Exactly solvable spin liquids in Kitaev bilayers and moiré superlattices

TL;DR

The paper addresses exactly solvable quantum spin liquids in Kitaev-type bilayer honeycomb systems and their moiré superlattices. It develops a Majorana-fermion framework with static $Z_2$ gauge fields to map ground states to flux sectors, using Monte Carlo sampling and variational analysis to chart flux patterns under zero and finite out-of-plane fields. The study reveals a robust $0-\pi$ ground state in the bilayer and, under twist, generalized interlayer flux configurations tied to local stacking, yielding both gapped and gapless spin liquids plus floating/edge/corner boundary modes. These results demonstrate tunable topological-like phenomena in spatially modulated bilayer systems, offering insight into exact solvability, flux ordering, and boundary physics in Kitaev-like moiré materials.

Abstract

Building on the recent advancements on moiré superlattices, we propose an exactly solvable model with Kitaev-type interactions on a bilayer honeycomb lattice for both AA stacking and moiré superlattices. Using Monte Carlo simulations and variational analysis, we uncover a rich variety of phases where the intra and interlayer $\mathbb{Z}_2$ fluxes (visons) are arranged in a periodic fashion in the ground state, tuned by interlayer coupling and out-of-plane external magnetic field. We further extend our model to moiré superlattices at various commensurate twist angles around two distinct twist centers represented by $C_{3z}$ and $C_{6z}$ of the honeycomb lattice. Our simulations reveal generalized arrangements of plaquette values that correlate with the AA or AB stacking regions across the moiré unit cell. Moreover, we find that, depending on the twist angle, twist center and interlayer coupling, moiré superlattices exhibit to a variety of gapped and gapless spin liquid phases and can also host corner and edge modes. Our results highlight the rich physics in bilayer and twisted bilayer models of exactly solvable quantum spin liquids.

Figures (12)

• Figure 1: Schematics of the bilayer Kitaev-like lattice. (a) The lattice is composed of two sublattices A and B represented by yellow and violet spheres respectively. From each site, four distinct Kitaev-type bonds are depicted in red ($x$), blue ($y$), green ($z$), and yellow ($v$) colors. The $c_i^x$ and $c_i^y$ below represent two free Majorana fermions obtained by decomposing the spin Hamiltonian into fermionic counterpart. (b) The lattice features one intralayer hexagonal plaquette $W_{\text{intra}}$ and three distinct interlayer square plaquettes ($W_{x \perp}, W_{y \perp}, W_{z \perp}$).
• Figure 2: (a) The ground state $0$-$\pi$ flux configuration of the bilayer model with AA stacking. The arrows on the bonds indicate the direction corresponding to $u_{\langle ij \rangle} = +1$ where $\vec{b}_{1,2}$ are reciprocal lattice vectors. The energy spectrum (b) shows Dirac points for $K = 0$, $h_z = 0$, (c) nodal ring for $K = 0$, $h_z = 0.4$ and (d) quadratic band closing for $h_z = K = 0.4$. For $h_z \ne K > 0$ the spectrum remains gapped. In (b)-(d) the bands above show the dispersion of the middle two bands near zero energy while the colour plot below shows the full energy dispersion in the full BZ.
• Figure 3: The $h_z/K$ vs. $J/K$ phase diagram of ground state configuration for the bilayer Hamiltonian. There exist four distinct phases separated by black colored phase boundary. These boundaries are evaluated variationally by calculating the energy of $192 \times 192$ system, for the flux configurations confirmed by Monte Carlo simulation. The brown dashed line in the $0-\pi$ phase shows the $h_z = J$ line, along which the band spectrum possesses quadratic band touching.
• Figure 4: Four figures in each row show intralayer, horizontal interlayer, vertical interlayer flux distribution and corresponding excitation spectrum of the lowest conduction band respectively for (a-d) $1/4 - \pi$, (e-h) $3/4 - 2/3$ and (i-l) $2/3 - 2/3$ flux phases. A clear pictorial scheme to identify horizontal and vertical interlayer flux distribution is illustrated in Appendix \ref{['Appendix_A1']}. We have chosen different colours of different flux phases for easy identification purposes in accordance with Fig. \ref{['fig:phase']}. The unit cell of every flux phase is also marked by black dashed line. While the light coloured plaquettes in each figure of first three columns denote zero flux, the dark coloured plaquettes represent $\pi$ flux sectors in all the ground state configurations.
• Figure 5: Interlayer plaquette distribution in the moiré unit cell for two different twist angles. (a) Square and (b) pentagon interlayer plaquettes at twist angle $21.79^{\circ}$ around $C_{3z}$ invariant center and (c) and (d) represent the same as (a) and (b) for the twist angle $32.21^{\circ}$. All the subfigures follow two different color schemes: one for background shades and other for plaquette types and plaquette eigenvalues. The background shades imply the stacking region: blue for AA region and yellow for AB region. For the plaquettes, the blue polygons depict square interlayer plaquettes, out of which light blue implies eigenvalue $+1$ and dark blue implies eigenvalue $-1$. Similarly, the red polygons depict pentagon plaquettes out of which light red pentagons imply $+i$ and dark red imply $-i$. The figure clearly shows all the $-1$ square plaquettes are in AA region, while $+1$ plaquettes are in AB region. The pentagon plaquettes are connected between AA and AB region. The plaquettes encircled in the middle illustrate the actual configuration of different plaquettes with their corresponding eigenvalues.