Gate-induced Dirac cones in multilayer graphenes
We study the electronic structures of ABA (Bernal) stacked multilayer graphenes in uniform perpendicular electric field, and show that the interplay of the trigonal warping and the potential asymmetry gives rise to a number of emergent Dirac cones nearly touching at zero energy. The band velocity and the energy region (typically a few tens of meV) of these gate-induced Dirac cones are tunable with the external electric field. In ABA trilayer graphene, in particular, applying an electric field induces a non-trivial valley Hall state, where the energy gap at the Dirac point is filled by chiral edge modes which propagate in opposite directions between two valleys. In four-layer graphene, in contrast, the valley Hall conductivity is zero and there are no edge modes filling in the gap. A nontrivial valley Hall state generally occurs in asymmetric odd layer graphenes and this is closely related to a hidden chiral symmetry which exists only in odd layer graphenes.
pacs:73.22.Pr, 81.05.ue, 73.43.Cd
Graphene is characterized with Dirac quasiparticles in the low-energy region, McClure (1956); DiVincenzo and Mele (1984); Semenoff (1984); Ando (2005) which give rise to anomalous physical properties due to the linear dispersion and non-trivial Berry phase. Shon and Ando (1998); Ando et al. (2002); Zheng and Ando (2002); Gusynin and Sharapov (2005); Novoselov et al. (2005); Zhang et al. (2005); Castro Neto et al. (2009) There are growing interests in multilayer variants of graphene such as bilayer Novoselov et al. (2006); Ohta et al. (2006, 2007); Castro et al. (2007) and trilayer, Güttinger et al. (2008); Craciun et al. (2009); Zhu et al. (2009); Bao et al. (2011); Taychatanapat et al. (2011) which also support chiral quasiparticles. Bilayer graphene has parabolic valence and conduction bands touching each other at Dirac point McCann and Fal’ko (2006); Guinea et al. (2006); Koshino and Ando (2006), while ABA(Bernal)-stacked trilayer graphene comes with a superposition of effective monolayer-like and bilayer-like bands. Guinea et al. (2006); Latil and Henrard (2006); Partoens and Peeters (2006); Lu et al. (2006); Aoki and Amawashi (2007); Koshino and Ando (2007, 2008); Koshino and McCann (2009); Koshino and Ando (2009); Koshino and McCann (2011). On top of these, the trigonal-warping deformation of the energy band, which is intrinsic to graphite-based systems Slonczewski and Weiss (1958); McClure (1960); Dresselhaus and Dresselhaus (2002), gives rise to small Dirac cones near Dirac point in these multilayers. The Lifshitz transition, in which the Fermi circle breaks up into separate parts, takes place at a small energy scale around a few meV. McCann and Fal’ko (2006); Koshino and Ando (2006, 2007, 2008); Koshino and McCann (2009)
On the other hand, it is possible to modify the band structure of multilayer graphenes by applying an electric field perpendicular to the layer, using external gate electrodes attached to the graphene sample. In bilayer graphene, a perpendicular electric field opens a band gap at Dirac point. Ohta et al. (2006); McCann and Fal’ko (2006); Lu et al. (2006); Guinea et al. (2006); Mccann (2006); Min et al. (2007); Castro et al. (2007); Oostinga et al. (2008) In contrast, for ABA trilayer graphene, previous works Koshino and McCann (2009); Koshino (2010) considered the gate-field effect on the band structure without the trigonal warping, and showed that the perpendicular electric field causes a band overlap of the conduction and valence bands at zero energy, rather than opening a gap. There two bands intersect each other on a circle around the K point whose radius is proportional to the potential asymmetry, causing an increase of the conductivity at charge neutral point. Craciun et al. (2009)
In this paper, we closely study the band structures of ABA-stacked multilayer graphenes in the presence of uniform perpendicular electric field, and find that the interplay of the trigonal warping and the potential asymmetry generally gives rise to a number of additional Dirac cones nearly touching at zero energy, as depicted in Fig. 1(b) for trilayer graphene. For these gate-induced Dirac cones, the band velocity and the energy region (i.e., the distance between Dirac point to the Lifshitz transition point) are tunable with gate bias voltage. The energy region is typically a few tens of meV, which is by order of magnitude greater than in the original non-biased multilayer graphene. In a magnetic field, there arise triply-degenerate Landau levels originating from off-center gate-induced Dirac cones, with wide energy spacings due to the linear dispersion.
The gate-induced Dirac cones are generally gapped at Dirac point by symmetry-breaking terms. When the Fermi energy is in the gap, the system is in a topologically non-trivial valley Hall state, where electrons at and valleys carry opposite Hall conductivities Castro et al. (2008); Jung et al. (2011). A manifestation of the valley Hall state is the emergence of chiral edge modes at a zigzag interface which transports valley pseudo-spins in an analogous way to the spin Hall effect Murakami et al. (2003); Kane and Mele (2005). The valley Hall state and the helical edge modes were previously studied for gapped monolayer and bilayer graphenes, Castro et al. (2008); Tse et al. (2011); Qiao et al. (2011a, b, c, 2012) and also for ABC (rhombohedral) stacked trilayer graphene. Li et al. (2012) We study the edge states a semi-infinite zigzag ribbon of asymmetric ABA multilayer graphenes, and relate the number of the edge modes to the valley Hall conductivity which is a bulk property. In trilayer graphene, in particular, we find that non-zero valley Hall state is realized in a small external electric field, and moreover, a topological transition takes place at a certain higher electric field, which is accompanied by a change of the number of edge channels inside the bulk gap. In four-layer graphene, in contrast, the valley Hall conductivity is always zero and there are no edge modes filling the energy gap. We show that the nontrivial valley Hall state generally occurs in asymmetric odd layer graphenes, and this is deeply indebted to an approximate chiral symmetry peculiar to odd layer graphenes.
Paper is organized as follows. We briefly introduce the effective mass model for graphite in Sec. II, and argue the trilayer graphene in Sec. III in terms of the gate-induced Dirac cones, the chiral symmetry, the Landau level structure and and the edge states. In Sec. IV, we study the four-layer graphene as an example of even-layer cases without the chiral symmetry. In Sec. V, we argue the chiral symmetry in general odd-layer graphenes, generalizing the trilayer’s argument. The conclusion is given in Sec. VI.
We describe the electronic properties of ABA-stacked multilayer graphene using Slonczewski-Weiss-McClure model Slonczewski and Weiss (1958); McClure (1960); Dresselhaus and Dresselhaus (2002) with hopping parameters described in Fig.2. The model includes intralayer coupling , nearest interlayer couplings , and , next-nearest layer couplings and , and on-site energy asymmetry , which are estimated in bulk graphite as Dresselhaus and Dresselhaus (2002): eV, eV, eV, eV, eV, eV, eV. is the energy difference between the sites which are involved in the coupling , and the sites which are not.
We consider ABA-stacked -layer graphene, where and represent Bloch functions at the point, corresponding to the and sublattices of layer , respectively. If the basis is arranged as ; ; ; , the Hamiltonian in the vicinity of the valley is written as Guinea et al. (2006); Partoens and Peeters (2006); Koshino and Ando (2007, 2008)
where , with being the vector potential arising from the applied magnetic field, and are the valley indeces for . The parameter is the band velocity for monolayer graphene, and a velocity related to the band parameter , where nm is the distance between the nearest sites on the same layer. describes the electrostatic potential on -th layer induced by the external electric field, where we assumed a uniform potential gradient in the perpendicular direction, This is valid in a few-layer graphene with typically . For thicker multilayers with , the potential drop occurs within a few layers near the external gate due to the screening by the charge carriers in graphene. Koshino and McCann (2009); Koshino (2010)
Iii Trilayer graphene
iii.1 Chiral symmetry and gate-induced Dirac cones
The Hamiltonian of ABA-trilayer graphene is given by Eq. (1) with , where the external potential is . In the absence of , the Hamiltonian can be block diagonalized into monolayer-like band and bilayer-like band. Koshino and McCann (2009) In finite , these sub-blocks are hybridized, and it is then useful to arrange the basis as
where the monolayer-like band correspond to 1st and 4th bases and bilayer-like band to 2nd, 3rd, 5th, and 6th. We categorize the first three bases in Eq. (2) as group , and the last three as group . The Hamiltonian is written in this basis as
If we keep only relevant band parameters , , and the potential , and neglect remaining parameters, the Hamiltonian Eq. (3) possesses the chiral symmetry (sublattice symmetry), in that the diagonal matrix blocks and all vanish, leaving the off-diagonal blocks which connects the bases of to the bases of . The reason for this can be understood in terms of reflection symmetry as follows: In the group (), the bases associated with and sublattices have odd and even (even and odd) parity, respectively, with respect to the reflection in the middle layer. The Hamiltonian without , i.e., the first term in Eq. (1) has even parity in the reflection, and has matrix elements only between and sublattices when only , and are kept. After the unitary transformation, it does not give any matrix elements in or because in each group ( or ), a base associated with and one associated with always have different parity from the definition. On the other hand, the potential term, i.e., the second term in Eq. (1) is odd in the reflection and matrix elements only connect the same sublattice. It gives no matrix elements in or either, because in each group, bases associated with the same sublattice always have the same parity.
The energy spectrum of this simplified Hamiltonian contains a single center Dirac cone and six off-center Dirac cones at zero energy, as depicted in Fig.1(a). The robustness of gapless spectrum is protected by the chiral symmetry. The extra terms with and in the diagonal blocks break the chiral symmetry and open small energy gaps at these Dirac cones as in Fig.1(b).
The positions of the Dirac points in the chiral Hamiltonian without and can be found by solving with . We obtain a Dirac point at , and six off-center Dirac points at
At each Dirac point, the degenerate zero-energy bases and are derived from the equation and , and the effective Dirac Hamiltonian is given by
keeping the lowest order in the momentum shift from the Dirac point. By rotating coordinate and the spinor space by angle at the same time, this is transformed to
where and are Pauli matrices, and is the momentum measured from each Dirac point. We find for the Dirac point at (no need for rotation),
with the valley index . For the off-center Dirac points at ,
and the velocities after rotation of by become
where and correspond to the radial and azimuthal directions respectively, with respect to . and of each Dirac points are plotted in Fig. 3(d). The velocities are mainly enhanced by applying , while for decreases only slowly. This indicates that the conductivity, which is roughly proportional to the square of the band velocity Shon and Ando (1998), is enhanced by , as is consistent with the previous transport measurement Craciun et al. (2009) and the theoretical estimation Koshino and McCann (2009).
The chirality for each Dirac cone can be defined by , and this coincides with the Berry phase in units of around the Dirac point. We find for , and for , so that the summation of chirality over seven Dirac points in the valley is . Since the chirality is a topologically protected number as long as the chiral symmetry is present, non-zero total chirality indicates that the conduction band and the valence band inevitably touch at some points in any value of .
Now we consider all the band parameters in Eq. (3) to argue the energy gaps at the Dirac points. The effective Hamiltonian of each gate-induced Dirac cone is modified to
where the mass and the energy shift are given by
The width of the energy gap is . Note that the approximation using the gapped Dirac Hamiltonian above is valid when the mass gap is smaller than the energy region of the gate-induced Dirac cone below the Lifshitz transition point. In Fig.3(c), we show the evaluated mass for the seven Dirac points. We see that the mass for changes from negative to positive when exceeds the critical value,
at which the energy gap closes.
When the Fermi energy lies in the gap in massive Dirac Hamiltonian Eq. (14), the Hall conductivity takes non-zero value even in zero magnetic field. Haldane (1988); Oshikawa (1994) Considering chirality and mass for each Dirac point in Table 1, the total Hall conductivity summed over the Dirac points at single valley becomes
We have a topological change at gap closing point, . The Hall conductivity have opposite signs between two valleys due to the time-reversal symmetry, so that the net Hall conductivity is always zero. Nevertheless, the single-valley Hall conductivity is directly related to the number of chiral edge modes appearing in zigzag edge, as we will see in Sec. III.3.
iii.2 Landau level structure
The Landau levels in the presence of a uniform magnetic field can be calculated by the Hamiltonian with and for and , respectively. Here and are raising and lowering operators, respectively, which operate on the Landau-level wave function as , and is the magnetic length. The Landau level spectrum at T is plotted as a function of potential asymmetry in Fig.3(b). Landau levels in and valleys are plotted as solid (blue) and dashed (red) lines, respectively. There we can see two distinct regions, a region around zero energy where the Landau level spacing are wide due to Dirac Landau levels, and a region above the Lifshitz transition where the Landau levels are densely spaced due to large density of states. In increasing , the gate-induced Dirac pockets accommodate more and more Landau levels as the energy region within the Dirac pockets expands.
The low-energy Landau levels inside the gate-induced Dirac pockets can be approximately described by the massive Dirac Hamiltonian, Eq. (14). There the Landau level spectrum is explicitly written as
We note that the Landau level of is sensitive to the chirality and the sign of the mass: it appears at the top of the valence band and the bottom of the conduction band when and , respectively. The two cases correspond to the mid-gap values of the Hall conductivity, and , respectively. Also, since the chirality is opposite between and valleys, the Landau level split in valleys, while all others are valley degenerate. Besides, we have additional triple-fold degeneracies for each of and .
In Fig.3(b), we actually see that the levels level is non-degenerate in valleys, appearing at either of the band edges of each massive Dirac band. At , we observe that one pair of levels from valleys crosses each other at the charge neutral point, in accordance with the topological change of from to . The levels are almost valley-degenerate while tiny splitting is due to the deviation from the massive Dirac Hamiltonian Eq. (14).
When we drop the band parameters other than and , the Hamiltonian becomes chiral symmetric and the zero-th Landau level of each Dirac cone comes exactly to zero energy as a chiral zero mode, of which wavefunction has amplitude only on and for , respectively. The index theorem then states that the difference between the number of the zero-modes belonging to and those to , is defined as chiral index, which coincides with the gauge flux penetrating the system. The chiral index in the present case is shown to be where is the magnetic flux penetrating the system area . This is, in units of , coherent with a summation of in each single valley. As in the conventional Dirac Hamiltonian, Jackiw and Rebbi (1977); Fujikawa (1979); Fujikawa and Suzuki (2004) the chiral index can be related to the geometric curvature of the gauge field, and the above relation between the chiral index and total magnetic flux stands in non-uniform magnetic field as well. The detailed argument is presented in Appendix. A.
iii.3 Edge modes
Non-trivial Hall conductivity in single valley indicates an existence of chiral edge modes localized at the interface, as long as the valley mixing is not present. There the number of emergent edge modes are directly related to the Hall conductivity, so that chiral edge modes as many as the number of should counterflow in opposite directions between and , as is analogous to the spin Hall insulator.Murakami et al. (2003) Here we numerically examined the edge modes in the asymmetric ABA trilayer graphene with zigzag interfaces, in which the valley mixing is absent. We consider a semi-infinite system with a zigzag boundary along direction as shown in Fig. 4(a), where is a good quantum number. The energy of edge modes in the bulk gap can be obtained by searching for the evanescent modes satisfying a boundary condition at the interface. The method is detailed in Appendix B.
First we consider the chiral symmetric case neglecting . Fig.4 illustrates the energy spectrum near point at meV, where we see that zero-energy edge modes appear between some of gate-induced Dirac points. The number of zero energy edge modes are closely related to the chirality of each Dirac cone. Ryu and Hatsugai (2002); Schnyder and Ryu (2011); Heikkilä and Volovik (2011); Burkov et al. (2011) When we regard a two-dimensional periodic system on -plane as a one-dimensional system with fixed as a parameter, we can define a winding number by integrating the Berry phase change all the way along on the Brillouin zone at the fixed . When we set the boundary perpendicular to axis (i.e., is still a good quantum number), the number of zero-energy edge modes appearing at the boundary coincides with except a constant.Ryu and Hatsugai (2002) In the present system, this bulk-edge relationship can be clearly seen in Fig.4.
Other hopping terms breaking the chiral symmetry give rise to mass to the Dirac points, and relative signs between these masses determines the connection of the chiral edge modes between different Dirac points. In Fig.5(a), we plot the band structure near including full band parameters at meV. We see that the left and right edge modes stick to either of the top or bottom of gapped Dirac cone depending on the sign of the mass. At the charge neutral point, we have three set of chiral edge channels crossing the Fermi energy, which all circulate in clockwise direction when viewed from direction. We also observe two edge states extending out of the plot and leading to the other valley . In , we have the exactly same spectrum with inverted to .
The correspondence to the single-valley Hall conductivity can be understood in a similar argument to that for integer quantum Hall effect.Laughlin (1981) Let us consider a cylindrical system which is closed in with circumference while finite in the axial direction with bound by the zigzag edges. When we adiabatically turn on a magnetic flux quantum penetrating into the cylinder (inducing an electric field along direction), every state at is shifted to . The single-valley Hall conductivity of then coincides with the total move of electrons in direction through this adiabatic process. At the Fermi energy, an electron moves from the left edge to the right for each of three pairs of counter-propagating channels, contributing to . A charge transfer also occurs below the Fermi energy, where a bulk state is pumped to an edge state in the left-edge channel going out of the valley. This yields a contribution to , which adds up to all together.
Fig.5(b) plots the band structure at larger bias, meV after the topological transition at . We find that the connection of the edge modes changes at , leaving only one clockwise and one anti-clockwise chiral edge modes crossing at the Fermi energy. The Hall conductivity from those two exactly cancel out, while we have the same contribution from the edge channel below the Fermi energy, giving the total Hall conductivity . This again coincides with bulk valley Hall conductivity estimated from the mass and chirality. Conversely, the single valley Hall conductivity gives the number of the counter edge modes crossing at the Fermi energy, when we appropriately exclude the half integer contribution from the edge modes connecting .
Iv Four-layer graphene
Unlike trilayer, a four-layer graphene with interlayer asymmetry does not possess the chiral symmetry even in the approximate model, and the band gap always open at the zero energy. The effective low-energy Hamiltonian can be obtained by excluding the high-energy bonding states at as ,McCann and Fal’ko (2006) where and represent diagonal blocks of the original Hamiltonian for low-energy bases spanned by , and for high-energy bases by , respectively, and and are off-diagonal blocks connecting them. This is explictly written in basis as
where we neglected the band parameters other than and . The approximation is valid only when .
If we even neglect term, the low-energy energy band is rotationally symmetric around points and its dispersion relation is given by
with . The band gap appears between , corresponding to an off-center momentum .
When we resume term and other parameters, six off-center pockets emerge at momentum and near , each of which are arranged in 120 degrees symmetry as illustrated in Fig.6, and Fig.7(a). The pre-existent gap never closes during this process. The pockets at are much deeper than those at , and the energy depth is about 15 meV at meV. Fig.7(b) describes an evolution of Landau level energies with increasing , where we observe the triply-degenerate Landau levels of pockets with wide energy spacing, similarly to trilayer graphene.
We can expand the Hamiltonian with respect to the center of each Dirac pocket, and obtain the effective Hamiltonian in the massive Dirac form of Eq. (14). The masses and the velocities are shown in Fig.7(c) and (d), respectively, where we set the basis so that the chirality becomes (i.e., positive). The masses at and are opposite in sign, so that the contributions to the Hall conductivity of Dirac cones cancel out in summation over a single valley. Therefore, the single valley system is a trivial insulator with zero Hall conductivity in contrast to trilayer graphene. Accordingly, we observe no Landau level crossing at charge neutral point in Fig.7(b), and if we look at the edge states in Fig.7(e), there are no edge channel crossing in the bulk gap, nor the counter-propagating flows of valley pseudo-spin.
V General odd-layer graphenes
The approximate chiral symmetry in the presence of the external electric field argued in trilayer actually holds in any odd-layer Bernal multilayer graphenes. As we see in the following, if we only consider terms , the Hamiltonian of odd layered graphene is chiral symmetric with a nonzero chiral index in the presence of magnetic field, which means that the band is gapless in limit of zero magnetic field.
The Hamiltonian for Bernal stacked -layer graphene is decomposed into block diagonal form with effective monolayer and bilayer blocks with a unitary transformation Koshino and Ando (2007, 2008, 2009); Koshino and McCann (2011). Let us define
is the layer index, is the block index given as
Then we take a basis
where or . A superscript indicates that the wavefunction is nonzero only on the sublattice on the layer odd/even.
With the basis set above, the Hamiltonian with is block diagonalized with . A block labeled by spanned by is dictated as with and
where we only left the terms with , For the case of , only two bases survive due to , and the corresponding block Hamiltonian is the first 2 by 2 component of Eq.(26).
Now let us consider an odd-layer graphene with the interlayer potential asymmetry , i.e., the second term of Hamiltonian, Eq. (1). When , we can easily show that the basis function belonging to the block is either symmetric or antisymmetric with respect to reflection in the central layer , and the parity is given by . Koshino and McCann (2011) If we write down in this basis, the matrix element ( even or odd) becomes non-zero only when , , and , because is an odd function in the reflection, and also diagonal in the original site representation.
Therefore, if we separate the basis functions into two groups as:
then the Hamiltonian including only has matrix elements only between and , and thus is chiral symmetric. Note that the chiral symmetry is not respected for even-layer graphenes since the basis function labeled by cannot be categorized to either even or odd parity, and the interlayer potential term gives rise to matrix elements connecting blocks and .
In the presence of magnetic field, the chiral index, i.e., the difference between the number of the zero-modes belonging to and those to , can be easily obtained by considering the Landau level spectrum with , and all switched off, since the chiral index never changes in such a continuous transformation. The bilayer-like Hamiltonian block, Eq. (26), is then consists of two monolayer-type diagonal blocks, giving two zero-energy Landau levels localized at the second and fourth (first and third) elements in the valley . Considering the base grouping in Eq. (27), the difference in zero energy states in block is found to be for and , respectively, where is the magnetic flux penetrating the system. The monolayer-type block lacks the third and fourth elements, giving . As a result, the total chiral index in odd-layer graphene for valley is finally given by
This states that at least one Landau level remains at zero energy, and thus in zero magnetic field, the conduction and valence band touch at one -point at least. The sum of Berry phases in a single valley coincides with times the chiral index in units of , that is for and , respectively. When the Dirac cones are gapped by including the additional band parameters, the Hall conductivity per single valley must be non-zero, because the number of Dirac cones per valley must be odd to achieve total Berry phase .
The result might seem to contradict with the well-known fact that the Berry phase is in -layer graphene at . As argued, the Hamiltonian with is block diagonalized into independent monolayer-like and bilayer-like subsystems. In each block, there is an ambiguity in the choice of bases for or , and the Berry phase of the block actually changes its sign when and are interchanged. If we simply assign all and sublattices to and , respectively, we obtain instead of Eq. (28). In the presence of nonzero , on the other hand, the subsystems are mixed with each other and then the grouping of Eq. (27) is the only possible way to make the Hamiltonian chiral symmetric.
In Bernal multilayer graphene more than three layers, an interplay of the gate electric field and the trigonal warping effect gives rise to emergent Dirac cones in the low energy bands, whose band velocity and Lifshitz transition energy are tunable by the gate voltage. In trilayer graphene, in particular, the low-energy effective theory shows that the valley Hall state is realized at the charge neutral point, where single valley Hall conductivity is quantized at a non-zero half integer. We have investigated the edge states at the zigzag interface, and demonstrated that the number of edge modes is closely related to the bulk single valley Hall conductivity. In four-layer graphene, gate-induced Dirac cones also appear, though the system is a trivial insulator with zero valley Hall conductivity. The non-trivial valley Hall state is generally found in odd-layer graphenes, where the approximate chiral symmetry is responsible for the emergence of non-zero valley Hall conductivity.
The gate-induced Dirac cones should be experimentally accessible directly by observing Landau levels with wide energy spacing. Sadowski et al. (2006); Jiang et al. (2007); Morimoto et al. (2012) Also, the single valley Hall conductivity argued here is expected to be observable in the transport through the edge modes at a zigzag interface, while a valley mixing caused by a concentration of atomic-scale scatterers or a presence of armchair edge would wash out the effect. There the conductance is related to the number of edge channels, and the topological transition at should be observed as a change in the conductance. Since the helical edge modes appearing in the gated multilayer graphene carry valley pseudospins, modulation of edge modes through the gate voltage could be a way to electrically control the valley polarized transport.
Authors thank helpful discussions with Akira Furusaki. This work was supported by Grants-in-Aid for Scientific Research, No.24840047 (TM), No.24740193 (MK) from JSPS.
Appendix A Index theorem for odd-layer graphenes
Here we show that the chiral index of general odd-layer graphenes can be written in terms of the total gauge flux penetrating the system, in a similar way to the argument for usual Dirac Hamiltonian. Jackiw and Rebbi (1977); Fujikawa (1979); Fujikawa and Suzuki (2004) The chiral symmetric Hamiltonian of odd-layer graphene with interlayer asymmetric potential is written as
where is a matrix. The chiral symmetry is then expressed by with the chiral operator ,
where is unit matrix. We can take zero modes of as an eigenvector of with eigenvalue , respectively. If we write the number of chiral zero modes as , the chiral index is defined as
where is a regularization function, which is smooth and monotonically decreasing function with and , and is an ultraviolet cutoff. The action of to the matrix is defined through its action onto the eigenvalues, as seen if we take .
If we take plane waves as a basis for the spatial direction, we have
where Tr means a trace over all the states while tr is a trace over the layer and site indeces or, equivalently, , and even/odd indices in Eq.(25).
The operator acts on a plane wave to give . Since contains at most first order derivative terms, we can write
where is a c-number matrix and of zero-th order in , but may depend on a polar angle of . Since is a square matrix, it can be written in a singular value decomposition
with unitary matrices and a diagonal matrix .
The action of on a plane wave is then described as
are c-number matrices of zero-th order in . are matrices including the operator , and up to first order of .
Having in mind that the ultraviolet behavior () is important for the contribution of term to the chiral index, we obtain
With a similar form for term, the chiral index with reduces to
As argued in Sec. V, the Hamiltonian of multilayer graphene with is decomposed into block diagonal form with effective monolayer and bilayer blocks labeled by , and when the number of layers is odd, always enters in the off-diagonal blocks. Then is block-diagonal because it is independent of , and so and are. Therefore, although block off-diagonal terms in arise from , they do not contribute to the sum of , and thus we only have to compute the chiral index for each block separately and add them up to obtain the overall chiral index.
Monolayer block . Using the commutation relation ,