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arXiv:2308.07965 · Physical review. B./Physical review. B

Kitaev-type spin liquid on a quasicrystal

Mehmet Akif Keskiner, Onur Erten, Mehmet Özgür Oktel

Published Sep 29, 2023• 18 citations
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Abstract

We develop an exactly solvable model with Kitaev-type interactions and study its phase diagram on the dual lattice of the quasicrystalline Ammann-Beenker lattice. Our construction is based on the $\mathrm{\ensuremath{\Gamma}}$-matrix generalization of the Kitaev model and utilizes the cut-and-project correspondence between the four-dimensional simple cubic lattice and the Ammann-Beenker lattice to designate four types of bonds. We obtain a rich phase diagram with gapped (chiral and abelian) and gapless spin liquid phases via Monte Carlo simulations and variational analysis. We show that the ground state can be further tuned by the inclusion of an onsite term that selects 21 different vison configurations while maintaining the integrability of the model. Our results highlight the rich physics at the intersection of quasicrystals and quantum magnetism.

DOI
10.1103/physrevb.108.104208

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We develop an exactly solvable model with Kitaev-type interactions and study its phase diagram on the dual lattice of the quasicrystalline Ammann-Beenker lattice. Our construction is based on the \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ-matrix generalization of the Kitaev model and utilizes the cut-and-project correspondence between the four-dimensional simple cubic lattice and the Ammann-Beenker lattice to designate four types of bonds. We obtain a rich phase diagram with gapped (chiral and abelian) and gapless spin liquid phases via Monte Carlo simulations and variational analysis. We show that the ground state can be further tuned by the inclusion of an onsite term that selects 21 different vison configurations while maintaining the integrability of the model. Our results highlight the rich physics at the intersection of quasicrystals and quantum magnetism.
Using unbiased Monte Carlo simulations and variational analysis, we present the ground state and finite-temperature phase diagrams of an exactly solvable spin-orbital model with Kitaev-type interactions on a square lattice. We show that an array of gapped and gapless vison crystals, characterized by the periodic arrangement of Z2{\mathbb{Z}}_{2}Z2​ flux excitations, can be stabilized as a function of external magnetic field and exchange anisotropy. In particular, we discover a variety of ``quarter phases'' wherein different 16-site periodic patterns emerge, with only a quarter of the fluxes adopting zero-flux configurations. In contrast, the rest remain in $\ensuremath{\pi}$-flux configurations. Vison crystals break translational symmetry and undergo finite-temperature phase transitions. We investigate the finite-temperature properties of these phases and report the corresponding critical and crossover temperatures. Our results reveal an array of novel phases in exactly solvable extensions of the Kitaev model, wherein local and topological orders can coexist.
Building on the recent advancements on moir\'e superlattices, we propose an exactly solvable model with Kitaev-type interactions on a bilayer honeycomb lattice for both AA stacking and moir\'e superlattices. Using Monte Carlo simulations and variational analysis, we uncover a rich variety of phases where the intra and interlayer Z2{\mathbb{Z}}_{2}Z2​ 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 moir\'e superlattices at various commensurate twist angles around two distinct twist centers represented by C3z{C}_{3z}C3z​ and C6z{C}_{6z}C6z​ of the honeycomb lattice. Our simulations reveal generalized arrangements of plaquette values that correlate with the AA or AB stacking regions across the moir\'e unit cell. Moreover, we find that, depending on the twist angle, twist center and interlayer coupling, moir\'e 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.
We theoretically studied an exactly solvable gamma matrix generalization of the Kitaev spin model on the ruby lattice, which is a honeycomb lattice with 'expanded' vertices and links. We find that this model displays an exceptionally rich phase diagram that includes (i) gapless phases with stable spin Fermi surfaces, (ii) gapless phases with low-energy Dirac cones and quadratic band touching points and (iii) gapped phases with finite Chern numbers possessing the values ±4,±3,±2 and ±1. The model is then generalized to include Ising-like interactions that break the exact solvability of the model in a controlled manner. When these terms are dominant, they lead to a trivial Ising ordered phase which is shown to be adiabatically connected to a large coupling limit of the exactly solvable phase. In the limit where these interactions are weak, we treat them within mean-field theory and present the resulting phase diagrams. We discuss the nature of the transitions between various phases. Our results show the richness of possible ground states in closely related magnetic systems.
We study an exactly solvable quantum spin model of Kitaev type on the kagome lattice. We find a rich phase diagram which includes a topological (gapped) chiral spin liquid with gapless chiral edge states, and a gapless chiral spin liquid phase with a spin Fermi surface. The ground state of the current model contains an odd number of electrons per unit cell, which qualitatively distinguishes it from previously studied exactly solvable models with a spin Fermi surface. Moreover, we show that the spin Fermi surface is stable against weak perturbations.
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.
In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.