Anyon Quasilocalization in a Quasicrystalline Toric Code

TL;DR

This paper explores how quasicrystalline geometry induces unique localization phenomena in a topologically ordered quantum spin liquid, revealing a hierarchy of coupling constants and localized anyonic excitations.

Contribution

It demonstrates that quasicrystal geometry naturally produces exponentially separated couplings and anomalous localization of anyons in a topologically ordered phase.

Findings

Hierarchy of exponentially separated coupling constants

Anomalous delocalization of anyonic excitations

Existence of strictly localized eigenstates

Abstract

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.