Topological Majorana flat bands in the Kitaev model on a Bishamon-kikko lattice

TL;DR

The paper addresses how topological flat bands of Majorana fermions can emerge in a Kitaev quantum spin liquid beyond the honeycomb lattice by introducing a $1/6$-depleted Bishamon-kikko pattern. They derive an effective Majorana Hamiltonian under a magnetic field using perturbation theory up to third order, and characterize topology via band Chern numbers $C_n$ and total $C$ calculated with the Fukui–Hatsugai–Suzuki method, while evaluating the thermal Hall response $\kappa_{xy}$ to reveal Berry-curvature effects. At zero field, flat Majorana bands arise from decoupled $c$ and $b$ sectors; a finite field hybridizes these sectors and yields topological flat bands with large $|C_n|$ (e.g., $|C_n|=3$ or $7$), producing phases with $|C|>1$ and, in some regions, Majorana Fermi surfaces. The large Chern numbers originate from the cooperative role of second-neighbor hoppings and $b$–$c$ hybridization, a feature rooted in spin fractionalization. These findings point to a new class of Kitaev materials where topology is governed by Majorana flat bands and can be probably probed via thermal Hall measurements.

Abstract

We unveil an interesting example of topological flat bands of Majorana fermions in quantum spin liquids. We study the Kitaev model on a periodically depleted honeycomb lattice, under a magnetic field within the perturbation theory. The model can be straightforwardly extended while maintaining the exact solvability, and its ground state is a quantum spin liquid as on the honeycomb lattice. As fractionalized excitations, there are unpaired localized Majorana fermions in addition to the itinerant Majorana fermions and $\mathbb{Z}_2$ fluxes. We show that in the absence of the magnetic field the Majorana fermions have completely flat bands at zero energy, and by applying the magnetic field, they turn into topological flat bands with nonzero Chern number. By varying the anisotropy of the interactions and the magnitude of the magnetic field, we clarify that the system exhibits a variety of topological phases that do not appear in the original model. We emphasize that the topological flat bands that give this rich topology come from the hybridization of the Majorana flat bands and unpaired Majorana fermions, which is unique to the flat bands of fractionalized excitations in quantum spin liquids. Our findings would stimulate the exploration of a new type of Kitaev materials exhibiting rich topology from topological Majorana flat bands.

Figures (10)

• Figure 1: (a) Japanese traditional Bishamon-kikko pattern. (b) Schematic picture of the Kitaev model on a Bishamon-kikko lattice. $J_{\mu}$ represents the coupling constant of the Kitaev interactions on the $\mu$ bond ($\mu=x$, $y$, and $z$). The black parallelogram indicates the unit cell of the model contains ten lattice sites (labeled with A to J). Blue arrows represent the primitive translation vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$. (c) The first Brillouin zones of the model (thick line) and the original Bishamon-kikko lattice (dotted line). $\Gamma$, $\mathrm{X}$, $\mathrm{Y}$, and $\mathrm{S}$ denote the high-symmetry points and the red thick line represents the path we show the band structures along it.
• Figure 2: Schematic pictures showing the difference in the fractionalization of quantum spins between the honeycomb and Bishamon-kikko lattices. Expressing a quantum spin as $S^{\mu}=\mathrm{i}b^{\mu}c/2~(\mu=x,~y,~\text{or}~z)$ with four types of Majorana fermions $b^{x},~b^{y},~b^{z},~\text{and}~c$ (spheres in the left panels). $b$ Majorana fermions make the $\mathbb{Z}_{2}$ gauge field $u^{\mu}$ in pairs. On the Bishamon-kikko lattice, there are unpaired$b$ Majorana fermions due to depletion (the right panels).
• Figure 3: Schematic pictures of the hoppings of the Majorana fermions included in each term in Eq. \ref{['eq:h_eff']}, which is obtained by the third-order perturbation theory: (a) nearest-neighbor $c$-$c$, (b) nearest-neighbor $b$-$c$, (c) second-neighbor $c$-$c$ and $b$-$b$, and (d) second-neighbor $b$-$c$ hopping processes. Hoppings from black sites to white sites are indicated by orange arrows, and those in the opposite direction are indicated by pink arrows.
• Figure 4: Majorana band structure in the isotropic case at zero magnetic field, shown along the path indicated by the thick red line in Fig. \ref{['fig:model']}(c).
• Figure 5: Majorana band structure in the isotropic case at (a) $h=0.001$, (b) $h=0.002$, and (c) $h=0.003$. The lower panels show an enlarged view of the nearly flat bands at low energy. Integers indicate the Chern number of each band, and $C$ indicates the total Chern number which is the sum of the Chern number of the occupied bands.