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arXiv:2005.13683 · Physical review. B./Physical review. B

Microscopic models for Kitaev's sixteenfold way of anyon theories

Sreejith Chulliparambil, Urban F. P. Seifert, Matthias Vojta C, Lukas Janssen, Hong-Hao Tu

Published Nov 11, 2020• 93 citations
arXivHTMLPDF

Abstract

In two dimensions, the topological order described by ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevb.102.201111

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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 \ensuremathν\ensuremath{\nu}\ensuremathν is classified by \ensuremathνmod16\ensuremath{\nu}\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}16\ensuremathνmod16 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 \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ matrices satisfying the Clifford algebra, enjoy a global SO(\ensuremathν)\mathrm{SO}(\ensuremath{\nu})SO(\ensuremathν) symmetry, and live on either square or honeycomb lattices depending on the parity of \ensuremathν\ensuremath{\nu}\ensuremathν. 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 \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 and \ensuremathν=3\ensuremath{\nu}=3\ensuremathν=3 models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.
A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nu ν species of Majorana fermions coupled to a static \mathbb{Z}_2 ℤ 2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1 𝒞 = 1 , yielding a total Chern number \mathcal{C}=\nu 𝒞 = ν . It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nu ν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.
In two dimensions, topological phases of free Majorana fermions coupled to a Z2{\mathbb{Z}}_{2}Z2​ gauge field are known to be classified according to the Chern number $\ensuremath{\nu}\ensuremath{\in}\mathbb{Z}$. Its value mod 16 specifies the type of anyonic excitations. In this paper, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}7$. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.
Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a Z2\mathbb{Z}_2Z2​ one-form symmetry, generated in each theory by a single nontrivial Z2\mathbb{Z}_2Z2​ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single Z2\mathbb{Z}_2Z2​ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the Z2\mathbb{Z}_2Z2​ one-form symmetry, which acquires a Z16\mathbb{Z}_{16}Z16​ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.
Emergent gauge theories take a prominent role in the description of quantum matter, supporting deconfined phases with topological order and fractionalized excitations. A common construction of \mathbb{Z}_2 ℤ 2 lattice gauge theories, first introduced by Wegner, involves Ising gauge spins placed on links and subject to a discrete \mathbb{Z}_2 ℤ 2 Gauss law constraint. As shown by Kitaev, \mathbb{Z}_2 ℤ 2 lattice gauge theories also emerge in the exact solution of certain spin systems with bond-dependent interactions. In this context, the \mathbb{Z}_2 ℤ 2 gauge field is constructed from Majorana fermions, with gauge constraints given by the parity of Majorana fermions on each site. In this work, we provide an explicit Jordan-Wigner transformation that maps between these two formulations on the square lattice, where the Kitaev-type gauge theory emerges as the exact solution of a spin-orbital (Kugel-Khomskii) Hamiltonian. We then apply our mapping to study local perturbations to the spin-orbital Hamiltonian, which correspond to anisotropic interactions between electric-field variables in the \mathbb{Z}_2 ℤ 2 gauge theory. These are shown to induce anisotropic confinement that is characterized by emergence of weakly-coupled one-dimensional spin chains. We study the nature of these phases and corresponding confinement transitions in both absence and presence of itinerant fermionic matter degrees of freedom. Finally, we discuss how our mapping can be applied to the Kitaev spin-1/2 model on the honeycomb lattice.
This pedagogical article is aimed to the beginning graduate students interested in broad field of frustrated magnetism. We introduce and present some of the exact results obtained in Kitaev model. The Kitaev model embodies an unusual two spin interactions yet exactly solvable model in two dimension. This exact solvability renders it to realize many emergent many body phenomena such as Z2Z_2Z2​ gauge field, spin liquid states, spin fractionalization, topological order exactly. First we present the exact solution of Kitaev model using Majorana fermionisation and elaborate in detail the Z2Z_2Z2​ gauge structure. Following this we discuss exact calculation of magnetization, spin-spin correlation function establishing its spin-liquid character. Spin fractionalization and de-confinement of Majorana fermion is explained in detail. Existence of long range multi-spin correlation function and topological degeneracy are discussed to elucidate the entangled and topological nature of any eigenstate. Some elementary questionnaires are provided in appropriate places for assimilation of the technical details.