Anyon Quasilocalization in a Quasicrystalline Toric Code

An exactly solvable model of a quantum spin liquid on a quasicrystal, akin to Kitaev’s honeycomb model, was introduced in Kim \textit{et al.}, \href{https://doi.org/10.1103/PhysRevB.110.214438}{\text{Phys. Rev. B} \textbf{110}, 214438 (2024)}. It was shown that in contrast to the translationally invariant models, such a spin liquid stabilizes a gapped ground state with a finite irrational flux density. In this work, we analyze the strong bond-anisotropic limit of the model and demonstrate that the aperiodic lattice geometry naturally generates a hierarchy of exponentially separated coupling constants in the resulting toric code Hamiltonian. Furthermore, a perturbative magnetic field leads to anomalous localization properties where an anyonic excitation sequentially delocalizes over subsets of sites forming equipotential contours in the quasicrystal. In addition, certain background flux configurations, together with the underlying geometry, give rise to strictly localized eigenstates that remain decoupled from the rest of the spectrum. Using numerical studies, we uncover the key mechanisms responsible for this unconventional localization behavior. Our study highlights that topologically ordered phases, in the presence of geometrical constraints can lead to highly anomalous localization properties of fractionalized charges.

💡 Research Summary

In this work the authors investigate the strong‑bond‑anisotropic limit of the recently introduced Kitaev‑type quantum spin liquid on a Penrose‑derived quasicrystalline lattice. By taking Jz≫Jx,Jy they map the model onto an effective toric‑code (TC) Hamiltonian defined on the dual lattice of the strong z‑bonds. Because the underlying quasicrystal contains vertices and plaquettes of many coordination numbers (n=2,3,4,5 for stars and n=4,6,10 for plaquettes), the TC Hamiltonian contains a hierarchy of star operators A⁽ⁿ⁾ and plaquette operators B⁽ⁿ⁾ with coupling constants λ⁽ⁿ⁾ that scale as Jⁿ/Jzⁿ⁻¹. Consequently the energy spectrum splits into exponentially separated gaps – a “hierarchical energy scale” that is a direct consequence of the aperiodic geometry.

The ground state in zero magnetic field already hosts a finite, irrational density of electric (e) and magnetic (m) anyons. The densities are determined by the golden‑ratio scaling of the various tile types and read ρe≈½φ⁻²+φ⁻⁵ and ρm≈½φ⁻². Thus, unlike the conventional square‑lattice toric code, the quasicrystalline version possesses a built‑in background of anyons dictated purely by geometry.

A perturbative Zeeman field HZ=−hz∑τᶻ−hx∑τˣ is then added. The longitudinal component hz breaks time‑reversal symmetry and does not commute with the star operators, thereby rendering the e‑anyons dynamical. As hz is increased, the gap ΔE⁽ⁿ⁾ₑ associated with each star‑operator family (n) shrinks proportionally to λ⁽ⁿ⁾A and vanishes when hz≈O(λ⁽ⁿ⁾A). This produces a sequence of “condensation” transitions: first the lowest‑order stars become gapless, then higher‑order ones, and so on. Importantly, the anyons do not delocalize over the entire lattice; instead they spread stepwise over subsets of sites that share the same coordination number. The authors term this behavior “quasilocalization”. Numerical evaluation of the mean‑square displacement ⟨r²(t)⟩ shows a characteristic staircase: flat plateaus when the anyon is confined to a small loop, followed by sudden jumps when it tunnels to a larger, geometrically similar loop. This is a novel dynamical regime absent in periodic toric‑code models.

In addition, certain background flux configurations—specifically π‑fluxes placed on square and octagonal plaquettes inherited from the isotropic Kitaev model—combine with the quasicrystalline geometry to produce strictly localized eigenstates. These states are eigenvectors of all star and plaquette operators with τᶻ=+1 on every bond, and they remain completely decoupled from the rest of the spectrum even in the presence of the Zeeman perturbation. Hence the model supports both quasilocalized mobile anyons and immobile, flux‑protected modes.

The authors corroborate their analytical findings with exact diagonalization and tensor‑network simulations on finite‑generation quasicrystals (generation ≤5). The numerics confirm the exponential hierarchy of coupling constants, the stepwise reduction of gaps with increasing field, the staircase dynamics of anyon propagation, and the existence of isolated flat‑band‑like states.

Overall, the paper demonstrates three key mechanisms by which a topologically ordered phase intertwines with aperiodic geometry: (i) the emergence of a hierarchy of energy scales set by the distribution of tile sizes, (ii) the appearance of quasilocalized anyon dynamics under a magnetic field, and (iii) the protection of strictly localized modes by specific flux patterns. These results open a new avenue for exploring unconventional localization phenomena in fractionalized systems and suggest that quasicrystalline platforms could host robust, geometry‑engineered anyonic excitations useful for quantum information applications.