Distant-Neighbor Hopping in Graphene and Haldane Models
DDistant-Neighbor Hopping in Graphene and Haldane Models
Doru Sticlet
1, 2, 3, ∗ and Fr´ed´eric Pi´echon Laboratoire de Physique des Solides, CNRS UMR-8502, Univ. Paris Sud, 91405 Orsay, France Max-Planck-Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany LOMA, CNRS UMR-5798, Univ. Bordeaux, F-33400 Talence, France
Large Chern number phases in a Haldane model become possible if there is a multiplicationof Dirac points in the underlying graphene model. This is realized by considering long-distancehopping integrals. Through variation of these integrals, it is possible to arrive at supermergingband touchings, which up to N7 graphene are unique in parameter space. They result from thesynchronized motion of all supplementary Dirac points into the regular ± K points of graphene.The energy dispersion power law is usually larger than the topological charge associated with them.Moreover, adding distant-neighbor hoppings in the Haldane mass allows one to sweep large Chernnumber phases in the topological insulator. I. INTRODUCTION
The Haldane model is the first topological insulatorthat presents the quantum Hall effect without an externalmagnetic field. It is a two-band system with bands thathas a nontrivial topology. The bands are characterized byChern numbers that are proportional to the conductancecarried by edge states. This model was a playground forideas that eventually led to the prediction and discoveryof the Z topological insulators. Here we revisit the Haldane model and show in prac-tice how the addition of hopping integrals between dis-tant neighbors can lead to a multiplication of topologicalphases with a large Chern number. Recent work suggestsa way to produce flat bands with arbitrary Chern num-bers in multilayer systems.
In contrast, we constrainourselves to the two-band Haldane system and we do notseek flatness of bands. Admittedly this is not a very phys-ical way to increase the topological index characterizinga band, because the contribution from distant neighborsare small. The conceptual advantage is that one can fullydescribe the phase diagram of such systems and analyti-cally predict its topological transitions. The system canbe understood from a decomposition of the model in anunderlying gapless graphene model and a Haldane mass.The variation of the topological index is related at onceto the multiplication of nodes in the energy dispersionfor the gapless system and to the rapid oscillations in theHaldane mass term. In general, if the two-band under-lying gapless system admits 2 n Dirac points, then theChern number can vary from − n to n . In Sec. II we present the theoretic tool to compute an-alytically the Chern number in a two-band system. Thatallows one to immediately discriminate the topologicalphases. In Sec. IV, we treat the underlying graphenewith long-distance hopping integrals. Up to N7 (next × ± K points in the graphene Brillouin zone (BZ). SatelliteDirac points can be found by perturbing around these special band touchings. The topological charge associ-ated with them can be immediately established from asum over the additional Dirac points. In Sec. IV, weconsider the effects of gapping the Dirac points with aHaldane mass term. The phase diagram for the modifiedHaldane model is shown. II. CHERN NUMBER IN TWO-BANDSMODELS
A generic two-band Hamiltonian on a two-dimensionalBravais lattice reads H = 14 π (cid:90) BZ d k (cid:88) α,β =1 , h αβ ( k ) c † α k c β k (1)with c † α k the creation operator of the Bloch state withwave vector k and where α constitutes a pseudospin indexresulting from either two sublattices or two orbitals perunit cell. The elements h αβ ( k ) form a 2 × h ( k ) that can be written h ( k ) = (cid:88) µ =0 h µ ( k ) σ µ , (2)with σ the identity matrix and σ , , the Pauli matri-ces. h µ =0 , ( k ) comes from intrasublattice contributions h αα and h µ =1 , ( k ) from intersublattice contributions h αβ .The real valued functions h µ ( k ) can be further split intoeven and odd components h µ ( k ) = h eµ ( k ) + h oµ ( k ), where h eµ ( k ) = h eµ ( − k ) and h oµ ( k ) = − h oµ ( − k ). For time-reversal symmetric, spinless Hamiltonians, h µ =0 , , ( k )are purely even and h ( k ) purely odd.The spectral decomposition of matrix h ( k ) reads h ( k ) = (cid:88) ± (cid:15) ± ( k ) P ± ( k ) , (3)with band energies (cid:15) ± ( k ) = h ( k ) ± | h ( k ) | and eigen-band projector P ± ( k ) = ( σ ± h · σ / | h | ), where h ( k ) =( h , h , h ). Component h ( k ) breaks particle-hole sym-metry by shifting the energy bands and it may also lead a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r to an indirect overlap of the two energy bands. Never-theless it does not intervene in the direct gap | h | or inthe projectors P ± which determine the topological prop-erties of the Hamiltonian. In the following we will neglectthe h ( k ) σ term and consider the system an insulator aslong as the direct gap | h | does not close; in this situationthe projection to the lower band is always well-defined.An insulating phase, in which the three components of h ( k ) never vanish simultaneously and | h | remains finitefor any k , can be characterized by a topological index, thefirst Chern number C . The integer C counts how manytimes the Brillouin zone wraps around the unit spheretraced by ˆ h = h / | h | . One can choose to index the sys-tem with the Chern number associated with the lowestband (cid:15) − ( k ), C = 14 π (cid:90) BZ d k ˆ h · ( ∂ k x ˆ h × ∂ k y ˆ h ) , (4)where the integral is over the Brillouin zone. The Chernnumber is zero unless one of the component of h ( k )breaks time-reversal symmetry. Furthermore, a non-zerovalue of C requires that any submodel obtained by con-sidering only two components of h must exhibits bandtouchings at some finite set of points in the BZ. In fact,an explicit calculation of C is made easy by consideringsuch a gapless system containing only two componentsof h . When the band touchings of the gapless submodelhave linear dispersion (i.e. they are Dirac points), C canbe calculated by treating separately the chirality of theDirac points and the sign the third component of h (the mass term that gaps the system) at these Dirac points.The Chern number then reads C = 12 (cid:88) k ∈ D i χ i ( k ) sgn( h i ) , (5)where D i is the set of Dirac points for a simplified two-component model without an h i term, and ∀ k ∈ D i , χ i ( k ) = sgn (cid:0) ∂ k x h × ∂ k y h (cid:1) i (6)is their corresponding chirality. Such formula permitsan analysis of the Haldane model by separately study-ing the underlying gapless graphene model and the signof the mass term. The caveat of the above formula isthat it works only for point band touchings with linearenergy dispersion (Dirac points). However, we shall seethat there can be Fermi lines or point band touchingswith higher power dispersion as well. The latter onescan be understood as the merging of many Dirac points.Then the topological charge of the merging points is justthe sum of the chiralities for the Dirac points that areconverging to it. This fast calculation of charge associ-ated with a band touching will be referred to as the sumrule. t t t t t t t t t FIG. 1. The possible hoppings in the graphene N9 model.From a central B atom, the neighbors are arranged in concen-tric circles. The hopping integrals from the central B atom toa site placed on a circle is denoted by t n , n growing with thedistance between sites. III. DISTANT-NEIGHBOR HOPPING INGRAPHENE
As seen in previous section, the possibility of a two-band insulator with a Chern number C = n requires oneto build a reduced two-band gapless model with at least2 n Dirac points. Let us consider the extended tight-binding graphene model, including distant N n [(next × ( n − A and B . Let us denote by t n the (isotropic) N n intra- and intersublattice hoppingterms (see Fig. 1). The parameter t corresponds to usualnearest-neighbor N1 graphene. In this section we consid-ered only intersublattice hoppings t n in units of t , suchthat there are n − σ and are neglected in the following. For real-valued t n , thematrix h ( k ) preserves time-reversal and inversion sym-metries [e.g. h ∗ ( − k ) = h ( k ) and σ h ( − k ) σ = h ( k ))].Moreover, when considering only intersublattice hopping t n , there is a sublattice symmetry characterizing the sys-tem. The symmetry is represented by the operator σ which anticommutes with h ( k ). Explicitly, h ( k ) reads h ( k ) = (cid:18) f ( k ) f ∗ ( k ) 0 (cid:19) , (7)with f ( k ) = h ( k ) − ih ( k ) or f ( k ) = (cid:88) n t n g n ( k ) , (8)where the functions g n ( k ) up to N9 are tabulated in Ta-ble I in which a = √ a (cid:0) , √ (cid:1) and a = √ a (cid:0) − , √ (cid:1) denote the primitive vectors of the triangular sublattices.The hexagonal lattice constant a is set to 1 from now on.The explicit form of the function g n ( k ) corresponds toa Bloch basis such that g n ( k + G ) = g n ( k ) with G areciprocal lattice vector. A. Nearest-Neighbor N1 graphene review
Before studying N n graphene, let us briefly review theusual properties of N1 graphene where f ( k ) = t g ( k ).The two energy eigenvalues are given by (cid:15) ± ( k ) = ±| f ( k ) | and there is a gap separating the two bands. Bandtouchings occur at isolated positions ± K correspond-ing to zeros of f ( k ). For N1 graphene the zeros cor-respond to the two nonequivalent Brillouin zone cor-ners ± K = ± ( a ∗ − a ∗ ) / a ∗ = π a (cid:0) √ , (cid:1) and a ∗ = π a (cid:0) − √ , (cid:1) . At each of these points, there aretwo degenerate zero-energy eigenstates. As illustrated inFig. 2, the bipartite property allows one to project onestate entirely on the A sublattice and the other on the B sublattice. Altogether there are four zero-energy states,each labeled by two indices: a valley index correspondingto ± K and a sublattice index A or B equivalent to theeigenvalues ± associated with sublattice symmetry op-erator σ . The time-reversal transformation exchangesvalley index without changing sublattice index. Inver-sion (represented by σ ) exchanges both valley and sub-lattice indices. Hence the product of the two operationsexchanges sublattice index only.In the neighborhood of the band touchings ± K , onecan expand the function f ( k ) in small momenta q = q (cos θ, sin θ ). It follows that f ( ± K + q ) (cid:39) qe ∓ iθ and thelinearity in q identifies the band touchings as masslessDirac fermions. More generally, if the band touching has f ∝ ( qe − iθ ) n , then its respective chirality is n . Thistranslates to the fact that the two dimensional vector( h , h ) ∝ q n (cos( nθ ) , sin( nθ )) rotates counterclockwiseby 2 πn for θ sweeping once the interval [0 , π ). Here,for n = 1, it follows immediately that Eq. (6) and thelow-energy expansion both predict χ ( ± K ) = ± B. Band touchings in N n graphene Let us study how the properties of the zero-energystates are modified when distant-neighbor hoppings are ǫ ∗ ǫǫ ∗ ǫ
111 1 1 ǫǫ ∗ ǫ ∗ ǫ ǫǫ ∗ ǫ ∗ ǫ ǫ ∗ ǫ ǫ ∗ ǫ ǫ ǫ ∗ ǫ ǫǫ ∗ ǫ ∗ ǫ (a) (b)FIG. 2. Real space representation of the four zero-energyeigenstates of N1 graphene. Filled (empty) bullets repre-sent A (B) sublattice sites. Wave functions componentson different lattice positions are related by Bloch theorem ψ k ( r + R ) = e i k · R ψ k ( r ) with R any Bravais lattice vector.Let us denote (cid:15) = e i K · a = e πi/ ; then e i K · a = (cid:15) ∗ with1 + (cid:15) + (cid:15) ∗ = 0 and (cid:15) = (cid:15) ∗ = 1. Left figure corresponds tovalley K and A sublattice and right figure to valley K andB sublattice. The wave functions in − K valley are obtainedby complex conjugating the amplitudes at K . Wave functionamplitudes are invariant under C rotation around a latticesite and under translations ( R ⊥ = m ( a + a )) perpendicu-lar to K and exhibit periodicity under translations parallel to K with a period R (cid:107) = 3( a − a ). Time reversal exchangesvalley index without changing sublattice index. Inversion ex-changes both valley and sublattice indices. The product ofthe two operations thus exchange sublattice index only. considered. For N n graphene, band touchings oc-cur at positions ± k corresponding to zeros of f ( k ) = (cid:80) n t n g n ( k ). Previous solutions, ± K , obey g n ( ± K ) = 0and thus remain zeros of f ( k ) regardless of the new hop-ping integrals t n> . To find the positions of other zeros,one can keep k on the three high-symmetry lines T join-ing Γ, ± K , and M . These lines are globally invariantunder time reversal, C , C and inversion with respectto the Γ point. Without loss of generality, let us ana-lyze the T line, k = k (1 ,
0) (see Fig. 4). Along it g n ( k )is a real function and the condition f ( k ) = 0 translatesinto a polynomial equation h ( x ) = 0 for the variable x = cos( √ k ). Up to N8 this polynomial reads as h ( x ) = 4 (cid:0) x + 12 (cid:1) p ( x ) , with p ( x ) = 4( t + t ) x + 2( t − t ) x + ( t − t − t ) x + 12 − t − t + t + 3 t k x direction.As anticipated x = − (at ± K ) is a solution regardlessof the value of t n . Other physically meaningful solu-tions must verify | x | ≤
1. For each such solution uponapplying a C rotation, one can associate three band-touching points at k = k (1 , k = k ( − , √ ) and k = k ( − , − √ ). From time-reversal symmetry, it fol-lows that there are three additional touching points at TABLE I. Properties of N n AB intersublattice hopping terms. Physical distance is counted in units of lattice constant a .Chemical distance is the smallest number of bonds passed while hopping between two sites. In the “neighbors” column arethe number of sites counted at a given physical distance from a chosen central site. In contrast, note that any site has 3 n neighbors located at a chemical distance n from it. The primitive vectors of the triangular sublattice are a = √ a (cid:0) , √ (cid:1) and a = √ a (cid:0) − , √ (cid:1) . N n Hopping Physical Chemical Neighbors g n ( k )distance distanceN1 t e − i k · a + e − i k · a N3 t e i k · ( a − a ) + e i k · ( a − a ) + e − i k · ( a + a ) N4 t √ e i k · a + e i k · a + e − i k · a + e − i k · a + e i k · ( a − a ) + e i k · ( a − a ) N7 t √
13 5 6 e i k · (2 a − a ) + e i k · (2 a − a ) + e i k · ( a − a ) + e i k · ( a − a ) + e − i k · (2 a + a ) + e − i k · (2 a + a ) N8 t e i k · ( a + a ) + e i k · ( a − a ) + e i k · ( a − a ) N9 t √
19 5 6 e − i k · a + e − i k · a + e i k · (2 a − a ) + e i k · (2 a − a ) + e i k · a + e i k · a − k , , . Hence a solution | x | ≤ x (cid:54) = {± , − / } implies at least six nonequivalent band-touching points at ± k , , . In contrast, a solution x = − x = − to the twononequivalent BZ corner ± K and a solution x = 1 to thethree nonequivalent M points. In a N n graphene modelwith a polynomial h ( x ) of degree m ≤ n the maximumnumber of nonequivalent band-touchings points per val-ley on a T line is thus [1 + 3( m − m = 4, andtherefore there are a maximum of ten Dirac points pervalley. As a final remark concerning the band-touchingpoints, we stress that we cannot exclude the possibilityof having additional touching points outside the high-symmetry T lines. However, the only case encounteredin the numerical simulations is that of Fermi lines (zeroenergy lines) which connect the zero-energy solutions lo-cated on the T lines. These are particular solutions thatcan be expected when a nondegenerate zero on the T line exhibits a vanishing chirality (see Sec. III F for anexample). C. Zero-energy state wave functions in N n graphene Similarly to N1 graphene, there are two degeneratezero-energy eigenstates that correspond to each band-touching point of N n graphene. The bipartite propertyis still valid for N n graphene and it allows one to projectone zero-energy state on the A sublattice and the other onthe B sublattice. The real-space representation of thesetwo energy states is illustrated in Fig. 3 for a generic k = k (1 ,
0) on a T line. The wave function exhibitstranslation invariance in the direction perpendicular to k and it is multiplied by a phase z = e i k · a on both A and B sublattices when translated by one unit along k .Figure 3 is especially useful as it allows one to quicklyconstruct the polynomials h ( x ) at all orders. z ∗ zzzz ∗ z ∗ z z z z ∗ z ∗ z ∗ z ∗ z z z z ∗ z ∗ z ∗ z ∗ z ∗ z z z FIG. 3. Real-space representation of a generic zero-energyeigenstate of N n graphene projected on the sublattice A (invalley K ). Wave functionscomponents on different lattice po-sitions are related by Bloch theorem ψ k ( r + R ) = e i k · R ψ k ( r )with R any Bravais lattice vector. For k on high-symmetrylines we have z = e i k · a and z ∗ = e i k · a (for k (cid:54) = ± K ,1 + z + z ∗ (cid:54) = 0 and z (cid:54) = 1 (cid:54) = z ∗ ). The three additional statesare obtained by performing a C rotation around the center ofa hexagon. Solutions in the opposite valley are obtained bycomplex conjugating the amplitudes. A similar picture canbe drawn for states projected of the B sublattice. D. Velocities and chirality of Dirac points in N n graphene When all the band-touching points on a given T lineare distinct, each one of them may correspond to a Diracpoint k . The energy dispersion near the touching pointis obtained by expanding to first order in small momenta q = q (cos θ, sin θ ) f ( k + q ) = q ( ∂ k x h cos θ − i∂ k y h sin θ ) + O ( q ) , (10)where on the T line [ k = ( k, ∂ k y h = ∂ k x h = 0. Let us define the velocities c x = ∂ k x h and c y = ∂ k y h . A band touching point k is aDirac point, if both velocities are nonvanishing at k , c x (cid:54) =0 (cid:54) = c y . More quantitatively c x and c y read as c x ( x ) = ∓ √ (cid:112) − x [ p ( x ) + ( x + 12 ) p (cid:48) ( x )] , (11) c y ( x ) = − p ( x ) − ( x + 12 ) r ( x )] , (12)where the sign ∓ of c x corresponds to band touchingsassociated with the ± K valley, and the polynomial r ( x )is given by r ( x ) = 16 t x + 4( t − t ) x + 4( t − t ) x +1 + t − t + 2 t + 5 t . (13)To simplify the above equations, it is opportune to studyseparately the velocities for band touchings at the ± K points ( x = − / ± K points on the T line for x (cid:54) = − / p ( x ) = 0.Let us take the band touchings only at the K valley,knowing that the c x changes sign at the opposite valley.The corresponding velocities read x = −
12 : c x = c y = −
32 (1 − t − t + 5 t + 4 t ) , (14) x (cid:54) = −
12 : (cid:40) c x = − √ √ − x ( x + ) p (cid:48) ( x ) c y = 3( x + ) r ( x ) . The above equations indicate that the two velocities areequal in magnitude and eventual Dirac points will haveisotropic cones at ± K . Also note that a merging of Diracpoints in the K valley creates an energy dispersion ofhigher order in q and this is equivalent to vanishing ofthe velocities to c x,y | K = 0. At band touchings differentfrom ± K one can use the condition p ( x ) = 0 to simplifythe expression of the r ( x ) polynomial: r ( x ) = 2(1 − x )[4( t − t ) x + 4( t − t ) x + t − t + t ] . (15)At time-reversal points Γ( x = 1) and M ( x = −
1) thevelocity c x is always zero. The Γ point (center of theBZ) is the band bottom and presents an isotropic energydispersion; therefore c y vanishes together with c x [as seenfrom Eq. (15)]. In contrast, at the M point, c y is notnecessarily zero. For example, in N3 graphene this allowsfor M band touchings with linear dispersion in k x andquadratic in k y . These semi-Dirac points correspond toa merging of two Dirac points with opposite chirality.If all the band touchings are Dirac points, then theirchirality (6) follows from Eqs. (14): x = −
12 : χ ( ± K ) = ± ,x (cid:54) = −
12 : χ ( ± k i ) = ∓ sgn (cid:2) p (cid:48) ( x ) r ( x ) (cid:3) , (16) where ± k i denote the position of the additional Diracpoints associated with the ± K valley. The next sectionsexemplify the above theory to the concrete cases of N3and N4 graphene. E. Dirac points and merging for N3 graphene
The isotropic N3 graphene was already investigatedin Ref. 10. Here the presence of a sufficiently strong t hopping integral was shown to produce three more satellite band-touching points orbiting around each reg-ular Dirac point ( ± K ). Indeed solving Eq. (9) (with t = t = t = 0), it follows that in addition to solution x = − (at ± K ), there is a solution x = t − t which maygive rise to up to six touching points at ± k = ± k (1 , , ± k = ± k (cid:18) − , √ (cid:19) , (17) ± k = ∓ k (cid:18) , √ (cid:19) , k = 2 √ (cid:18) t − t (cid:19) , where ± k i points are associated with the ± K valley. Aphysically meaningful solution corresponds to | x | ≤ t ∈ ( −∞ , − ∪ (1 / , ∞ ) . (18)For t ∈ ( − , −∞ ) satellite touching points appear at Γ( t = −
1) and move along the T line and reach the Σpoint ( x = 1 / t = −∞ ), midway between K and Γ.For t ∈ (1 / , ∞ ) satellite touching points appear at M ( t = 1 /
3) and move along the T line and reach againΣ ( x = 1 / t = ∞ ) (see Fig. 4). For t (cid:54) = 1 /
2, thesatellites are Dirac points away from the regular Diracpoints ± K , x = t − t (cid:54) = − /
2. The chirality associatedwith the three satellite Dirac point k , , in valley K reads χ (cid:18) x = t − t (cid:19) = − sgn[ t (1 + t )] . (19)The chirality χ is always opposite to points associatedwith the − K valley.As already emphasized, there is a particular value, t = 1 /
2, that corresponds to a merging of three satel-lite Dirac points with a central regular Dirac point. Thiscase is realized when x = − / h ( x ). Here p ( x ) = 0 and therefore the ve-locities c x,y vanish simultaneously, indicating the for-mation of a band touching with a higher than lineardispersion. Note, however, that at the merging point, χ ( x = − /
2) = sgn( c x c y ) is not well defined. Neverthe-less, from Eq. (19) it is apparent that the satellite pointsclose to merging at ± K have an opposite chirality fromthe central Dirac point χ ( ± K ) = ±
1. Then the sum ruledictates that the chirality at the merging point is thesum of chiralities over the colliding Dirac points. At ± K merging this yields χ ( t = 1 /
2) = ± (1 −
3) = ∓ Γ K M − K Σ
FIG. 4. (Color online) Evolution in BZ of a satellite Diracpoint in N3 graphene on the high-symmetry T line: Γ- K - M .The evolution of the satellite point is represented in blue when t varies from −∞ to − t varies from 1 / ∞ . The chirality of the merging point can be equally de-termined by expanding the energy dispersion at ± K . Itsuffices to find it at K , knowing that time-reversal sym-metry demands opposite chirality at − K . Expanding at t = 1 / q = q (cos θ, sin θ ) it followsthat f ( ± K + q ) = 98 q e ± iθ + O ( q ) . (20)This indicates that the band touching at the merging ofall the Dirac points in a valley has a quadratic dispersionand a topological charge of ∓ ± K . F. Dirac points for N4 graphene
For N4 graphene, solving Eq. (9) yields, besides thesolution x = − (at ± K ), two additional solutions x ± = − t [ t ± ( t + 8 t + 4 t t − t ) / ] ( x + ≤ x − ) such thatthere are up to seven band-touching points per valley.More quantitatively, for 0 ≤ t , t ≤
1, one obtains theexistence domains for additional solutions when | x ± | < | x + | < t ≥
110 and 2( (cid:113) t − t − t ) ≤ t ≤ t . (21)Similarly, | x − | ≤ (cid:18) t ≤
110 and t ≥ t (cid:19) or (cid:18) t ≥
110 and t ≥ (cid:113) t − t − t ) (cid:19) . (22)The existence domains are represented graphically inFig. 5. Note that the two solutions coexist when t ≤ t . In the coexistence region one can generally ex-pect to have seven Dirac points per valley (see Fig. 6).In their existence domain, Eq. (16) determines the chi-rality in the K valley χ ( x ± ) = ± sgn( t − t ) . (23) . . . . . . t . . . . . . t . . . . . . t (a) (b)FIG. 5. (Color online) (a) | x − | ≤
1. (b) | x + | ≤
1. Existencedomains and corresponding chirality of solutions | x ± | ≤ t , t ) parameter space and in the K valley. The regionwith positive (negative) chirality is represented in red (blue).The green line t = 2 t where chirality changes is associ-ated with the existence of Fermi lines instead of Dirac points.The supermerging point t = and t = at the inter-section of the t = 2 t line with the domain border curve t = 2( (cid:112) t − t − t ) is indicated in yellow. − π − π/ π/ π − π − π/ π/ π FIG. 6. (Color online) The zero lines of h ( k ) = 0 (in green)and h ( k ) = 0 (in red) for N4 graphene. A small perturbation(+0 . t at the merging point t = 2 / t = 1 / K . In the insetthere is a zoom around K . The Dirac points are representedby full circles, • ; there is a central K Dirac point in black,and two sets of satellite Dirac points, in blue and red.
However, the chirality information is exact when the so-lutions x ± stand for Dirac points. The model presentsa rich phenomenology and the investigation of the so-lutions indicates that for particular parameters there arealso band touchings different from the simple Dirac pointcase. .
24 2 .
32 2 .
40 2 . k x − . − . . . . k y . . . . . . . . . . FIG. 7. Cross section slice through the energy dispersion ofthe conduction band near zero energy. A triplet of semi-Diracpoints is formed around the central Dirac point at K . On theparameter line t = 2( (cid:112) t − t − t ) (here with t = 0 .
25) thetriplets of satellite points merge on the high-symmetry linesto form the semi-Dirac points.
Remember that each solution x ± stands for a tripletof solutions at each valley. Then there are different sce-narios for the behavior of Dirac points. There are casessimilar to the N3 graphene where there is a single tripletof solutions merging to the central Dirac point to yield apoint with high-energy dispersion. There are cases wherethe two triplets merge with one another to yield a newtriplet of band touchings with quadratic dispersion in onedirection and linear in the other. There is also a uniquesupermerging point where all Dirac points in a valleymerge. A completely new feature to the phenomenologyof band touchings in N4 graphene is the formation ofFermi lines for specific values of parameters.The first case is that of a line in parameter space whereonly one triplet given by the | x ± | -solutions merges withthe central Dirac point. These are obtained under thecondition that either x + = − / x − = − / x ± = − ⇐⇒ t = 1 − t , t ≶ t . (24)A different case is that of the triplet satellite Diracpoints merging two by two to form semi-Dirac points,i.e. band touchings with quadratic dispersion in the di-rection of merging and linear in the direction perpen-dicular to it. They correspond to a scenario wheretwo Dirac points with opposite chirality collide. Fromthe condition x + = x − , they are determined on the line t = 2( (cid:112) t − t − t ). This case is represented in Fig. 7At the intersection of line t = 2 t with the domainborder curve t = 2[ (cid:112) t (1 − t ) − t ] ( t = and t = ),there is a supermerging point where there is a unique band touching per valley that can be understood as acollision of all additional Dirac points into the central( ± K ) one. This point in parameter space has a topologi-cal charge given by the sum of all Dirac point chiralities.Because of the cancellation of the triplet charges, the finalpoint will have a charge ± ± K . Expandingin small momentum q = q (cos θ, sin θ ) around the super-merging point at ± K yields an effective f function in the K valley, f ( ± K + q ) = − q e ∓ iθ + O ( q ) . (25)This result reinforces the sum rule calculation by showinga band touching with cubic dispersion, but with a lowtopological charge ± ± K .Finally, there is a phenomenologically new situationthat is absent in the previously studied N3 graphene.Note that on line t = 2 t the chirality (23) is zero eventhough there are nondegenerate solutions x − (cid:54) = x + , awayfrom the supermerging. This case corresponds to the ex-istence of closed Fermi lines in the Brillouin zone thatlink the two solutions. Hence, in contrast with the casesstudied until now, here the energy dispersion exhibits aline of zeros outside the T line. One of the cases is repre-sented in Fig. 8, where the x ± solutions are connected bya Fermi line. However, even for a vanishing x + solution,Fermi lines subsist and link band touchings associatedonly to x − solution. Aside from these numerical observa-tions of the Fermi line at t = 2 t , it remains a dauntingtask to analytically solve for general solutions away fromthe high-symmetry lines. However, one can investigateanalytically the peculiarity of this case by considering thebehavior of the energy dispersion near the T lines. Theabsolute value of the energy for t = 2 t is E = | t x + 4 t xy − t + 1 | (cid:112) x + 4 xy + 1 , (26)where x = cos( √ k x /
2) and y = cos(3 k y / k y directionfor a zero-energy solutions x ± on the T line k (1 ,
0) van-ish at all orders. This indicates that the solution x ± arenot longer pointlike band touchings, but extend as Fermilines in the k y direction. G. Supermerging at ± K in N n graphene In the two preceding sections it was shown that for N3and N4 graphene it is possible to adjust the parameters t , t so that for each valley all the additional touchingpoints merge with the usual Dirac points at ± K (a super-merging point). This means that x = − / h for N3 graphene, t = 1 / t = 2 / , t = 1 / t n for which all the additionaltouching points merge with the usual Dirac points at K appears to be valid for all N n graphene and relies essen-tially on the fact that the polynomial h ( x ) is of a degree FIG. 8. An expansion in small momentum q around the K point of N4 graphene illustrates the formation of Fermi lines(lines of zeroes for the energy dispersion) around the regularDirac point K in graphene, on the parameter line t = 2 t .The hopping parameters are chosen near the supermerging at t = 2 / δ and t = 1 / δ with δ = 0 . equal to or less than the number n − t n . [More precisely, it can be proven that a model withhopping terms at a chemical distance m will result inpolynomial h ( x ) of maximum order m .] Note that be-cause the number of free parameters grows faster than thedegree of the polynomial there are no longer unique su-permerging points for graphene N m , with m >
7. At thissupermerging the components c x,y vanish and thereforeone needs to go beyond a linear expansion to characterizethe neighborhood of K . As an example, it was shown inRef. 10 that for N3 graphene at the supermerging one ob-tains f ( ± K + q ) (cid:39) q e ± iθ which now identifies a gaplessquadratic dispersion, with a phase that is understood asresulting from the sum of the respective chirality of allthe merging Dirac points. Similarly, for N4 graphene, itfollowed that f ( ± K + q ) (cid:39) q e ∓ iθ . The location of theunique supermerging band touching and their associatedtopological charge are given in Table II. Note that theenergy dispersion of supermerging band touchings hasa higher than linear dispersion. However, the topolog-ical charge of the converging triplets of satellite pointsis alternating and hence the resulting topological chargeremains low.Finally, note that the above scenario of a unique super-merging together with an alternating ± ∓ ± K (see Table II) is not generally validin N n graphene, and in fact it already breaks down inthe N8 model. For N8 graphene, the supermerging isno longer unique, but becomes a line in ( t , t , t , t ) pa-rameter space. Nevertheless, Eq. (9) implies that in N7and N8 graphene there is the same number of satelliteDirac points per valley, because p ( x ) remains a third or-der polynomial. An expansion near the supermerging TABLE II. Supermerging characteristics at K . The function f from the effective low-energy Hamiltonian H eff = σ + f +H . c . is written as a function of small momenta π = q x + iq y and up to a multiplicative constant which is neglected.Graphene Supermerging f ( K + q ) Charge t t t t N1 1 0 0 0 π ∗ / π − / / π ∗ π /
12 1 / / π ∗ π − point for N7 graphene (see Table II) yields in K valley f ( K + q ) = 2764 π ∗ (cid:2) π − t ( π − π ∗ ) (cid:3) , (27)with π = q x + iq y . For vanishing t , one recovers atopological charge − K , atthe supermerging in N7 graphene. However, when t reaches the critical value 1 /
12 the band touching clearlyexhibits the topological charge 4. This scenario canbe explained by a change in chirality for a triplet ofDirac points in the vicinity of the supermerging line(1 − − → (1 − n graphene model was shown to exhibit morethan one touching point in each valley. Now it remains toanswer the question whether large Chern number phasesbecome possible when gapping them with a Haldanemass. As long as that the position of the band touch-ings and their respective chirality is known, determiningthe topological phase diagram is within analytical grasp. IV. CHERN NUMBER PHASE DIAGRAM FORTHE LONG DISTANCE HOPPING HALDANEMODEL
The Haldane model is built on the hexagonal latticefor N1 graphene by adding N2 (intrasublattice) hop-ping t , such that when hopping is performed clockwisein the unit cell an electron gains a phase φ . Howeverthere is no net magnetic flux in the unit cell. The N2hopping term leads to two contributions of respectiveform h ( k ) σ with h ( k ) = h ( − k ) and h ( k ) σ with h ( k ) = − h ( − k ). These two contributions break chiralsymmetry, but do not break inversion symmetry. Thefirst contribution breaks particle-hole symmetry, whilethe second breaks time-reversal symmetry. As noted be-fore, the first contribution does not weight on the Chernnumber calculation and therefore can be discarded, pro-vided the second contribution produces the necessarymass term from Eq. (5) that gaps the Dirac points. Thatis to say, the topological properties of each band are un-affected by smooth deformations that preserve a finite TABLE III. The first hopping integrals t n contributing to theHaldane mass. The hopping distances are expressed in unitsof lattice constant.Hopping Physical distance Chemical distance t √ t t √ direct gap at all momenta. As we shall see later, the sec-ond contribution h σ allows for a Chern phase diagramwith only odd (even) Chern number phases when addedto the N n graphene model. In order to have a Chernphase diagram allowing for transition between even andodd Chern number phases, it is necessary to add a massterm that breaks inversion symmetry. The simplest suchterm is of the form M σ and corresponds to a differenton-site potential energy on each sublattice.The mass term h σ in the original Haldane modelbreaks time-reversal and inversion symmetry. It reads h = M − t sin φ { sin( k · a ) − sin( k · a )+sin[ k · ( a − a )] } . (28)When intrasublattice hopping between distant sites is al-lowed, the generalized mass term reads h = M − (cid:88) n t ( n ) sin( nφ ) { sin( n k · a ) − sin( n k · a )+ sin[ n k · ( a − a )] } , (29)where n is an integer that indicates that hopping takesplace between AA or BB sites situated at a distance of n √ a . Here will be considered only the first two termsin this expression, corresponding to a hopping across twounit cells (see Fig. 1 and Table III). The term containingthe hopping integral t just multiplies the identity Paulimatrix and is neglected. Interesting for the topology ofthe problems are hoppings along the links where the elec-trons gain the phase φ . Here only the first two terms inthe mass term are considered: t and t .The goal of this part is to illustrate how gapping thegraphene system with 2 n Dirac points can yield Z topo-logical phases characterized by a large Chern number (upto C = ± n ). The following sections investigate caseswhere different mass term gaps the previously obtainedN n graphene. The strategy will be to illustrate the pos-sibility of large Chern phases by considering first the ac-tion of t Haldane mass on different models of N n inSec. IV A. In Sec. IV B it is shown that the addition of t terms allows one to further increase the absolute valueof the Chern number. A. t Haldane model
N1 graphene with a hopping t constitutes the origi-nal Haldane model. The phase diagram is obtained byobserving that h changes sign between the Dirac points − π − π/ π/ πφ − − − M / t − FIG. 9. (Color online). Chern number phase diagram for theHaldane Hamiltonian as a function of the on-site energy M divided by the hopping integral t as a function of the flux φ . The topologically nontrivial insulating phases are coloridentified and have the topological index denoted inside therespective regions. The topologically insulating regions, C =0, are white. ( ∓ K ) of graphene. Therefore the Hamiltonian exhibitsthree topological phases: a trivial insulating phase andtwo C = ± C = 12 (sgn M − − sgn M + ) , (30)where M ± = M ∓ √ t sin φ is the mass term at ∓ K .The phase diagram is represented in Fig. 9. The lines M ± = 0 represent topological transition lines where thebulk gap closes at least at one of the ± K points.Larger Chern phases become possible when the under-lying model is N3 graphene. Now the mass term takesdifferent values between a regular Dirac point and itssatellites. Therefore the topological charges can add upto yield Chern |C| = 2 phases.Momentum ± k i locates any satellite point of ± K and,manifestly, the expression for χ ( k i ) holds in the range ofexistence of separate satellite points.Let us define the mass at the regular Dirac points M ± = h (cid:0) ∓ ( π √ , (cid:1) . Similarly, the mass at the satel-lite Dirac points k (cid:54) = K is denoted by m ± = h ( k ) invalley ∓ K . Then from Eq. (5) it follows that the Chernnumber is C = 12 (cid:20) (sgn M − − sgn M + ) − m − − sgn m + ) (cid:21) (31)where the mass of the Dirac points read M ± = M ∓ √ t sin φ,m ± = M ∓ t t (1 + t ) (cid:115) − (cid:18) − t t (cid:19) sin φ. (32)0 − π − π/ π/ πφ − − − M / t − − − FIG. 10. (Color online) Chern number phase diagram for the t Haldane model on N3 graphene. The hopping parametersare t = 1 / t = 0 .
35 in units of t . Equation (31) yields the phase diagram for the systemwhen all eight Dirac points are present. When there areno satellite Dirac points [ t ∈ ( − , / t = 0 and therefore it has the phase diagram in Fig. 9.When t is varied to go outside the region ( − , / M = 0 line. For example, from Eqs. (32), we see that at M = 0 a regular Dirac point and its satellites will havethe same mass. Therefore the Chern number reduces to C = sgn M + − sgn M − . This yields topological phases in-dexed by ±
2. By increasing | M | , one crosses a transitionline where the Haldane mass of all satellite points in thesystem becomes identical, while it remains different forthe regular Dirac points. This transition is given by m ± = 0 . (33)This region extends up to the the last topological transi-tion line given by M = ± √ t sin φ . In this region theChern number reduces again to the original case ( t = 0)with C = 1 / M − − sgn M + ). When M is increasedeven further, all Dirac points are gapped identically andtherefore this is the topologically trivial region. In Fig. 10is represented a typical phase diagram for the case wheresatellite Dirac points are present.Note that at the merging point t = 1 / C = ± C = ± M = ± √ t sin φ .Then at the topological transition from the |C| = 2 phaseto the trivial insulator, there is a quadratic band touchingthat is represented in Fig. 11(a).The phase diagram in the N3 Haldane model (Fig. 10)has the nice feature that it accommodates lines of tran-sition where the Chern number changes by three units.This is realized by the formation of three Dirac points atthe topological transition. These band touchings come from the vanishing of the Haldane mass at the three satel-lite Dirac points previously found in N3 graphene. Forexample, let us take parameters t = 1, t = 1 / t = 0 .
35 from the phase diagram in Fig. 10. Then fixing φ = π/
2, there are two transition points between C = − C = 1 phases near ± K . In particular, near − K ,the Dirac points form at the satellites where m + = 0.The energy dispersion at the topological transition is il-lustrated in Fig. 11(b).Similarly one can take as the underlying model the N4graphene model which contains the t hopping. This wasshown to produce seven Dirac points per valley. Henceone can expect the presence of larger Chern phases. Thisis exemplified in Fig. 12, where a choice of particularparameters yields |C| = 4 QAH phases. Note the presenceof multiple Dirac points is reflected in the phase diagramas a multiplication of transition lines in the M directionfor fixed magnetic flux φ ( (cid:54) = 0 , π ). B. t Haldane model
The existence of 2 n Dirac points for a submodel con-taining only two sigma matrices allows one, in principle,to build topological insulators with Chern phases C = n .For the N3 graphene model with eight Dirac points, onecan have a large Chern number C = ±
4. To actualizeall possible topological phases it is sufficient to add a t mass term. It has the effect to produce oscillations inthe phase dependent Haldane mass, such that the termchanges sign between a regular graphene Dirac point andits satellites in N3 graphene. As expected, all phases areattainable under this modification of the Hamiltonian.The mass term becomes h = M − t sin φ { sin( k · a ) − sin( k · a )+ sin[ k · ( a − a )] } − t sin(2 φ ) { sin(2 k · a ) − sin(2 k · a ) + sin[2 k · ( a − a )] } . (34)The new phase diagram is computed by considering themass term (34) at the eight N3 graphene Dirac points.Then the topological transition lines are given by thezeros of the new mass terms, M (cid:48)± and m (cid:48)± expressed asa function of the previous mass terms from Eq. (32), M (cid:48)± = M ± ± √ t sin 2 φm (cid:48)± = m ± ∓ t sin 2 φ (2 sin 2 κ − sin 4 κ ) , (35)where κ = arccos[( t − / (2 t )] in the domain of exis-tence of the satellite Dirac points in N3 graphene.The dependence of the mass term on sin 2 φ makes pos-sible large Chern number phases |C| = ± − k x − − − k y − − − E − − k x − − − k y − − − E − − − − FIG. 11. (a) Energy dispersion at the topological transition between C = − C = 0 phases at the merging point betweenthe regular − K and its three satellites − k i in N3 graphene. The energy dispersion in N3 Haldane shows a quadratic bandtouching at − K . The parameters are chosen φ = π/ M = √ t = 1 / t = 1 /
2. (b) Energy dispersion for the N3Haldane model at the transition between C = 1 and C = − − K for t = 1 / t = 0 .
35 in units of t . The change in Chern number by three units is reflected in the presence of three Dirac conesat the topological transition. − π − π/ π/ πφ − . − . − . − . . . . . . M − − − − − − FIG. 12. Chern number phase diagram showing the existenceof 2 sets of satellite Dirac points and large QAH phases in t t = 1, t = 1 / t = 0 .
59 and t = 0 . by numerical integration over the BZ in Eq. (4) and theresults were in agreement.Let us consider briefly the case of N4 and N7 grapheneby adding, respectively, t and t hopping terms. Withhopping integral t fixed as before, there are two freeparameters t and t . The parameter space becomes toolarge to describe analytically the dynamics of the Diracpoints and to track at the same time the sign of themass at the Dirac points. The general thesis,however,remains correct. Larger and larger QAH phases becomepossible. In the case of N4 graphene there is a maximumof six Dirac points near a K point; for N7 graphene thereare nine possible Dirac points per valley. That indicatesthat with a proper mass term one can have the largest − π − π/ π/ πφ − − − M − − − − − − − − FIG. 13. All QAH phases possible for N3 graphene with t Haldane mass; here the phase diagram for the parameterchoice t = 1, t = 1 / t = 0 .
35, and t = 0 .
26 illustratesthis point. For M = 0, the possible Chern phases have onlyeven Chern numbers. Chern phases |C| = 7 (in N4 graphene) or |C| = 10 (in N7graphene). In Fig. 15 is represented a Haldane t masson a N4 graphene with QAH phases |C| ≤
5. It appearsthat one needs even longer hopping terms in the Haldanemass to realize the largest |C| = 7 phase.Note that in all the cases the presence of distant-neighbor hoppings in Haldane mass potentially leads tomore bulk gap closings at a given on-site energy M for φ varying from − π to π . This is reflected in the struc-ture of the phase diagrams, which present oscillations ofthe topological phase boundaries in the φ direction. Thisaccounts for the oscillatory nature of the Haldane mass,2 − π − π/ π/ π − π − π/ π/ π K − K FIG. 14. (Color online). A Dirac point that is represented by • ( ◦ ) has chirality + ( − ). The colored lines represent lines ofzeros for h (green), h (red), and the mass term h (blue).The regular Dirac points placed at (cid:0) ± π √ , (cid:1) are gapped bya Haldane mass that has opposite sign. Also the mass termchanges sign between the regular Dirac point and its satellites.For parameters t = 1, t = 1 / t = 0 . t = 0 . M = 0and φ = π/ C = − which can pass more times through zero (as a function ofthe flux), when it contains strong distant-neighbor hop-ping terms. V. CONCLUSION
We have shown that in a graphenelike system addingdistant-neighbor hopping integrals leads to the appari-tion of satellite Dirac points in the spectrum near the reg-ular ± K points of graphene. The number of additionalDirac points grows as more distant-neighbors are con-sidered. Here, Dirac points up to N7 (next × AB sites potentially yieldsa triplet of Dirac points near K (and because TRI, atriplet at − K ). For N7 graphene there is a maximum ofthree triplets of satellites created.The position of the nodes in the dispersion requiressolving a polynomial whose degree grows as more dis-tant neighbors are considered. Analytically, one canhope to determine their position only for a limited num-ber of added neighbors (here N4 graphene). Already,for N4 graphene, the investigation revealed a rich phe-nomenology for band touchings in the system. BesidesDirac point band touchings, there are semi-Dirac points − π − π/ π/ πφ − − − − M − − − − − − − −
22 1 − − −
11 00 00 FIG. 15. Haldane model from N4 graphene with a t massterm. Hopping integrals t = 1, t = 1 / t = 0 . t = 0 . t = 0 .
35. For M = 0, the possible Chern phases have onlyodd Chern number. (band touchings with a linear dispersion in one directionand quadratic in the other), or higher-energy dispersionpoints. Among the latter, we show that there is a uniquesupermerging band touching at ± K that can be under-stood from a collision scenario of all possible Dirac pointsunder a variation of the hopping integrals. Their unique-ness in hopping integral parameter space indicates thatthey are extremely unstable. Moreover, the peculiarityof this point resides in the fact that is characterized bya high-energy dispersion, but a low topological charge.This is due to the fact that the supermerging points resultfrom a union of Dirac points organized in triplets withalternating chirality. Numerical and analytical investiga-tions also revealed a new phenomena in N4 graphene: theformation of Fermi lines for a particular choice of param-eters. The particular constraints to obtain them indicateagain that they are unstable band touchings.The creation of multiple Dirac points is a preconditionto achieve phases with a large Chern number. This isput to test by implementing the Haldane model in thedistant-neighbor hopping graphene. The Haldane massterm gaps the Dirac points such that new QAH phasesappear. We have presented various Chern number phasediagrams to illustrate the role of distant hoppings in theHaldane mass term. The flux dependence allows one toresolve neighbor Dirac points with the different chiralityby gapping them with an opposite mass. Said differently,the mass term now changes sign not only between K and − K , but also between the satellite created near theregular Dirac points. In principle, for 2 n Dirac points inthe modified graphene, phases with Chern number |C| = n can be created.As a final remark concerning these N n graphene-Haldane models, we stress that we do not claim thatsuch long-range hopping is relevant to graphene physics.We believe, however, that the phenomenology of com-3plex band touchings and large Chern number phases thatappears in this two-band long-range hopping model israther universal and might appear as the effective lowenergy physics of a more realistic nearest-neigbor modelwith N orbitals or N atoms per unit cell. In support ofthis view there is a recent work that establishes a map-ping of the low energy physics of the bilayer graphene(four atoms per unit cell) with that of N3 graphene near supermerging. ACKNOWLEDGMENTS
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