Majorana edge magnetization in the Kitaev honeycomb model

Abstract

We propose an approach to detect the peculiarity of Majorana fermions at the edges of Kitaev magnets. As is well known, a pair of Majorana edge modes is realized when a single complex fermion splits into real and imaginary parts which are, respectively, localized at the left and right edges of a sample magnet. Reflecting both of this peculiarity of the Majorana fermions and the ground-state degeneracy caused by the existence of the Majorana edge zero modes, the spins at the edges of the sample magnet are expected to behave as a peculiar "free" spin which exhibits a unidirectional magnetization without any transverse magnetization when applied a sufficiently weak external magnetic field. For the Kitaev honeycomb model, we obtain the expression of the Majorana edge magnetization by relying on standard techniques to diagonalize a free fermion Hamiltonian. The magnetization profile thus obtained indeed shows the expected behavior. We also elucidate the relation between the Majorana edge flat band and the bulk winding number from a weak topological point of view. I. INTRODUCTIONKitaev introduced a seminal quantum spin model [1] which realizes the desired properties of quantum spin liquids [2, 3], whose study was initiated by Anderson. More precisely, Kitaev's model is defined on the honeycomb lattice, and mapped to a free Majorana ferminon model. In consequence, the model is exactly solvable, and shows short-range spin correlations [4]. Although Kitaev's model is fairly artificial, some materials, e.g., A 2 IrO 3 (A = Na, Li) [5, 6] and α-RuCl 3 [7], are expected to exhibit very similar properties to those of the Kitaev honeycomb model [8][9][10][11][12]. In particular, detecting the evidence of the Majorana fermions is one of the central issues in condensed matter physics [13][14][15][16][17][18][19].On the other hand, the Kitaev honeycomb model with open boundaries shows many Majorana zero modes at the edges [20]. In fact, the zero modes form a flat band. This is nothing but a consequence of the weak topological character [21][22][23][24][25][26] of the model. In fact, the celebrated bulk-edge correspondence [27,28] ensures the relation between the number of the Majorana edge zero modes and the winding number for the bulk Hamiltonian when the Fermi energy, which equals zero in the present system, lies in the spectral or mobility gap of the Hamiltonian. Interestingly, similar Majorana edge flat bands were found to appear also in the gapless regime of the Hamiltonian, depending on the geometry of the edges [20].As is well known, a pair of Majorana edge modes appears when a single complex fermion splits into real and imaginary parts which are, respectively, localized at the left and right edges of the system. Therefore, a single Majorana fermion has only a real degree of freedom as

Molecular-orbital representation of generic flat-band models

“…(13), thus one can easily implement the fartherneighbor hoppings retaining the flat band by introducing, e.g., "the NN hopping term" of the MOs. Such an implementation of the farther-neighbor hoppings was discussed by one of the authors in the context of tuning of the flatband energy [27]. We hope that the MO-representation sheds light on the physics of flat-band models in various contexts.…”

“…The translationally-invariant case, i.e., t R = t , t R = t for all R, is known as a breathing Kagome model [24][25][26][27]. In this model, the band touching between flat and dispersive bands occurs [20,21,27] [ Fig. 1(b)].…”

Dissipationless Spin Current Generation in Kitaev's Chiral Spin Liquid

“…, L x , where L x is the number of unit cells, from left to right. For this configuration, the gauge-field Majorana fermions included in the top and bottom spins at open edges, b z l,1 and b z l,N , can not form the Z 2 gauge fields (54)(55)(56)(57). Thus, the perturbative calculations within the vortex-free sector described in the previous section are not applicable to these sites.…”

Oriented propagation of magnetization due to chiral edge modes in Kitaev-type models

“…One finds that the pairs of d i d j in Eqs. (8) and (9) commute with H K and H 3−spin , thus they are the conserved quantities. Consequently, we can replace d i d j with the classical number, as d i d j = ±i.…”

“…To rewrite the Hamiltonian of Eqs. (10) and (11) in the BdG form, we introduce a complex fermion 8 ,…”