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arXiv:1910.00601 · Physical Review Research

Toward Kitaev's sixteenfold way in a honeycomb lattice model

Shang-Shun Zhang, Cristian D. Batista, Gábor B. Halász

Published Jun 15, 2020• 70 citations
arXivHTMLPDF

Abstract

The authors realize a large class of topologically ordered phases in an exactly solvable spin model on the honeycomb lattice. These strongly entangled topological phases display a variety of nonlocal anyonic quasiparticles and are characterized by different quantized values of the thermal Hall conductivity.
DOI
10.1103/physrevresearch.2.023334

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In two dimensions, the topological order described by Z2{\mathbb{Z}}_{2}Z2​ gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number $\ensuremath{\nu}$ is classified by $\ensuremath{\nu}\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}16$ as predicted by Kitaev [Ann. Phys. 321, 2 (2006)]. Here, we provide a systematic and complete construction of microscopic models realizing this so-called sixteenfold way of anyon theories. These models are defined by $\mathrm{\ensuremath{\Gamma}}$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}(\ensuremath{\nu})$ symmetry, and live on either square or honeycomb lattices depending on the parity of $\ensuremath{\nu}$. We show that all these models are exactly solvable by using a Majorana representation and characterize the topological order by calculating the topological spin of an anyonic quasiparticle and the ground-state degeneracy. The possible relevance of the $\ensuremath{\nu}=2$ and $\ensuremath{\nu}=3$ models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.
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In this work we study the phase diagram of the Kekul\'e-Kitaev model. The model is defined on a honeycomb lattice with bond-dependent anisotropic exchange interactions, making it exactly solvable in terms of Majorana representation of spins, in close analogy to the Kitaev model. However, the energy spectrum of Majorana fermions has a multiband structure characterized by Chern numbers 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We obtained the phase diagram of the model in the plane of exchange couplings and in the presence of a magnetic field and found chiral topological and trivial spin-liquid ground states. In the absence of magnetic field most of the phase diagram is a trivial gapped phase continuously connected to an Abelian phase, while in the presence of the magnetic field a topological phase arises. Furthermore, motivated by recent thermal measurements on the spin-liquid candidate $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$, we calculated the thermal Hall conductivity at different regimes of parameters and temperatures and found the latter is quantized over a wide range of temperatures.
Three-dimensional Kitaev models -- generalizations of the well known exactly solvable Kitaev honeycomb model -- exhibit a rich variety of quantum spin liquid phases. Their fractionalized excitations are Majorana fermions that may form topological band structures. These can be enriched by lattice symmetries, as shown in the present work. In particular, the authors propose two new instances of Kitaev spin liquids that exhibit nodal chains and 3D Dirac nodes in the bulk. Such crystalline Kitaev spin liquids are direct generalizations of crystalline topological insulators, albeit in an interacting spin system. A potential material realization in metal-organic frameworks is also discussed.
We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with Abelian and non-Abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.