sides of the ribbon, and thus . When the Rashba coupling is dominant (, see below), and are on the same side of the ribbon. The quantity in the Rashba dominant system, however, would not reach a saturated value owing to the finite size effect as the perfect edge states is obtained only when the ribbon width is infinite. During the phase transition, the wave function starts to mix with each other in the central region of the ribbon, and thus, is expected to deviate from either or .
Let us consider the case of , and ranging from to in Eq. (10), as an example. For , we find that decreases to zero near some magnitude of (see below) and does not change sign, as shown in Fig. 3(a). The pattern of in the region is the parity symmetry of that in . We find that the expectation value changes sign when the direction of the chiral current is reversed.
Based on the bulkboundary correspondence Thouless ; KaneRMP10 , the existence of the phase transition is supported by evaluating the critical values of the exchange field. The critical value of the exchange field () for the occurrence of the phase transition is determined by the bulk gapclosing phenomena, where the bottom of the bulk conduction band () and the top of the bulk valence band () become degenerate, namely, at . It can be shown that the degenerate point is located at . The critical value for the exchange field is given by
(11) 
which is obtained for a nonzero KaneMele coupling that satisfies and . The presence of causes the critical value for the exchange field to shift from (Rashba system) to a nonzero value. The magnitude corresponds to the location where the bulk valence and conduction bands become degenerate and the Chern number starts to jump from one integer to another.
For a system with given and , if (), the system is in the chiral current state (pseudoQSH state). When (or ), the conduction and valance bands touch at and simultaneously. Because the current direction is opposite to , the exchange of the locations of and results in a change of chirality [see Fig. 3(a)]. The corresponding variation of the Chern number is also shown in Fig. 3(a). Therefore, we find that the Chern number jumps from to . In this case, the critical values are in agreement with the numerical result.
V Phase transition in the HaldaneRashba system
In this section, we will neglect the KaneMele coupling . We will show that the presence of the Hamiltonian creates two new edge modes between the two phases. Interestingly, these intermediate states are either the hyperchiral state () or antihyperchiral state (). The Hamiltonian is given by
(12) 
As described in Sec. III, the Haldane system has and the Rashba system has when the exchange field is positive. If the phase transition occurs in this system, the bulk gapclosing phenomena must take place. In the following, we show that the HaldaneRashba system has two critical values of the exchange field.
For the sake of discussion, we consider the region with , and focus on the region because the behavior of the corresponding degenerate points in is the mirror symmetry of that in . In the presence of only, there are two degenerate points. One is at (i.e., point), and the other is at
(13) 
Eq. (13) shows that the degenerate point depends on the strength of the coupling and thus varies with the magnitude of . However, the two degenerate points appear simultaneously, namely, there is only one critical value for the exchange field, , even when the coupling is considered [as shown in Fig. 3(a)].
Unlike the KaneMele coupling , we find that the coupling will close the bulk gap twice at two different magnitudes of the exchange field. Namely, there are two degenerate points appearing at two different magnitudes of the exchange field. Therefore, the Chern number varies discontinuously from to through an intermediate state. One of the degenerate points is at the point; the corresponding critical value of the exchange field is
(14) 
However, the second degenerate point for the Hamiltonian Eq. (12) is different from expressed in Eq. (13). The second degenerate point is determined by the condition and can be expressed as
(15) 
where satisfies the following equation:
(16) 
Numerically, Eq. (16) can be solved for a given set of and . When , Eq. (13) is the solution of Eq. (16) and the critical value of the exchange field is , which is in agreement with that in the case of the Rashba system. For the present case, and . The two critical points are and . Therefore, the conduction and valence bands first touch at , and the bulk gap would reopen when (referred to as the intermediate state). The two bands would touch the second time at . The bulk gap is open when . The calculated Chern number as a function of is shown in Fig. 3(b).
Surprisingly, the intermediate state between the critical values and shows . The band structure and current distribution of the intermediate state are shown in Fig. 4. We find that in the presence of Haldane orbital effect, the system establishes two new edge modes: one is the pair and , and the other is the pair and . Furthermore, the current distribution (and ) also shows that the four currents in have the same chirality. Importantly, we find that unlike the quantum Hall plateau, the Chern number is not restricted in changing from one integer to the next integer . Instead, a higher Chern number can exist in a system with a spinorbit interaction if the orbital effect is also taken into account.
The Chern number [see Eq. (9)] can be written as . The bulk band structure and the corresponding Berry curvature along the profile are shown in Fig. 4(c) and (d), respectively, where we use , and . It can be shown that if the linear term is taken into account, the Chern number is still . This clearly shows that the anomalous Chern number is due to the second nearestneighbor hopping in graphene. We believe that if the third nearestneighbor hopping is considered in Haldance orbital effect and spinorbit interaction, a higher Chern integer of, e.g., , may be obtained.
Let us define for the intermediate state. The calculated variation of with is shown in Fig. 3(b). Apart from the occurrence of the intermediate state, the expectation value changes sign as the exchange field is swept through the phase transition, and this is accompanied by a change of the Chern number from to , as shown in Fig. 3(b). The Chern number obtained from the bulk Hamiltonian represents the number of the perfect edge states. Note, however, that because of the finite size effect, cannot reach the saturated value [see Fig. 3(b)]. Interestingly, we find that it is not necessary to reverse the direction of the exchange field in order to flip the current chirality in this case. Therefore, the graphene can be brought to either the paramagnetic phase or the diamagnetic phase by adjusting the magnitude of the exchange field.
Very recently, Tse et al. Tse11 also proposed
that the Hall conductance can be quantized as ,
, in bilayer graphene with the Rahsba coupling under the
influence of an external gate voltage. In the present work, in contrast,
we show that in singlelayer grapnene, the quantized Hall conductance can
be when the Haldane orbital effect is
considered.
Furthermore, we find that the change
in the quantized Hall conductance can be achieved by varying the
exchange field instead.
Vi Conclusions
In summary, we find that the edge current chirality in a graphene ribbon can be flipped by varying the exchange field. The resultant phase transition of the current chirality is caused by the bulk gapclosing phenomena; that is, the phase transition is due to the topological effect of the bulk band structure of graphene. We show that the paramagnetic response in the Rashba system can exhibit ether the chiral or antichiral state, and thus, the Hall conductance is quantized as . We find that the KaneMele system has the pseudoQSH state. For the KaneMeleRashba system, the transition between the chiral (or antichiral) current state and the pseudoQSH state can be achieved by varying the strength of the exchange field.
Unlike the Rashba system, the Haldane system exhibits a diamagnetic response to the exchange field, and the quantized Hall conductivity is . However, the competition between and leads to a phase transition between the diamagnetic and paramagnetic responses, and hence an intermediate phase. Interestingly, this intermediate phase has two new edge modes and is thus a new quantum anomalous Hall state with high Chern number . The corresponding quantized Hall conductance is in the graphene ribbon in the absence of Landau levels.
Acknowledgments
T.W.C thanks S. Murakami for valuable discussions about the construction of the Chern number. We thank the National Science Council and NCTS of Taiwan for supports.
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