Topological states in the Hofstadter model on a honeycomb lattice
TTopological states in the Hofstadter model on a honeycomb lattice
Igor N.Karnaukhov G.V. Kurdyumov Institute for Metal Physics, 36 Vernadsky Boulevard, 03142 Kiev, Ukraine
We provide a detailed analysis of a topological structure of a fermion spectrum in the Hofstadtermodel with different hopping integrals along the x, y, z -links ( t x = t, t y = t z = 1), defined ona honeycomb lattice. We have shown that the chiral gapless edge modes are described in theframework of the generalized Kitaev chain formalism, which makes it possible to calculate the Hallconductance of subbands for different filling and an arbitrary magnetic flux φ . At half-filling thegap in the center of the fermion spectrum opens for t > t c = 2 φ , a quantum phase transition inthe 2D-topological insulator state is realized at t c . The phase state is characterized by zero energyMajorana states localized at the boundaries. Taking into account the on-site Coulomb repulsion U (where U << t << t ∼ U . Thus, in the case of U >
PACS numbers: 73.22.Gk;73.43.-f
INTRODUCTION
In the framework of the Hofstadler model [1], we studythe conductance of the 2D Chern insulator (CI), in whichthe Hall conductance σ xy = e h C γ [2] is realized in the in-sulator state at the rational magnetic flux φ = pq , wherep and q are relatively prime integers, C γ is the Chernnumber of γ -filled subbands. For a square lattice andodd q, the spectral q − φ isknown as the colored Hofstadter butterfly. The coloredHofstadter butterfly has been calculated for the honey-comb [3], triangular [4] lattices. Wiegmann and Zabrodinestablished that the Hofstadter model is related to thequantum group U q ( sl ) [5], the model Hamiltonian isdetermined in terms of the generators of the quantumgroup. Using the exact solution of the Hofstadter modeldefined on a square lattice for the rational flux [5], theauthors [6] have calculated the wave function of spin-less fermions at zero energy in the semi-classical limit at p = 1 and q → ∞ . Near the boundaries of the samplethe wave function of fermions is localized, power-low be-havior of the modulus of the wave function is critical andunnormalizable. In the center of the spectrum, the chiralMajorana fermion liquid is realized at q=N (N is the sizeof the sample), the Bloch states disappear in the bulk ofthe system [7]. In [8, 9] the authors associate peculiari-ties of topological behavior at zero energy with the VanHove anomalies, that exist in each band in the butterflylandscape. In [10] the solution [5] has been generalizedon the honeycomb lattice.At half-filling the gapless state of the spinless fermionsdetermined by the tight-binding model defined on thehoneycomb lattice is unstable [11]. In the case of tak-ing into account the next-nearest-neighbor hoppings offermions, nontrivial stable solutions for the phases, thatdetermine these hopping integrals along the links, lead to spontaneous breaking of time reversal symmetry and CIstate [11]. We show, that in an external magnetic fieldand at half-filling the topological phase transition occursfrom gapless state to gapped topological insulator state.Topological insulators (superconductors) can be char-acterized alternatively in terms of bulk or edge proper-ties, so-called the bulk-edge correspondence. So by cal-culating the winding number of the gapless edge modesalong a loop in the Brillouin zone it is possible to assignto each gap its Chern number or the Hall conductancefor given filling. In the case of an irrational flux the Bril-louin zone is not defined, as a result, we can not calculatethe Chern number integrating the Berry curvature overthe Brillouin zone. We circumvent this problem by com-puting the Hall conductance using a different approach,calculating the total number of gapless edge modes in thegap [12]. We show that the topological properties of theband spectrum reflect the nature of the correspondingchiral gapless edge modes localized at the boundaries ofthe sample, their total number in the gap is conservedunder both rational and irrational magnetic fluxes.In this paper we investigate the topological structureof the spectrum of spinless fermions defined on the hon-eycomb lattice. It is shown that the state of an insulatorwith nontrivial topological properties is realized in a mag-netic field. This state is determined by zero-energy Majo-rana states localized at the boundaries of the 2D system.The proposed approach allows to calculate topologicalnumbers in the case of irrational magnetic fluxes. De-spite numerous theoretical calculations, there is no uni-versal criterion for realizing the topological state in theinsulator with allowance for the short-range Coulomb re-pulsion. Such a criterion should connect the value ofthe gap of a topological insulator with the magnitude ofthe Coulomb repulsion between fermions. Taking intoaccount the exact solution of fermion chain model [13],we calculate the stability of topological state in the Hof-stadler model of interacting electrons. Result does not a r X i v : . [ c ond - m a t . s t r- e l ] A ug stripe geometry FIG. 1. (Color online) Stripe geometry for the 2D sample,chiral edge currents localized at the boundaries are markedwith red arrows. depend on the symmetry of the lattice, we believe thatit is generic and can be applicable to different 2D topo-logical insulators.
MODEL HAMILTONIAN
We consider CI insulator defined on a honeycomb lat-tice within the Hofstadter model [1]. In the presence ofa transverse homogeneous magnetic field H e z the modelHamiltonian has the following form H = (cid:88) x − links (cid:88) j t x ( j ) a † j a j +1 + (cid:88) y − links (cid:88) j t y ( j ) a † j a j +1 + (cid:88) z − links (cid:88) j t z ( j ) a † j a j +1 + H.c. (1)where a † j and a j are the fermion operators determinedon a honeycomb lattice with sites j . The one-particleHamiltonian (1) describes the nearest-neighbor hoppingsof spinless fermions with different amplitudes along the x-links t x ( j ) = t and along the y-and z-links t y ( j ) = t z ( j ) =exp[ iπ ( x j − φ ]. A magnetic flux through an unit cell φ = H Φ is determined in the quantum flux unit Φ = h/e ,a homogeneous field H is represented by its vector poten-tial A = Hx e ξ , e ξ = { , √ } , while the X-axis coincideswith the direction of the x-links, is perpendicular to the ξ -direction. The x -links determine coupling between zig-zag ξ -chains, the vector potential is directed along e ξ .We consider the 2D fermion system in the stripe geome-try Fig.1 with open boundary conditions for the bound-aries along the zig-zag chains (a sample has size N × N). The magnetic field enters the Hamiltonian (1) as themagnetic flux φ through an unit cell. The Hamiltonianis defined both rational and irrational magnetic fluxes. a)b) k - k k - k k - k k k - k
18 14 38 12 58 34 78 k π - - c)FIG. 2. (Color online) Energy levels calculated on a cylinderwith open boundary conditions along zig-zag ξ − chains (thewave vector k along the chain) for φ = , t = 1 a), t = b)and t = 0 c) ( k = π ). Colored lines indicate the numberedchains in the cell, ellipses show the gaps in which the chiralgapless edge modes are realized. TOPOLOGICAL STRUCTURE OF THESPECTRUMThe rational flux φ = Let us consider formation gaps in the butterfly land-scape at a rational magnetic flux φ = . This case cor-responds to a simple butterfly landscape in which theChern number is different for different gaps. Numeri-cal calculations of the fermion spectrum, obtained fora stripe geometry (see in Fig.1), are shown in Fig.2. FIG. 3. (Color online) Phase diagram defined in the coordi-nates q, t for half-filling (anisotropic hopping integral t , mag-netic flux φ = q ), t = 1 corresponds to an isotropic case. Thelow energy spectra near the zero energy and the imaginarypart of the wave vector k x as function of the wave vectoralong the ξ -chain, calculated at q = 4, t = 1 and t = 1 . The spectrum is symmetric with respect to zero energy,it includes seven isolated subbands. The topologicalstructure of the subbands is determined by the followingChern numbers { , , − , , − , , } , here two (gapless)subbands in the center of the spectrum are one topolog-ical subband with the Chern number 2. Below we showthat the state of fermions in the center of the spectrum(at (cid:15) = 0) is nontrivial for an arbitrary rational magneticflux. At (cid:15) = 0 the spectrum is gapless for t < t c = 2 , at t > t c the gap opens. We recall that for φ = 1 the gapopens at t > t > t c . From calculations of the spectrafor an arbitrary rational flux φ = q it follows that t c = 2 q (numerical calculations of the spectra for different t werecarried out in [15]).Stability of zero energy states, localized at the bound-aries, is determined by the imaginary part of the wavevector k x (wave vector with direction perpendicular tothe boundaries), a change in its sign leads to the instabil-ity of this state. The imaginary part of k x depends on thevalue of the wave vector along the zig-zag chains k and t , changes sign at the singular points (see in Fig.3). At t = 1 the points p π ± π , p = 1 , , , t and at t > t c zero-energyMajorana states are realized in the entire range of k . Foran arbitrary rational flux φ = q the bulk spectrum is πq -periodic. The wave function of zero energy Majoranastates is localized at the boundaries of the sample, theirbehavior is the similar (but not the same) to chiral gap-less edge modes that realized in CIs. At half-filling for t < t c , the wave function is localized at different bound- FIG. 4. (Color online) The real part of wave function of zeroenergy Majorana fermions (an imaginary part has the samebehavior) defined on the lattice sites l (along the x -direction,N=5000). The edge modes are shown at t = 1 for k = π, . π and at t = 1 . k = π, π/
4, they are localized at theboundaries of the sample. aries for the wave vectors k which belong to the regionsbetween the Dirac points. For the wave vectors outsidethese regions they are delocalized. In the insulator statefor t > t c , the zero energy wave function is localized atdifferent boundaries for any k (see in Fig.4).To calculate the edge states, we consider cylindricalsystem with bearded edges at both boundaries (the zig-zag boundaries along the ξ -direction). Zero energy Ma-jorana states are realized at (cid:15) = 0, they are localizednear the boundaries. The real part of the wave functionof zero energy Majorana fermions as function of latticesite l along the axis of cylinder is shown in Fig.4, thewave function is calculated for different k (behavior ofthe imaginary part of the wave function is the same).Thus, the 2D topological insulator state with zero en-ergy Majorana fermion states is realized. At the sametime, the Chern number and the Hall conductance arezero, which corresponds to trivial insulator state, in otherwords, the state of the 2D topological insulator with zeroChern number is realized.Consider the behavior of the edge modes in the gaps[17], which are realized at the energies (cid:15) = ± π and (cid:15) = ± π in the t → k p , here k p = (2 p − π p = 1 , ...
4. Numerical calculations show that the val-ues of the gaps ∆ are equal to ∆ = | t | + 0( t ). In the t → ξ chains crosse inthe points k p , the energies of the chains are shifted in k = πq = π . In Fig 2c) it is shown, that two statesof fermions of the ξ − n -chain with k n = k ( n − ± k (where k = π ), tunnel between the chains at the en-ergy (cid:15) . These tunneling processes are realized due tothe conservation of the momentum and energy. The cor-responding states are determined by the Fermi opera-tors a † n = a † ( k n ) , a n = a ( k n ). Taking into account thetunneling of fermions in the gaps we can write the low-energy Hamiltonian H eff = τ (1) (cid:80) N − n =1 a † n a n +1 + H.c. ,where τ (1) is a constant that determines the tunnel-ing of fermions between the nearest-neighbor ξ -chains, n define a lattice site along the x -direction or the num-ber of the ξ -chain. Using the hermitean and antiher-mitean parts of the a n , leading to a Clifford algebra χ n = a n + a † n and γ n = a n − a † n i , we redefine the Hamil-tonian as i τ (1)2 (cid:80) N − n =1 ( χ n γ n +1 − γ n χ n +1 ). The Majoranaoperators γ n are defined by the algebra { γ n , γ m } = 2 δ n,m and γ n = γ † n . Unlike traditional fermion hoppings, whenthe result of a hopping does not depend on the chiral-ity (or the velocity of the particles), the conservationlaws allow only hoppings of fermions between differentchains with different velocities (for different signs of k see in Fig.2c)) and with a given sequence of changes inthe chirality of the movement of fermions along the x -chain (left-right, right-left,... for (cid:15) and right-left, left-right,... for − (cid:15) ). We define the effective Hamiltonian,which takes into account the low energy excitations ofMajorana fermions in the gap near the energy (cid:15) H ( (cid:15) ) = i τ (1)2 N − (cid:88) n =1 χ n γ n +1 , H ( − (cid:15) ) = i τ (1)2 N − (cid:88) n =1 χ n +1 γ n (2)In the case N = qN c ( N c is the number of the q-unit cells)from the lattice periodicity n → n + q , we can considerthe states of fermions of one q-cell or N c cells, taking intoaccount the tunneling of fermions belonging the nearest-neighbor cells. The total Hamiltonian H ( (cid:15) )+ H ( − (cid:15) ) doesnot break the U (1) symmetry of the Hamiltonian (1). Ac-cording to Kitaev [16] the Hamiltonian (2) is describesthe chain of isolated dimers of pairing Majorana fermionswith the energy ± τ (1)2 and two free zero energy Majoranafermions localized at the boundaries. The value of thegap, equal to t , corresponds to τ (1) (cid:39) t . At the givenenergies ± (cid:15) two gapless edge modes with different chiral-ity are localized at the boundaries, they determine theHall conductance σ xy = ( e /h ) C with the Chern num-ber C = 1. In the center of the spectrum at (cid:15) = 0, theHamiltonians (2) define gapless fermion states with thebandwidth τ (1) .The generalization of the approach is based on the ob-servation, that the Kitaev chains with the next-nearestneighbor hoppings between fermions describe two zeroenergy states of Majorana fermions localized at eachboundaries. According to numerical calculations, thegaps are equal to ∆ = t + 0( t ) at (cid:15) = ±√
2. Thesegaps are formed due to the hoppings of fermions betweenthe next-nearest neighbor ξ -chains (see in Fig.2). Thestructure of the effective Hamiltonian, which defines thestates of fermions in the gaps is the same, it is deter-mined as sum of two chains, defined on even and odd lattice sites H ( (cid:15) ) = iτ (2)2 N − (cid:88) n =1 χ n γ n +2 + iτ (2)2 N − (cid:88) n =2 χ n γ n +2 , H ( − (cid:15) ) = iτ (2)2 N − (cid:88) n =1 χ n +2 γ n + iτ (2)2 N − (cid:88) n =2 χ n +2 γ n , (3)where tunneling constant τ (2) (cid:39) t determines the ef-fective hopping between fermions located at the next-nearest neighbor ξ -chains.The Hamiltonian (3) defines the zero energy states ofMajorana fermions located at the ends of the chains witheven and odd lattice sites (see in Fig.2). The values ofthe gaps in the spectrum depend on distance between ξ − chains δ , the energies of which intersect for given k and (cid:15) . An effective tunneling constant for fermions ofdifferent chains is equal to τ ( δ ) (cid:39) t δ + 0( t δ +1 ). Thenumber of chiral edge modes localized at a boundary ofthe sample is equal to δ , the dimension of a chiral modeis (they propagate only in one direction). Calculatingthe energy of intersections of the spectra of the chainswith the spectrum of the first chain and the distancebetween these chains and the first chain for an arbitraryrational flux φ = pq at t <<
1, we obtain the diophantineequation pC γ = q · s + γ (here s is an integer) [3, 15], thesame equation as for a square lattice [2, 18, 19]. The Hallconductance σ xy = e h C γ is defined for γ -filled subbandsor fixed filling.Finally, for given φ the equations for determining theHall conductance in the gap ∆, given with the energy (cid:15) , have the following form: (cid:15) ( δ ) = ± πδφ/
2) and (cid:15) ( δ ) = ± πδφ/
2) and (cid:15) ( δ ) (cid:54) = 0, these equations wereobtained in the limit t →
0, but they are valid for t ≤ (cid:15) ) in thefermion spectrum with their Hall conductance, when theFermi energy lies in the gap ∆. The spectrum is de-formed with increasing t , but the sequence of the gapsdoes not depend on the value of t . For a rational fluxthese equations reduce to the Diophantine equation pre-sented above, where γ is the analog of (cid:15) .In the case of experimentally realizable magnetic fields,that corresponds to q ∼ − , p = 1, in the semi-classical limit we obtain (cid:15) ( δ ) = π δq , δ = 1 , , , ... forthe states near the center of the spectrum. This result,obtained in the t → t < t c . In the case of t > t c (see Fig.3), the lowest energy gap corresponds to ainsulator state with σ xy = 0 at half filling. This set of thegaps corresponds to the levels with the energies (cid:15) ( δ ) = πq ( δ + 1 /
2) [20]. The experiments [21, 22] confirm aninteresting behavior of the Hall conductance in graphene,which is quantized. This is an interesting new phenomenacompletely explained by the relativistic Dirac spectrumof graphene [23].Let us try to explain the behavior of the Hall conduc-tance in the semi-classical approximation. The regionsof the spectrum near the edges of a band, that corre-spond to small filling of particles or holes, and the centerof spectrum, that correspond to a half-filling, are sepa-rated. The fermion states that determined by the Lan-dau levels with the energies E = ± [3 − √ πφ ( n + 1 / ± / π √ φn [24] near theDirac points have different the Hall conductance. Atsmall filling the Chern number of n-filled subbands ofspinless fermions is equal to n. Taking into account thespin degenerate and the Zeeman splitting we obtain theexpression for the Hall conductance σ xy = e h (2 n + 1).For the fermion states near the center of spectrum theZeeman splitting is small, so you can disregard it. Wemust take into account the degeneracy of states with en-ergies near the center of the spectrum. The lowest Diraclevel, separated by a space, in which two gapless edgesare localized at the boundaries at x = 1 and x = N, re-sulting in C = 1. The Chern number for the next leveldetermined by ’new’ two chiral modes localized at x=2and x=N-1 and modes localized at x=1 and x=N withfilling 1 (they form the pair of spinless fermions withdifferent chiralities and momenta near the point k = π see in Fig.5). For the Fermi energy, that correspondsthe second gap, the states of gapless edge modes with k − < k < k + ( k ± = π − πq ± πq ) are fulfilled, this statecorresponds to a pair of spinless fermions. In this case theChern number is equal to C = 1 + 2 = 3, the structureof the Chern number C is as follows C = 1 + 2 ×
2. Itis considered that two ’new’ modes localized at x=3 andx=N-2 plus two pairs of spinless fermions at x=1,x=Nand x=2,x=N-1. For the states of spinless fermions nearthe center of the spectrum we obtain the expression forthe Chern number C n = 2 n − σ xy = e h (2 n −
1) and σ xy = 2 e h (2 n −
1) for electrons.
The irrational flux φ = √ We study the spectrum and topological numbers inthe case of an irrational magnetic flux. As an illus-tration, we consider the evolution of the simplest ra-tional flux q = 3 (see in Fig.6a)) to an irrational flux q = √ . ... (see in Fig.6b),c)). Although theflux values differ slightly, the fine structure of the spec-trum with other topological numbers corresponds to theirrational flux. For an arbitrary flux φ in the t → πφ and intersect at the points k p = πpφ , (cid:15) p = ± πpφ ), (cid:15) p = ± πpφ ), here p = 1 , ..., ∞ . The set k p , (cid:15) p (cid:54) = 0determines the gaps in the fermion spectrum for givenirrational flux φ .Numerical calculations of the excitation spectrum fora rational flux in the Hofstadter strips with open bound- - -
2 N - k π FIG. 5. (Color online) Energies of fermions (shifted by πq ) inatomic layers at the boundaries (1;N) near the center of thespectrum as a function of the wave vector, the spectrum issymmetric around zero energy. Dashed lines define the Fermienergies or the center of the gaps in insulator states in t → ary conditions were obtained for samples of sizes Nq is anatural number. When calculating the spectrum for anirrational flux, which is approximated by rational num-bers φ (cid:39) pq , we assume that N < q . In this case nu-merical calculations correspond to a spectrum with anirrational flux to within N . Calculations of the fermionspectrum for φ = √ , t = 1, obtained on the latticewith size N = 200, are shown in Fig.6b),c). Compar-ing the spectra with q = 3 Fig.6a) and q = √ < (cid:15) ≤ t → (cid:15) = 0 . ⇒ δ = 2, (cid:15) = 0 . ⇒ δ = 5, (cid:15) =0 . ⇒ δ = 9, (cid:15) = 0 . ⇒ δ = 8, (cid:15) = 0 . ⇒ δ = 6, (cid:15) = 0 . ⇒ δ = 11, (cid:15) = 0 . ⇒ δ = 3,..., where theenergy (cid:15) determine the position of the gap in the finestructure of the subband and δ is the number of gaplessedge modes localized at a boundary of the sample. Wehave considered all possible gaps in the fine structure ofthe low energy subband when δ changes from 1 to 11.The topological properties of the fine structure of thesubband are conserved with an increase in the value of t ,therefore it is fair to expect that the Hall conductance ofeach subband is determined by these values of δ . The cal-culation of the fine structure of a low energy subband isshown in Fig.6 c), the energies corresponding to the gapsvary, but their sequence and the total numbers of gaplessedge modes in the gaps, that define the Hall conductanceof the subbands, are conserved. The fine structure of the a) b)c)FIG. 6. (Color online) Energy levels calculated on a cylinderwith open boundary conditions along ξ -direction for t = 1, φ = a), φ = √ N=200 b). The fine structure of the low-energy subband c), δ denotes the total number of gapless edgemodes localized at a boundary of the sample (the modes inthe gaps are numerated). subband is determined by the gaps with δ = 2 , , , , (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 .
379 and (cid:15) = 0 . THE HOFSTADTER MODEL OF INTERACTINGELECTRONS
The interaction between fermions ’kills’ the state ofthe topological isolator, it is obvious that in the case of strong interaction a trivial insulator state is realized.Below, we consider in detail the stability of the topolog-ical insulator state in the presence of the on-site inter-action U (cid:80) j n j ; ↑ n j ; ↓ . In graphene the distance betweenthe Landau levels is very large compared to the Zeemansplitting [23], so we will not consider this term. Thetotal Hamiltonian takes into account also the terms (1)for fermions with different spins. In the limit t → ξ − chains. We consider the caseof a weak on-site interaction U <<
1, when U ∼ t or U ∼ ∆. In the case of a weak interaction the electronspectrum of the Hubbard ξ -chain is renormalized slightly.We accent our consideration on the behavior of edgemodes in the gaps for an rational flux. As we notedabove, only two states of spinless fermions into the gapswith fixed wave vectors ± πδ/q , defined at one latticesite n , tunnel between the ξ − chains, these states havedifferent chirality (see in Fig.2c)). Taking into accountthat the density of fermions at the site j = { n, l } ( l nu-merates the lattice sites along the ξ -chains) and the en-ergy (cid:15) is determined as n n,l ; σ = n n ; σ ( (cid:15) ) = i γ n ; σ χ n ; σ + . Taking into account that (cid:80) l ( n n,l ; ↑ − )( n n,j ; ↓ − ) = N (cid:80) k m ( n, k ; ↑ ) m ( n, − k ; ↓ ), where m ( n, k ; σ ) = (cid:80) l ( n n,l ; σ − ) exp( ikl ), we obtain N (cid:80) k m ( n, k ( n − ↑ ) m ( n, k ( n − ↓ ) = − γ n ; ↑ χ n : ↑ γ n ; ↓ χ n : ↓ , here m ( n, k ( n − σ ) = n n ; σ ( (cid:15) ) − . We used that the spec-trum is periodic with the period k . Taking into accountthe interaction term, we define an effective low-energyHamiltonian H eff = i τ ( δ )2 (cid:88) σ N − δ (cid:88) n =1 γ n ; σ χ n + δ ; σ − U (cid:88) n γ n ; ↑ χ n ; ↑ γ n ; ↓ χ n ; ↓ , (4)where δ denotes also the number of the chains of electronswith the hopping integral τ ( δ ) between electrons locatedat the sites on the distance δ , in (4) Majorana fermionsare defined for fermions with different spins.Taking into account the spin freedom of fermions, theChern number for electron subbands and the gapless edgemodes doubles. The Hamiltonian (4) with the on-siteHubbard interaction has be diagonalized exactly by Mat-tis and Nam [13]. The model is reduced to chain of spin-less fermions (2) with the chemical potential U [13]. Ac-cording to [13], the ground state degeneracy is dependenton whether κ = Uτ ( δ ) > | κ | ≤
4. The number of chiralgapless edge modes is associated with the topological or-der, so for | κ | = 4 in the thermodynamic limit a topologi-cal phase transition between phases with different Chernnumbers is realized. The value τ ( δ ) decreases with in-creasing of δ , depends on (cid:15) . The phase state, for which | U ∆ | >
4, is topological trivial state, when the interactionis taking into account. The topological ambitions of thesubbands are limited by a weak interaction
U <
CONCLUSION
We have studied the behavior of 2D fermions in theHofstadter model, defined on the honeycomb lattice, fo-cusing our attention on the structure of the chiral gap-less edge modes in the gaps. In contrast to the tradi-tional approach, when the states of the bulk fermions aremainly taken into account, we pay attention to the studyof fermion states in the insulator gaps, since these statesdetermine the Hall conductance. This approach makes itpossible to calculate the Hall conductance in CI for an ar-bitrary flux. The structure of chiral gapless edge modes isdescribed within the framework of the Kitaev chain, witheffective hopping integral between Majorana fermions.For an irrational flux the fine structure of the subbandstransforms to the super fine structure with infinite num-ber of subbands and different numbers of chiral gaplessedge modes. We have shown also, that at half-filling agapless state, is unstable in the case an isotropic Hofs-tadter model; an external magnetic field leads to gapped(insulator) state. The 2D topological insulators that sup-port chiral gapless edge modes are extremely susceptibleto short range electron-electron interactions, for
U > [1] D.Hofstadter, Energy levels and wave functions ofBloch electrons in rational and irrational mag-netic fields, Phys.Rev.B, 14 (1976), 2239-2249.https://doi.org/10.1103/PhysRevB.14.2239[2] I.Dana, Y.Avron and J.Zak, Quantised Hall conduc-tance in a perfect crystal, J.Phys.C: Solid State Phys.,18 (1985), L679-684. https://doi.org/10.1088/0022-3719/18/22/004[3] A.Agazzi, J.-P.Eckmann and G.M.Grafm, Thecolored Hofstadter butterfly for the honey-comb lattice, J.Stat.Phys., 156 (2014), 417-426.https://doi.org/10.1007/s10955-014-0992-0[4] J.E.Avron, O.Kenneth and G,Yehoshua, A study ofthe ambiguity in the solutions to the Diophantineequation for Chern numbers, J.Phys.A: Math.Theor.,47 (2014), 185202. https://doi.org/10.1088/1751-8113/47/18/185202[5] P.B.Wiegmann and A.V.Zabrodin, Bethe-ansatz for the Bloch electron in magneticfield, Phys.Rev.Lett. 72, (1994), 1890-1893.https://doi.org/10.1103/PhysRevLett.72.1890[6] Y.Hatsugai, M.Kohmoto and Y.-S. Wu, Explicit solu-tions of the Bethe ansatz equations for Bloch electrons ina magmnetic field, Phys.Rev.Lett. , 73 (1994), 1134-1137. https://doi.org/10.1103/PhysRevLett.73.1134[7] I.N.Karnaukhov, Chern insulator with large Chern num-bers. Chiral Majorana fermion liquid,
J.Phys.Commun. ,1 (2017), 051001. https://doi.org/10.1088/2399-6528/aa9541[8] G.G.Naumis, Topological map of the Hofstadter but-terfly: Fine structure of Chern numbers and VanHove singularities, Phys. Lett A. 380 (2016), 1772.https://doi.org/10.1016/j.physleta.2016.03.022[9] I.I.Satija, Topology and self-similarity of the Hofstadterbutterfly, arXiv:1408.1006 [cond-mat.dis-nn] (2014).[10] M.Eliashvili, G.I.Japaridze and G.Tsitsishvili, Thequantum group, Harper equation and the struc-ture of Bloch eigenstates on a honeycomb lat-tice, J.Phys.A:Math.Theor., 45 (2012), 395305.https://doi.org/10.1088/1751-8113/45/39/395305[11] I.N.Karnaukhov, Spontaneous breaking of time-reversal symmetry in topological insulators,Physics Letters A, 381 (2017), 1967-1970.https://doi.org/10.1016/j.physleta.2017.04.014[12] J.C.Avila, H.Schulz-Baldes and C.Villegas-Blas,Topological invariants of edge states for periodictwo-dimensional models, Math. Phys. Anal. Geom. , 16(2013), 137-170. https://doi.org/10.1007/s11040-012-9123-9[13] D.C.Mattis and S.B.Nam, Exactly soluble model of in-teracting electrons. J.Math.Phys., 13 (1972), 1185-1189.https://doi.org/10.1063/1.1666120[14] A.Yu.Kitaev, Anyons in an exactly solved modeland beyond, Annals of Physics, 321 (2006), 2-111.https://doi.org/10.1016/j.aop.2005.10.005[15] M.Sato, D.Tobe and M.Kohmoto, Hall conduc-tance, topological quantum phase transition,and the Diophantine equation on the honey-comb lattice, Phys.Rev.B, 78 (2008), 235322.https://doi.org/10.1103/PhysRevB.78.235322[16] A.Yu.Kitaev, Unpaired Majorana fermions inquantum wires, Phys.- Usp., 44 (2001), 131-136.https://doi.org/10.1070/1063-7869/44/10S/S29[17] I.N.Karnaukhov, Edge modes in the Hofstadter modelof interacting electrons,
Europhysics Letters , 124 (2018),37002. https://doi.org/10.1209/0295-5075/124/37002,see also C.Malciu, L.Mazza, and C.Mora, 4 π and π dual Josephson effects induced by symmetry defects ,arXiv:1901.03342v1 [cond-mat.mes-hall] 2019.[18] I.Dana, Topologically universal spectral hierarchies ofquasiperiodic systems, Phys.Rev.B (2014) 205111.https://doi.org/10.1103/PhysRevB.89.205111[19] G.Amit and I.Dana, Topological phase transitions fromHarper to Fibonacci crystals, Phys.Rev.B, 97 (2018),075137. https://doi.org/10.1103/PhysRevB.97.075137[20] B.A.Bernevig, T.L.Hughes, S.-C.Zhang, H.-D.Chen andC.Wu, Band collapse and the quantum Hall effectin graphene, Int.J.Mod.Phys.B, 20 (2006), 3257-3278.https://doi.org/10.1142/S0217979206035448[21] K.S.Novoselov, A.K.Geim, S.V.Morozov, D.Jiang,M.I.Katsnelson, I.V.Grigorieva, S.V.Dubonos andA.A.Firsov, Two-dimensional gas of massless Diracfermions in graphene, Nature, 438 (2005), 197-200.https://doi.org/10.1038/nature04233[22] Y.Zhang, Y.W.Tan, H.L.Stormer and P.Y.Kim, Exper-imental observation of the quantum Hall effect andBerry’s phase in graphene, Nature, 438 (2005), 201-204.https://doi.org/10.1038/nature04235(2014) 205111.https://doi.org/10.1103/PhysRevB.89.205111[19] G.Amit and I.Dana, Topological phase transitions fromHarper to Fibonacci crystals, Phys.Rev.B, 97 (2018),075137. https://doi.org/10.1103/PhysRevB.97.075137[20] B.A.Bernevig, T.L.Hughes, S.-C.Zhang, H.-D.Chen andC.Wu, Band collapse and the quantum Hall effectin graphene, Int.J.Mod.Phys.B, 20 (2006), 3257-3278.https://doi.org/10.1142/S0217979206035448[21] K.S.Novoselov, A.K.Geim, S.V.Morozov, D.Jiang,M.I.Katsnelson, I.V.Grigorieva, S.V.Dubonos andA.A.Firsov, Two-dimensional gas of massless Diracfermions in graphene, Nature, 438 (2005), 197-200.https://doi.org/10.1038/nature04233[22] Y.Zhang, Y.W.Tan, H.L.Stormer and P.Y.Kim, Exper-imental observation of the quantum Hall effect andBerry’s phase in graphene, Nature, 438 (2005), 201-204.https://doi.org/10.1038/nature04235