Compact localized fermions and Ising anyons in a chiral spin liquid

TL;DR

This work demonstrates that the Yao-Kivelson chiral spin liquid hosts perfectly flat bands formed by compact localized states (CLS) due to destructive interference on the star lattice. By formulating a Majorana hopping model and solving across flux sectors, the authors derive exact CLS wavefunctions, including zero-energy Majorana zero modes bound to π flux that realize compact localized Ising anyons with no hybridization at minimal separation. They compute spin-spin correlations showing signatures of CLS and analyze bound-state configurations, including vortex-induced MZMs, with implications for non-Abelian braiding in quantum simulators. The results establish a framework for flat-band phenomena in quantum spin liquids and guide experimental exploration of topological order and anyon braiding in programmable quantum devices.

Abstract

Quasiparticle hybridization remains a major challenge to realizing and controlling exotic states of matter in existing quantum simulation platforms. We report the absence of hybridization for compact localized states (CLS) emerging in the chiral spin liquid described by the Yao-Kivelson model. The CLS form due to destructive quantum interference at fine-tuned coupling constants and populate perfectly flat quasiparticle bands on an effective kagome lattice. Using a formalism for general Majorana-hopping Hamiltonians, we derive exact expressions for CLS for various flux configurations and both for the topological and trivial phases of the model. In addition to finite-energy matter fermions with characteristic spin-spin correlations, we construct compact localized Majorana zero modes attached to $π$-flux excitations, which enable non-Abelian braiding of Ising anyons with minimal separation. Our results inform the quantum simulation of topologically ordered states of matter and open avenues for exploring flat-band physics in quantum spin liquids.

Figures (5)

• Figure 1: (a) Star lattice with nine bond types $\alpha_\chi$. The lattice sites $j_1$, $j_2$ and $j_3$ wind around the triangular plaquette $p$ following the convention in Eq. \ref{['eq:wilson']}. Itinerant Majoranas on the orange-shaded sites $(m, \bigtriangledown)$ and $(m, \bigtriangleup)$ are paired in Eq. \ref{['eq:matter_pairing']}. (b) The paired Majoranas form a fermion located on site $m$ of the kagome lattice.
• Figure 2: (a)-(d) Gauge configurations of uniform flux patterns (a) $\mathbf w=(i,i,-1)$, (b) $(-i,i,1)$, (c) $(i,i,1)$ and (d) $(-i,i,-1)$. A solid arrow pointing from site $j$ to site $k$ indicates $u_{jk}=+1=-u_{kj}$. White (blue) shaded dodecagons are pierced by zero ($\pi$) flux. Yellow (red) shaded triangles are pierced by $\pi/2$ ($-\pi/2$) flux. The black dashed arrows mark the primitive lattice vectors of the resulting Majorana hopping model \ref{['eq:majorana_hopping']}. (e)-(h) Single-particle energies $\varepsilon$ of corresponding quasiparticle bands as function of the coupling ratio $g=J_0/J_\bigtriangledown$. The gray (white) shaded background represents the trivial (topological) phase. Black circles indicate the flat bands specified in (e) Eq. \ref{['eq:flat1']} and (g) Eqs. \ref{['eq:flat2']} and \ref{['eq:flat3']}. The energy bands are symmetric with respect to $g\rightarrow -g$.
• Figure 3: (a) Energy difference $\Delta E_{\mathbf w}=E_{\mathbf w}-E_{(i,i,-1)}$ per unit cell of uniform flux sectors $\mathbf w$ and the ground-state sector $(i,i,-1)$ as function of the coupling ratio $g$. (b) Chern numbers $\mathcal{C}$ of quasiparticle ($\varepsilon>0$) and hole ($\varepsilon<0$) bands in ground-state sector as a function of $g$. The total Chern number of the ground state is the sum of the Chern numbers of all negative-energy bands and equal to unity for $|g|<\sqrt{3}$.
• Figure 4: (a) Cluster $\Lambda$ on a dodecagonal plaquette. An arrow pointing from site $j$ to site $k$ indicates $u_{jk}=+1$. (b) Coefficients $\varphi_j=\pm$ (up to normalization) of compact localized MZMs on clusters $\Lambda$ and $\Lambda'$, see Eq. \ref{['eq:creation_op']}. The thick green loop hosts a CLS created by $(A^\dagger_\Lambda+A^\dagger_{\Lambda'})/\sqrt{2}$.
• Figure 5: Spin correlations near a dodecagonal plaquette for a system with periodic boundary conditions and linear system size $L=37$. (a) & (b) Compact contributions $\mathcal{S}^{\mathrm{CLS}}_{jk}-\mathcal{S}^{\mathrm{vac}}_{jk}$ in Eq. \ref{['eq:spin_CLS']} indicated by the color of the corresponding bond $\braket{jk}$ for (a) a CLS in the ground-state sector $\mathbf w=(i,i,-1)$ for $g=g^\star_1$ and (b) a CLS for $g=g^\star_3=2$ in the excited flux sector described in the main text. Numerical values of finite contributions are annotated at the corresponding bonds. (c) Contribution $\mathcal{S}^{\mathrm{BS}}_{jk}-\mathcal{S}^{\mathrm{vac}}_{jk}$ of the bound state in the excited flux sector as a function of the coupling ratio $g$ for the spins connected by the bonds indicated in (b).