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arXiv:2209.09054 · SciPost Physics

Lacing topological orders in two dimensions: Exactly solvable models for Kitaev's sixteen-fold way

Jin-Tao Jin, Jian-Jian Miao, Yi Zhou

Published May 1, 2023• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.14.5.087

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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, 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.
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.
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.
We construct spin models with bond-dependent anisotropic interactions on the penta-coordinated maple-leaf and trellis lattices, which yield exact Z2\mathbb{Z}_2Z2​ quantum spin liquids akin to the Kitaev honeycomb model. We characterize the resulting ground states by their flux sectors, Chern numbers, and topological excitations. Using replica exchange quantum Monte Carlo simulations, we find that the gauge fluxes are ordered at sufficiently low temperatures such that every unit triangle has ±π/2\pm π/2±π/2-flux, where the sign is uniform across the system, and every unit hexagon (square) has 000-flux (πππ-flux) within the parameter space of interest. We map out the topological phase diagram for each model, which reveals parameter regimes hosting Z2\mathbb{Z}_2Z2​ and Ising topological orders, and we derive an analytical expression for the mass term of the Majorana fermions, the vanishing of which indicates a transition between these phases. We further establish the correspondence between individual vortices (i.e., flux excitations) and two species of anyons in the dimer limit, overcoming the obstacle faced by degenerate perturbation theory in treating odd-length elementary plaquettes. Interestingly, we find that two dimer limits of the maple-leaf model with distinct assignments of anyon species can be smoothly connected to each other in the vortex-free sector, but they are separated by fermion-gap-closing transitions in certain two-vortex sectors.
A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the two-dimensional (2D) Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) flux attachment. For a correct fermionization of lattice quantum spin-1/21/21/2 magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange---this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge fields. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons field theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to specific combinations of dual lattice gauge fields that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman field in the zzz-direction. Our CS gauge theory framework provides an understanding of why the topological phase is degraded at lower (higher) critical fields for the ferromagnetic (antiferromagnetic) Kitaev interaction. Additionally, we observe an effectively one-dimensional character of the low excitations at higher fields in the zzz-direction, which we also confirm by spin-wave calculations.
The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.