Edge modes in the Hofstadter model of interacting electrons
We provide a detailed analysis of a realization of chiral gapless edge modes in the framework of the Hofstadter model of interacting electrons. In a transverse homogeneous magnetic field and a rational magnetic flux through an unit cell the fermion spectrum splits into topological subbands with well-defined Chern numbers, contains gapless edge modes in the gaps. It is shown that the behavior of gapless edge modes is described within the framework of the Kitaev chain where the tunneling amplitude of Majorana fermions is determined by effective hopping of fermions between chains oriented in the perpendicular direction. In the case of a strong on-site Hubbard interaction, ( is a gap), the topological state of the system, which is determined by the corresponding Chern number and chiral gapless edge modes, collapses. The strength of the Hubbard interaction, as well as the dependence of the gaps in the fermion spectrum on filling, leads to a phase transition between topological and trivial topological phases.
Exotic phases in the condensed matter physics may arise in topologically nontrivial superconductors, insulators. Topological states are described in the framework of the band theory, while the topological invariant characterizes the class of a topological insulator or a topological superconductor. Topological order in topological insulators and superconductors is called topological order because it is associated with a topological invariant 9 (); 10 ().
A ground state of the topological Chern systems is characterized by the Chern number, it is an integer topological number which is well defined in insulator (superconductor) state for the subbands isolated from all other bands H (); K1 (); K2 (). A non-trivial topology of the ground state provides the quantization of the Hall conductivity, which can be interpreted as the Chern number. Filled bands with the Chern numbers yield a Hall conductance . The value of the Chern number for a given occupied band is far from obvious without numerical calculations, so another approach to investigating non-trivial topological properties of the system is the existence of chiral gapless edge modes. In the 2D Chern systems the nontrivial topological structure is also intrinsically connected with the existence of robust gapless chiral edge modes.
Initially, the Chern insulators were solved in the framework of noninteracting fermion models with periodic and open boundary conditions. Most of the topological states found in condensed matter systems belong to different classes of noninteracting topological insulators. Weak interactions are not expected to significantly change the stability of topological states. The behavior of topological phases is much more complicated, however, once interactions are taken into account. The interaction between particles can lead to a transition from topological to topological trivial phase, as a result of which a classification of noninteracting fermion systems can collapse in real system. The slave-particles mean-field theory 11 (); 12 (); 13 (), the cellular dynamical mean-field theory 14 (), quantum Monte Carlo simulations 15 (); 16 () and other methods of analytical or numerical calculations of interacting fermion system do not always work in low dimensional and 2D, in particular, systems with interaction. In 17 (); 18 (); 19 () the authors have explored the topological order in 1D Majorana fermion models in context of the Kitaev p-wave chain model with interactions, which exhibits topological order. Strong interactions in the 1D electron systems destroy the superconducting gap that stabilizes the Majorana edge states. Unlike the 2D fermion models, some 1D models with interaction are exact solvable KOR (); 20 (); Kar (), which allows us to investigate the effect of strong interaction on the phase state. The goal of the work is to make a non-trivial step in the study of topological systems with allowance for interaction - to obtain a stability criterion for the phase of the 2D topological insulator with the case of strong short-range interaction between fermions. The question of how the interaction between particles affects the stability of the state of topological insulators remains open.
The question arises - How important is the interaction between fermions for the stability of chiral gapless edge modes, and hence also the topological state? Within the Hofstadter model, as the model of the 2D topological Chern insulator, in which an external magnetic field breaks a time reversal symmetry, we found the answer to this question. We illustrate the effect of short-range electron-electron interaction on the Majorana gapless edge modes using the exact solution of fermion chain model of MN (). The breaking of the topological state in 2D systems at increasing interaction can be regarded as a topological phase transition between the topological and topologically trivial phases. We show that the interaction leads to the topological phase transition at a critical value of the on-site Hubbard interaction, the point of the phase transition separates the state with gapless Majorana edge modes and topological trivial phase. We propose the effective Hamiltonian which describes the low energy fermion states near and into the dielectric gaps, that has exact solution for arbitrary value of the on-site Hubbard interaction. The state of topological insulator and Majorana gapless edge modes are stable only at weak interaction between electrons. The obtained results of calculations shed light on the stability of topological phases in real materials, in which interactions are taken into account.
Ii The model
We consider the 2D Chern insulator defined on a square lattice within the Hofstadter model Hof (). In the case of a rational magnetic flux per unit cell (p and q are coprime integers), the Hofstadter model is reduced to one-particle model of spinless fermions with the q-magnetic cell, which leads to commensurability of magnetic and lattice scales Har (). In the presence of a transverse homogeneous magnetic field the model Hamiltonian has a well-known form Hof ()
where and are the fermion operators determined on a square lattice with sites . The one-particle Hamiltonian (1) describes the nearest-neighbor hoppings of spinless fermions with different amplitudes along the -direction () and along the -direction . A magnetic flux through the unit cell is determined in the quantum flux unit , a homogeneous field is represented by its vector potential . The value changes in the interval , the limiting cases of weak and strong couplings correspond to noninteracting y-chains and the original Hofstadter model Hof (), respectively. We consider the 2D fermion system with periodic boundary conditions along the -direction and open boundary conditions along the -direction (a sample has size along the -direction).
Iii Topological structure of the spectrum
iii.0.1 The flux
The behavior of the fermion spectrum of the Hamiltonian (1) in a transverse homogeneous magnetic field is studied in IK (). We will use this approach to study stability of topological states of interacting fermion systems. In contrast to IK (), where a magnetic flux with odd is analyzed, we consider the case of even . Such detailed consideration is necessary for an understanding of the solution of the problem. First of all we consider the case , the one-particle spectrum splits into three isolated subbands (see in figure 1 a),c)), which correspond to strong and weak coupling between chains. The state of fermions in the centrum of the spectrum (at ) is gapless for an arbitrary rational magnetic flux 2 (); 3 (). Three subbands, each of which is isolated from all other bands, are topological with the following Chern numbers . The topological subbands are connected by the chiral edge modes, the total number of edges modes in the gap with allowance for their chirality, determines the Chern number of -filled subbands .
For detailed study of the fermion spectrum of the Hamiltonian (1) we consider an anisotropic variant of the Hofstadter model with an arbitrary . The values of gaps are equal to at (see in figure 1c)). In the case the fermions form weak coupling chains along the y-direction. In the limit the energies of fermions in the chains are shifted by , intersect at the points (), . The fermion spectrum at the energies is determined by tunneling of Majorana fermions between corresponding nearest-neighbor chains; so Majorana fermions of the second chains and , which determine the chiral modes with opposite velocities, tunnel to the first chain at and into the third chain for , the corresponding Majorana fermions of the third chain tunnel into the fourth chain for . Considering a sample with open boundary conditions, the size of which is an -integer, we find that Majorana fermions located at the first and last chains are free at , therefore zero energy edge states are realized (see in figure 1c)). The tunneling of fermions located at the nearest-neighbor chans opens the gaps in the spectrum at the points ,, so energy excitations of spinless fermions near the energy are determined by their hoppings along the direction . Using the presentation for the Majorana fermions and we redefine this term in the following form . Let us introduce the effective hopping of fermions between the nearest-neighbor chains and define the effective Hamiltonian, which takes into account the low energy excitations of Majorana fermions near the energy
where Majorana operators , describe the chiral modes (with opposite velocities) of the chain , Majorana state operators defined by the algebra and , or , or , or , or , numerate the chains for (the sets values of for the energies or , see in figure 1 c).
The terms determine the fermion spectrum in the region of the energy , the total Hamiltonian does not break the symmetry of the Hamiltonian (1). Taking into account the numerical calculations of the spectrum, we can increase the region of the spectrum near the energy which determined by the Hamiltonian (2), assuming . The Hamiltonian (2) defines the -chains of noninteracting dimers, it was proposed by Kitaev K () for describing topological superconductivity in the chain of spinless fermions. The Majorana fermions are paired in the dimers with energy , the operators and are free Majorana fermions at the energy equal to , they remain unpaired and form zero energy edge states. The gapless edge modes existing in the energy range of about are associated with Majorana operators and . These Majorana fermions have opposite chirality (see in figure 1 c)), at energies and the Chern numbers have opposite signs -1 and 1, respectively.
iii.0.2 The flux
Below we consider more general case of a rational flux, namely . Numerical calculations of the fermion spectrum are shown in figure 1 c),d). The topological structure of the subbands is determined by the following Chern numbers here two (gapless) subbands in the centrum of the spectra are one topological subband with the Chern number . The behavior of the edge modes in the first gaps, which corresponds to the energies at , , is described by the Hamiltonian (2) analogously to the flux . The value of the gaps, calculating for this case is equal to
and at .
The generalization is based on the observation that the Kitaev chains with the next-nearest neighbor hoppings between fermions can describe two zero energy states of Majorana fermions located at the each boundaries. In the limit the energies of the fermions in the y-chains are shifted in the phase , the crossing points , correspond to the energies of the next-nearest neighbor chains. The value of is equal to
The corresponding Hamiltonian describing the energy states in the gaps () is determined by two noninteracting chains
where at is determined the hopping of fermions between the next-nearest neighbor -chains.
The Hamiltonian (3) defines the zero energy states of Majorana fermions located at the ends of the chains with even and odd lattice sites (see in figure 1 b),d)). Considering the structure of the edge modes for arbitrary rational fluxes at we obtain the diophantine equation D1 (); D2 () (where is an integer).
Iv The Hofstadter model of interacting electrons
The nontrivial topological properties of the 2D systems are manifested in the existence of a nontrivial value of the Chern number and chiral gapless edge modes. Below, we consider in detail the stability of edge modes with allowance for the on-site Hubbard interaction. By introducing spin degrees of freedom, it is possible to add the on-site Hubbard interaction in the Hamiltonian (1)
where determines the spin of electron and the first term in (4) takes into account the fermions with different spins, is the density operator.
We use a single band approach, considering the splitting of one band into two subbands, so for , the splitting of a band with positive or negative energy into two subbands divided by a gap for or . The chemical potential lies in the middle of the band. In the limit the model (4) is reduced to the noninteracting Hubbard chains. In the case of weak interaction and filling of the unequal or , the electron spectrum of the Hubbard chain KOR () is renormalized slightly. In should be noted, that in the Hofstadter model with an arbitrary rational flux, the state with is gapless. We accent our consideration on the behavior of edge modes in the gaps. In this case an effective Hamiltonian has the following form
where denotes also the number of the chains of electrons with the hopping integral between electrons located at the sites on the distance .
Taking into account the spin freedom of fermions, the Chern number for electron subbands and the gapless edge modes doubles. As a result, at and the Hamiltonian (5) determines the two zero energy edge modes on each boundary which correspond to fermions with different spin. The Hamiltonian (5) with the on-site Hubbard interaction is mapped to a noninteracting fermion model, which can be diagonalized exactly MN (). The model is reduced to the Kitaev chain of spinless fermions (2) with the hopping integral and the chemical potential MN (). Indeed, the ground state energy and the energy of the excited states are calculated exactly. For arbitrary and , the energy spectrum of the chain (5) is gapped, having the form
For the gap vanishes, in this case we have for a spectrum (6) linear in . According to MN (), the ground state degeneracy is dependent on whether or otherwise. Interestingly, for the equation for wave vectors (which follows from the boundary conditions) has one real root less than in the previous case along with an imaginary root associated with the ends of the chain. In the thermodynamic limit there are zero energy Majorana states. These zero mode Majorana states are absent for and its presence for increases the degeneracy by two, since excitation of the Majorana mode does not change the energy of the system. The low energy spectrum is shown in figure 2 as a function of . The zero energy edge states are realized in the interval . The number of the chiral gapless edge modes follows the Chern number, so in the thermodynamic limit the system has a quantum phase transition at . The phase transition is realized between the phase states with different Chern numbers. The value decreases with increasing of or energy decrease , so the phase states with the Chern numbers, for which , are topological trivial states when the interaction is taking into account. The topological ambitions of the low energy subbands are limited a weak coupling . In this case, the topological phase transition is realized also when the filling is changed. The criterium of realization of the topological state at the -filled band separated by a gap is defined as .
We have studied the behavior of 2D fermions in the Hofstadter model with a rational magnetic flux through an unit cell, focusing on the realization of the chiral gapless edge modes. The fine structure of the bands is characterizes by the Chern numbers, which are well defined for insulator phase. The chiral gapless edge modes are described in the framework of the Kitaev chain with effective hopping between Majorana fermions. The 2D topological insulators that support chiral gapless edge states are extremely susceptible to short range electron-electron interactions. Strong interactions generically destroy the Majorana edge states, in the case of weak interaction, a regime is realized in which the Majorana edge states persist. Exactly solvable Kitaev chain of interacting fermions proposed by Mattice and Nam MN () provides an opportunity to solve the problem of the stability of topological state taking into account the on-cite Hubbard interaction between fermions. The solution problem follows from this exact solution MN (). The stability of the chiral edge modes and, consequently, of the topological phase is determined by the following relation between the value of gap and the Hubbard interaction . The subbands near the centrum of the fermion spectrum with smaller values of gaps are topological trivial due to the presence of the interaction between fermions. We find that moderate short-range interaction did not affect the phase diagram qualitatively but lead to nontrivial quantitative changes in the phase boundary. Obtained results determine the stability of the topological state against interactions, they are generic and can be applicable to different 2D topological insulators.
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