Multilayer Haldane model
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Multilayer Haldane model
Xi Wu, ∗ C.X. Zhang, † and M. A. Zubkov ‡ § Physics Department, Ariel University, Ariel 40700, Israel
We propose the model of layered materials, in which each layer is described by the conventionalHaldane model, while the inter - layer hopping parameter corresponds to the ABC stacking. Wecalculate the topological invariant N for the resulting model, which is responsible for the conduc-tivity of intrinsic quantum Hall effect. It has been shown that in a certain range of the values ofinterlayer hopping parameter, the value of N is equal to the number of layers multiplied by thetopological invariant of each layer. At the same time this value may be calculated using the lowenergy effective theory. I. INTRODUCTION
Quantum Hall Effect (QHE) [1–3] is one of the most remarkable phenomena in solid state physics. The quantizationof Hall conductivity is so precise that it may be used as an etalon of its physical unit. The quantization is providedby the topological properties of matter. Originally the QHE has been considered in the presence of external magneticfield. Later the models of intrinsic anomalous QHE have been proposed, in which the QHE exists without externalmagnetic field. The very first model of this type is the so - called Haldane model [4]. It reveals correspondencewith the tight - binding model of single - layer graphene [5]. Several extra terms are added to the latter model thatprovide a non - trivial band topology resulted in the QHE. The Haldane model itself does not describe any realmaterials. However, qualitatively it describes the wide class of two - dimensional solids - state systems called nowChern insulators [6, 7].The quantum Hall conductivity of intrinsic QHE may be expressed through the topological invariant N composedof the two - point Green function [8–10]. This invariant is also relevant for the description of topological phenomenain Helium-3 superfluid [11]. As well as the ordinary QHE the intrinsic QHE may be both integer and fractional [6, 7].In the majority of solid state systems the values of N are 0, 1, −
1. Larger values of N are more rare. The systemswith arbitrary values of the topological invariant have been considered, for example, in [12, 13].In the present paper we propose the model of two - dimensional layered systems based on the analogy to the multi- layer graphene [14–16]. Various multi - layered systems are well - known in solid - state physics (see, for example,[17] and references therein). The model considered in our present paper may be called the multilayer Haldane model(with ABC stacking). We demonstrate that its value of N is equal to the topological invariant of conventional (mono- layer) Haldane model multiplied by the number of layers, in a certain range of inter - layer hopping parameter. Asit was mentioned above, the Haldane model is defined on the hexagonal lattice. Therefore, our multi - layer HaldaneHamiltonian is related to the mono - layer Haldane Hamiltonian in the way similar to the models of multi - layergraphene [14–16]. II. THE MULTI - LAYER HALDANE MODEL
Thus in the case of ABC stacking, the multi - layer Haldane Hamiltonian is given by H n = H t T t H t T t H t T ... t H n × n , (1) ‡ On leave of absence from NRC ”Kurchatov Institute” - ITEP, B. Cheremushkinskaya 25, Moscow, 117259, Russia ∗ Electronic address: [email protected] † Electronic address: [email protected] § Electronic address: [email protected]
Here t = (cid:20) t ⊥ (cid:21) , while t ⊥ is the inter - layer hopping parameter. H is the Hamiltonian of mono - layer Haldanemodel H = 2 t cos φ X i cos( p · b i ) σ + t X i [cos( p · a i ) σ + sin( p · a i ) σ ] + [ M − t sin φ X i sin( p · b i )] σ , (2)where | a | = | a | = | a | = a cos h a , a i = cos h a , a i = cos h a , a i = − b i = 12 ǫ ijk ( a j − a k ) . (3)We can also represent the Hamiltonian of the multi - layer system as follows H n = h σ + X i =1 , , h i σ i + ˆ Y t ⊥ , (4)with h = 2 t cos φ X i cos( p · b i ) (5) h = t X i cos( p · a i ) (6) h = t X i sin( p · a i ) (7) h = [ M − t sin φ X i sin( p · b i )] . (8)Here ˆ Y = y T y y T y y T ...y n × n , (9)with y = (cid:20) (cid:21) while by σ i we denote the ”repeated” Pauli matrices: σ i = σ i σ i σ i ... σ i n × n , (10)Commutation relations follow: { σ i , σ j } = 2 δ ij , i, j = 0 , , , { σ , σ j } = 2 σ j , j = 1 , , { σ , ˆ Y } = 0 { σ , ˆ Y } = Σ { σ , ˆ Y } = Σ { σ , ˆ Y } = 2 ˆ Y ˆ Y = 1 ± (11)where 1 ± = − ... + n × n , (12)with 1 − = (cid:20) (cid:21) + = (cid:20) (cid:21) while Σ = ... n × n , (13)and Σ = − ii − ii − i...i n × n , (14)Propagator may be represented as G n = (( E − h ) σ − X i =1 , , h i σ i − ˆ Y t ⊥ ) − = (( E − h ) σ + X i =1 , , h i σ i + ˆ Y t ⊥ ) (cid:16) ( E − h ) − X i =1 , , h i − t ⊥ ± − t ⊥ Σ h − t ⊥ Σ h (cid:17) − (15)The energy levels E are given by solutions of equation0 = det (( E − h ) σ − X i =1 , , h i σ i − ˆ Y t ⊥ )= det σ (( E − h ) σ − X i =1 , , h i σ i − ˆ Y t ⊥ ) σ = det (( E − h ) σ − h σ + X i =1 , h i σ i + ˆ Y t ⊥ ) (16)It gives 0 = det (( E − h ) σ − X i =1 , , h i σ i − ˆ Y t ⊥ )(( E − h ) σ − h σ + X i =1 , h i σ i + ˆ Y t ⊥ )= det (cid:16) ( E − h ) − X i =1 , , h i − t ⊥ ± − t ⊥ Σ h − t ⊥ Σ h (cid:17) (17)or 0 = d n ( E )where d n ( E ) = det F − ( E ) − t ⊥ ( h − ih ) − t ⊥ ( h + ih ) F ( E ) − t ⊥ ( h − ih )0 − t ⊥ ( h + ih ) F ( E ) − t ⊥ ( h − ih ) ... − t ⊥ ( h + ih ) F + ( E ) n × n , (18)and F ( E ) = ( E − h ) − X i =1 , , h i − t ⊥ F + ( E ) = ( E − h ) − X i =1 , , h i − t ⊥ + F − ( E ) = ( E − h ) − X i =1 , , h i − t ⊥ − Let us consider, for example, the particular case n = 2. Then d ( E ) isdet ( E − h ) − P i =1 , , h i − t ⊥ ( h − ih ) 00 ( E − h ) − P i =1 , , h i − t ⊥ − t ⊥ ( h − ih ) − t ⊥ ( h + ih ) 0 ( E − h ) − P i =1 , , h i − t ⊥ − t ⊥ ( h + ih ) 0 ( E − h ) − P i =1 , , h i Let us denote F ( E ) = ( E − h ) − P i =1 , , h i . We come to the following equation0 = F ( E )( F ( E ) − t ⊥ ) − F ( E )( F ( E ) − t ⊥ ) t ⊥ ( h + h ) + t ⊥ ( h + h ) (19)The energy bands can be found analytically: E = h ± vuut X i =1 , , h i + t ⊥ (cid:16) ± s h + h t ⊥ (cid:17) (20)One can see, that when the value of t ⊥ is increased from zero, the gap is not closed as far as h remains nonzero.However, it becomes infinitly close to zero at t ⊥ → ∞ at the points, where h ( p , p ) = 0. Such points do exist ifthe corresponding monolayer Haldane model has nonzero topological invariant. We do not exclude, however, that forthe larger number of layers the gap might be closed for a certain finite critical value t ( c ) ⊥ . But even so, there shouldobviously exist the region of the parameter 0 < t ⊥ < t ( c ) ⊥ , where the gap remains open. III. TOPOLOGICAL INVARIANT FOR HALL CONDUCTIVITY
The topological invariant [8–10] responsible for the conductivity of intrinsic QHE, is defined as follows N [ G ] = 13! Z d p (2 π ) ǫ ijk Tr( G ∂ i G − G ∂ j G − G ∂ k G − ) . (21)It is proportional to Hall conductivity: σ H = N [ G ]2 π . In these expressions G is the two - point Green function ofelectrons. Starting from Eq. (19), the inverse of Green’s function in multilayer Haldane model (with n layers) can bewritten as G − = Q t T t Q t T t Q t T ... t Q n × n , (22) FIG. 1: Dependence of energy on momenta in the units of t (the lowest bands) for the 5 - layer Haldane model at 2 t cos φ =2 t sin φ = 0 . t , M = 0 . t , t ⊥ = t . where Q = iω − H .If a matrix G is a direct tensor product of the two other matrices G and G , then the topological invariant, or thewinding number, of G will be the sum of the topological invariant of G and that of G . Namely, if G = (cid:20) G G (cid:21) ,then N [ G ] = N [ G ] + N [ G ] [18].The proof is trivial. Namely, we have G − = (cid:20) G − G − (cid:21) , N [ G ] = 13! Z d p (2 π ) ǫ ijk Tr( G ∂ i G − G ∂ j G − G ∂ k G − )= 13! Z d p (2 π ) ǫ ijk Tr( (cid:20) G G (cid:21) ∂ i (cid:20) G − G − (cid:21)(cid:20) G G (cid:21) ∂ j (cid:20) G − G − (cid:21) (cid:20) G G (cid:21) ∂ k (cid:20) G − G − (cid:21) )= 13! Z d p (2 π ) ǫ ijk Tr( G ∂ i G − G ∂ j G − G ∂ k G − + G ∂ i G − G ∂ j G − G ∂ k G − )= N [ G ] + N [ G ] . (23)When t ⊥ = 0, Eq. (22) becomes G − = Q Q Q ... Q n × n , (24)Therefore, N [ G n ] = n N [ G ] . (25)One can see that without the inter - layer hopping, the n-layer Haldane model would have the topological invariantwith the value equal to the sum of the topological invariants of each layer. Although above we considered the case,when all layers have the same Hamiltonian, this is true even when the Hamiltonians of the layers differ from eachother.According to remark of the previous section when the value of t ⊥ is increased starting from zero, the value of theHall conductivity remains the same if the Fermi energy is tuned in order to remain inside the gap. This value remainsconstant until the gap is closed (which may occur for a certain critical value of t ⊥ ). IV. EFFECTIVE LOW ENERGY THEORY
Let’s consider the case when the interlayer hopping parameter is much larger than the gap of the monolayer Haldanemodel. This corresponds qualitatively to the real situation that takes place in multilayer graphene [5]. In such a case,there is an effective description in which the topological invariant can be directly computed. At the would - be Fermipoints K and K ′ of graphene the off-diagonal parts of the Hamiltonian for the monolayer Haldane model vanish. Inthe small vicinity of K we can write the Hamiltonian of monolayer Haldane model as (the similar expression will beat the K ′ point): H = (cid:20) h + h vπ † vπ h − h (cid:21) , (26)where π = ( p − K ) + i ( p − K ), and h = M − t sin φ X i sin( p · b i ) ,h = 2 t cos φ X i cos( p · b i ) ,h is nonzero in a vicinity of K . The interlayer hopping parameter is much larger than the gap, which gives that t ⊥ >> | h | or t ⊥ >> t , M . At this point we assume that the gap remains open in the given model, and the Fermienergy is inside the gap.Let us show now that the effective 2 × H effn = " h ( n )3 h ( n ) ( vπ † ) n h ( n ) ( vπ ) n − h ′ ( n )3 , (27)using the mathematical induction similar to that of [16]. Here certain functions h ( n ) , h ′ ( n )3 , h ( n )3 depend on the sizeof the matrix. First, indeed the monolayer Haldane model has such a form. Next, suppose that the n-layer Haldanemodel has the effective 2 × h ( n )3 h ( n ) ( vπ † ) n h ( n ) ( vπ ) n − h ′ ( n )3 t ⊥ t ⊥ h vπ † vπ − h . (28)The transformation based on the following matrix S = brings the Hamiltonian to the following form h ( n )3 h ( n ) ( vπ † ) n − h vπvπ † h t ⊥ h ( n ) ( vπ ) n t ⊥ − h ′ ( n )3 = (cid:20) ( H P P ) × ( H P Q ) × ( H QP ) × ( H QQ ) × (cid:21) . (29)We can consider H P Q and H QP as perturbations in a vicinity of π = 0, when t ⊥ >> | h | . Then according to thedegenerate state perturbation theory the effective 2 × H eff ≈ H P P − H P Q H QQ H QP . (30)Applying Eq. (30) to Eq. (29) we get the the effective 2 × n + 1) - layer Haldanemodel of the form of Eq. (27), in which functions h ( n ) , h ′ ( n )3 , h ( n )3 are defined by recursive relations: h (1)3 = h + h , h (1) = 1 , h ′ (1)3 = h − h h ( n +1)3 = h ( n )3 + h + | vπ | n ( h ( n ) ) ( h + h ) t ⊥ + ( h + h )( h ′ ( n )3 − h ) h ( n +1) = − t ⊥ h ( n ) t ⊥ + ( h + h )( h ′ ( n )3 − h ) h ′ ( n +1)3 = h − h + | vπ | t ⊥ + ( h + h )( h ′ ( n )3 − h ) ( h ′ ( n )3 − h ) (31)Recall that t ⊥ ≫ h . In the following we consider small vicinity of K , where | vπ | ≪ t ⊥ . One can see, that function h ( n ) in this vicinity is close to the constant − t ⊥ ) n − , while h ( n )3 is close to function h + h , which remains almostconstant in this vicinity. V. TOPOLOGICAL INVARIANT FOR THE LOW ENERGY EFFECTIVE THEORY
If the gap is not closed when the value of t ⊥ is increased from zero to the given value, we may deform the Hamiltoniansmoothly to the form, in which it is equal to H ′ effn +1 = (cid:20) h h ( n +1) ( vπ † ) n +1 h ( n +1) ( vπ ) n +1 − h (cid:21) . (32)It is possible to make this deformation in such a way that the poles of the Green function do not appear. Therefore,the given deformation cannot lead to modification of the topological number under consideration [12, 19]: N [ H ] = 14 π Z d p ǫ abc ˆ h a ∂ p x ˆ h b ∂ p y ˆ h c , (33)where ˆ h a = g a pP a g a and g = h , g + iq = h ( n +1) ( vπ † ) n +1 . From Eq.(33) we can understand that by stripping the same positive functionfrom all the coefficients, we do not change the value of the topological invariant. Thus, we can use an even moresimple Hamiltonian H ′′ effn +1 = " h | h ( n +1) | sgn( h ( n +1) )( vπ † ) n +1 sgn( h ( n +1) )( vπ ) n +1 − h | h ( n +1) | . (34)Moreover, the simultaneous change of the signs of h and h will not affect the topological invariant. Therefore, wedo not need to worry about the sign of h ( n +1) , and get a further simplified Hamiltonian with the same value of thetopological invariant as that of Eq.(34) H ′′′ effn +1 = " h | h ( n +1) | ( vπ † ) n +1 ( vπ ) n +1 − h | h ( n +1) | . (35)This can already be put into Eq. (33) for the direct computation, i.e. we set g = h | h ( n +1) | , g + iq = ( vπ † ) n +1 . Theresult is N [ H effn +1 ] = ( n + 1) N [ H ] . (36)The similar procedure can be applied to the vicinity of the K ′ point. Finally, we obtain that the topological invariant(and thus the Hall conductivity) for the low energy effective theory is equal to the number of the layers times thevalue of topological invariant (the Hall conductivity) of monolayer Haldane model. This pattern is illustrated by Fig.2. - - - - - p y E FIG. 2: Dependence of energy on momentum p y at p x = 0 in the units of t for the 5 - layer Haldane model at 2 t cos φ =2 t sin φ = 0 . t , M = 0 . t , t ⊥ = t . The orange dashed line represents the energy bands of effective low energy theory.The green dotted line represents the line of zero energy E = 0 counted from the Fermi level. VI. CONCLUSIONS
In this paper, we have studied the topological invariant responsible for the intrinsic QHE conductivity of themultilayer Haldane model with ABC stacking. We showed that the value of the topological invariant is equal to thenumber of layers times the value of the topological invariant of monolayer Haldane model for the sufficiently smallvalues of interlayer hopping parameter. It has the same value also when the value of this parameter is increased untilthe gap is closed if Fermi energy remains in the gap. It appears that the same value is given also by the effective lowenergy model containing only two energy bands.It is worth mentioning, that in our analysis we disregard completely the consideration of both inter - electroninteractions and disorder. According to rather general considerations of [20] the interactions cannot affect the valueof Hall conductivity, at least, on the level of perturbation theory. The similar statement may also be drawn for therole of disorder [21] as long as we consider the value of conductivity averaged over the sample area. In the latter case,however, the Hall current is expected to be concentrated along the boundary.The multi - layered system qualitatively similar to the one discussed here, in principle, may be realized experimen-tally [22, 23] via the appropriate crystal growth. Such a growth being performed would represent an engineering ofthe system with arbitrary large integer value of the topological invariant responsible for intrinsic anomalous quantumHall effect. [1] K. v. Klitzing, G. Dorda, and M. Pepper, Physical Review Letters , 494 (1980).[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Physical Review Letters , 405 (1982).[3] X.-L. Qi, Physical Review B (2008).[4] F. D. M. Haldane, Physical Review Letters , 2015 (1988).[5] M. I. Katsnelson, Materials Today , 20 (2007), ISSN 1369-7021, URL .[6] C. C. Titus Neupert, Luiz Santos and C. Mudry, Physical Review Letters (2011).[7] D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature Communications (2011), URL https://doi.org/10.1038%2Fncomms1380 .[8] K. Ishikawa and T. Matsuyama, Nuclear Physics B , 523 (1987), ISSN 0550-3213, URL .[9] M. F. L. Golterman, K. Jansen, and D. B. Kaplan, Phys. Lett. B301 , 219 (1993), hep-lat/9209003.[10] G. E. Volovik,