Chern insulator with large Chern numbers. Chiral Majorana fermion liquid

In the framework of Hofstadter`s approach we provide a detailed analysis of a realization of exotic topological states such as the Chern insulator with large Chern numbers. In a transverse homogeneous magnetic field a one-particle spectrum of fermions transforms to an intricate spectrum with a fine topological structure of the subbands. In a weak magnetic field $H$ for a rational magnetic flux, a topological phase with a large Chern number is realized near the half filling. There is an abnormal behavior of the Hall conductance ${\sigma_{xy}\simeq \frac{e}{2 H}}$. At half-filling, the number of chiral Majorana fermion edge states increases sharply forming a new state, called the chiral Majorana fermion liquid.


Introduction
In the case of a rational magnetic flux through the unit cell f = p q (p and q are coprime integers), a new physics of the 2D model systems subjected to a perpendicular magnetic field arises from its two magnetic and lattice competing scales [1]. Astrong periodic potential leads to an intricate spectrum of the topological system, the problem brings to play the commensurability of these two scales [2]. For the particular case, when the magnetic flux assumes either the continued fraction approximations towards the golden mean or the golden mean itself, the multifractal properties of Harper's equation, the winding numbers were discussed in [3]. The problem of realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices is addressed, for example, in [4,5].
Wiegmann and Zabrodin pointed out [6] that Hofstadter's model responsible is closely related to the quantum group ( ) U sl q 2 , the model Hamiltonian is determined in terms of the generators of the quantum group (see also the approach of Faddeev and Kashaev [7]). Using the exact solution of the Hofstadter model [6,7] for the semi-classical limit at p=1 and  ¥ q , the behavior of the wavefunction was calculated at zero energy (for the center of the spectrum) y = = ( j is the site of the lattice) [8]. Near the edge j=0, the finite size correction is given by y = 256 . The authors of [8] noted that power-low behavior of y | | j 2 is critical and un-normalizable. For the golden mean flux, the wavefunction is multifractal and critical, has a clear self-similar branching structure. Harper et al [9] shown that the Hofstadter model converges to continuum Landau levels in the limit of small flux per plaquette (in the  ¥ q limit). Filled r bands with the Chern numbersC γ (   g r 1 ) yield a Hall conductance s s s . The relationship between the Hall conductance s xy and energy spectrum, the Chern numbers of isolated subbands are discussed in [10]. Dana et al [11] proposed that magnetic translational symmetry yields the diophantine equation s + = | | p q s r 2 2 r (two deuces in the equation take into account spin degeneracy). The solution of the equation can be uniquely determined by imposing the condition  s | | q r , that is realized in the model.
The dynamics of a charged particle in a magnetic field applied perpendicular to the plane of the lattice and to the electric field in the plane of the lattice is considered in [12]. A non-zero magnetic field crucially modifies this Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. dynamics of fermions along the edges. Exact results [6,7,13] and numerical calculations [14,15] show that the behavior 2D fermion systems in a transverse magnetic field does not depend on the symmetry of the lattice.
However strange as it may seem at first glance, the behavior of 2D fermions in a transverse magnetic field largely depends on the boundary conditions. We show that in the case of a rational magnetic flux the Hall conductivity has an anomalous behavior in a small magnetic field near the half-filling due to large number of chiral gapless modes localized in wide regions near boundaries. In the case of an irrational magnetic flux, the magnetic scale tends to ¥ (in reality, > q N , where N is the size of the system), therefore it is necessary to take into account the boundary conditions. We show that the number of chiral modes is anomalously large, they form the chiral Majorana fermionic liquid state.
Non-trivial topological order in a band is indicated by non-zero Chern number. Thetopological insulating phase state is determined by the Chern number of all fermion bands below the Fermi energy. TheChern number is a topological invariant which can be easily defined for the γ-band isolated from all other bands by the formula integrating the Berry curvature   = ǵ g ( ) ( ) k k k over the Brillouin zone(BZ). The Berry potential

The model
We will analyze a model of the 2D CI determined in the framework of Hofstadter's approach [1]. In the presence of a transverse homogeneous magnetic field He z the model Hamiltonian is determined according to [1]  where † a n m , and a n m , are the spinless fermion operators on a site { } n m , with the usual anticommutation relations. The Hamiltonian describes the nearest-neighbor hoppings along the x-direction with the magnitude t ( = ( ) t n m t , x ) and along the y-direction with the hopping integral , a homogeneous fieldH is represented by its vector potential = Hx A e y . The case t=1 corresponds to the original Hofstadter model [1]. We focus on the 2D system in the form of a hollow cylinder, with periodic boundary conditions along the y-direction and size N along the x-direction.
3. Topological structure of the spectrum 3.1. The flux f = 1 3 We consider an evolution of the one-particle spectrum of the Hamiltonian (2) in a transverse homogeneous magnetic field with rational magnetic flux f = p q . In the case > q 2 the magnetic field breaks a time reversal symmetry [16,17], leads to topological states of fermions [18]. Ourstarting point is q=3, when the oneparticle spectrum consists of three topologically non-trivial subbands with the Chern numbers - . A detailed calculation of the Chern numbers for a rational flux is given in [19,20]. In an external field the spectrum of fermions is intricate, the band is split into the topological subbands γ with nontrivial topological index C γ . The structure and number of subbands in the fine structure of the spectrum depend on the value of the magnetic flux. The excitation spectrum of the sample in stripe geometry consists of the chiral edge modes, which are indicated in red color (see in figures 1(a), (b)).
They are localized near the boundaries of the sample, the amplitudes of the wave functions decrease exponentially with receding from the boundaries. The energies of the edge modes are determined by the wave vector component that is parallel to the boundary, they intersect each other at the Dirac point, merge with the bulk states. The edge modes with different chirality are localized at the different boundaries, they are associated with Majorana operators that belong to the boundaries. The chiral gapless edge modes do exist in the gap if the Chern number of isolated bands located below the gap is non-zero. The phase state of fermions is characterized by a chiral current along the boundary of the 2D system, the direction of the current is determined by the sign of the Chern number.
The structure of the spectrum does not depend on the value of t, the gaps close at t=0 and open for ¹ t 0. The values of gaps are equal to D = + - . The value of Δ is shown in figure 1(c). The topology of the spectrum is not changed at the point t=0. The gaps close at t=0, but the spectrum of the excitations has not the Dirac-type in the k x -direction (k is a wave vector). Let us consider the spectrum of the excitations in the case of a weak coupling along the x-direction at small values of t (see in figure 1(b)). In the  t 0 limit the fermions form the non-interacting chains along the y-direction (see in figure 1(d)). The energies of fermion excitations in the chains intersect at the energies equal to ±1 for = p k y p 3 = ¼ p 1, , 6 (see in figure 1(b)). Due to the hoppings of fermions between the nearest chains t the gaps open at the energies equal to ±1. The tunneling of fermions between chains dominates in the regions of the crossings of energies of the nearest chains, since the conservation of the energy and the momentum of fermions are realized automatically. The Hamiltonian, which takes into account the low energy excitations of Majorana fermions, is determined as follows  figure 1(b)). They are localized near the boundaries of the sample [21]. The chiral gapless edge modes do exist in the gaps, connect the lower and upper fermion subbands (see in figures 1(a), (b)).      Consider the excitation spectrum, calculated at t=1, f =  , analyzing the results of calculations presented in figure 5. Three bands separated by wide gaps, connected by the chiral gapless edge modes, are topological ones with the Chern numbers -{ } 1, 2, 1 , as in the case q=3 (see in figure 1(a)). The number of gapless edge modes, indicated in red lines, determines the values of the Chern numbers of the isolated subbands. The number of these gapless edge modes is conserved which is confirmed by numerical calculations of the spectrum for a set of rational fluxes that correspond to an irrational flux. For an irrational flux the structure of the fermion spectrum differs from that in the case of a rational flux. The gaps in the spectrum are determined by crossings of the energies of noninteracting chains (in the case of weak t-interaction see in figure 6(a) and for t=1 see in figure 6(b)). In contrast to the rational flux the number of crossing points n c depends on N ( =n q 1 c in the case of the rational flux). The tunneling of fermions between chain opens the gaps in the spectrum near the crossing points, in the gaps chiral localized modes are formed. The corresponding low energy effective Hamiltonian has the following form