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arXiv:2609.29004

Exact selection of a toric-code vison crystal in a flux-conditioned Kitaev model

Jiucai Wang, Chuan Chen

Published Sep 24, 2026• 0 citations
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Abstract

We construct an exactly solvable extension of the spin-$1/2$ Kitaev honeycomb model in which conserved $\mathbb{Z}_2$ fluxes determine not only the signs but also the connectivity of nearest-neighbor Majorana hopping. At a tuned loop point, hopping survives only across opposite-flux plaquettes, so every vertex has active degree zero or two and the matter Hamiltonian fragments in each flux sector into independent Majorana rings and isolated zero modes. Exact ring spectra and bond counting then bound the matter energy over all local flux configurations and all four Wilson-loop sectors, and split it exactly into a frustrated triangular-lattice Ising term, whose extensively degenerate ground states are the fully packed loop coverings, and a non-negative Majorana residual. On admissible commensurate tori, the residual selects precisely the three translation-related $2/3$-vison crystals, which saturate the bound, and an exact fermion-parity identity shows that for antiferromagnetic coupling their vacua also survive projection, making them rigorous ground states of the spin model. Within a single crystal, a depth-one local unitary then maps the ground space onto that of a sheared square-lattice toric code, giving fourfold topological degeneracy. The model thus realizes exact nonperturbative gauge-matter feedback: the flux fixes where the Majorana fermions may move, and their zero-point energy selects in return a topologically ordered vison crystal with spontaneously broken translation symmetry.

DOI
10.48550/arxiv.2609.29004

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We construct an exactly solvable extension of the spin-1/21/21/2 Kitaev honeycomb model in which conserved Z2\mathbb{Z}_2Z2​ fluxes determine not only the signs but also the connectivity of nearest-neighbor Majorana hopping. At a tuned loop point, hopping survives only across opposite-flux plaquettes, so every vertex has active degree zero or two and the matter Hamiltonian fragments in each flux sector into independent Majorana rings and isolated zero modes. Exact ring spectra and bond counting then bound the matter energy over all local flux configurations and all four Wilson-loop sectors, and split it exactly into a frustrated triangular-lattice Ising term, whose extensively degenerate ground states are the fully packed loop coverings, and a non-negative Majorana residual. On admissible commensurate tori, the residual selects precisely the three translation-related 2/32/32/3-vison crystals, which saturate the bound, and an exact fermion-parity identity shows that for antiferromagnetic coupling their vacua also survive projection, making them rigorous ground states of the spin model. Within a single crystal, a depth-one local unitary then maps the ground space onto that of a sheared square-lattice toric code, giving fourfold topological degeneracy. The model thus realizes exact nonperturbative gauge-matter feedback: the flux fixes where the Majorana fermions may move, and their zero-point energy selects in return a topologically ordered vison crystal with spontaneously broken translation symmetry.
We study the spin-1/21/21/2 Kitaev-Heisenberg model on the square-hexagon-dodecagon lattice with a symmetry-preserving tri-coloring of the Kitaev exchanges. We focus on two cases: (i) the pure Kitaev model with anisotropic exchanges and (ii) the Kitaev-Heisenberg model with isotropic Kitaev exchange. In the absence of Heisenberg interactions, we use the standard mapping of the Kitaev model onto a quadratic Majorana-fermion problem in a Z2\mathbb{Z}_2Z2​ gauge field. We find a gapped energy spectrum throughout the positive-coupling phase diagram, except at a single point where the twofold-degenerate energy bands form a Dirac cone. This unique gapped region corresponds to a toric code phase. However, depending on the couplings, vortex excitations on square, hexagonal, and dodecagonal plaquettes realize three distinct relative assignments to the eee and mmm anyons. We determine these relative assignments from the physical fermion parity of the vortex sectors and locate their boundaries by tracking zero crossings of vortex-bound Majorana levels. For the isotropic Kitaev-Heisenberg model, we combine graph-based projected entangled-pair-state calculations, exact diagonalizations, and linear spin-wave theory to study the entire phase diagram. Apart from the toric code topological phases that are robust around the Kitaev limits, we obtain four collinear magnetically ordered phases that can be characterized by their local and relative orderings of hexagonal plaquettes. We also identify a special point at which the ground state is a product state.
The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static Z2\mathbb{Z}_2Z2​ gauge field, giving the possibility of πππ-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength κκκ, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy εεε in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling εεε in the two limits of small or large κκκ. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.
We study Z 2 topologically ordered states enriched by translational symmetry by employing a recently developed two-dimensional (2D) bosonization approach that implements an exact Z 2 charge-flux attachment in the lattice.Such states can display "weak symmetry breaking" of translations, in which both the Hamiltonian and ground state remain fully translational invariant but the symmetry is "broken" by its anyon quasiparticles, in the sense that its action maps them into a different superselection sector.We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of one-dimensional Kitaev wires, leading to the amusing property that there is no local operator that can transport the π -flux quasiparticle across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant.We explain why this phenomenon occurs hand in hand with other previously identified peculiar features such as ground-state degeneracy dependence on the size of the torus and the appearance of dangling boundary Majorana modes in certain Z 2 topologically ordered states.Moreover, by extending the Z 2 charge-flux attachment to open lattices and cylinders, we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes.We also review the Z × (Z 2 ) 3 classification of 2D BdG Hamiltonians (class D) enriched by translational symmetry and provide arguments on its robust stability against interactions and self-averaging disorder that preserve translational symmetry.
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.
We present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner-type fermionization and find that the Hamiltonian takes a BCS-type form, allowing the system to be solved by Bogoliubov transformation. Our fermionization does not employ nonphysical auxiliary degrees of freedom and the eigenstates we obtain are completely explicit in terms of the spin variables. The ground state is obtained as a BCS condensate of fermion pairs over a vacuum state which corresponds to the toric-code state with the same vorticity. We show in detail how to calculate all eigenstates and eigenvalues of the model on the torus. In particular, we find that the topological degeneracy on the torus descends directly from that of the toric-code, which now supplies four vacua for the fermions, one for each choice of periodic vs antiperiodic boundary conditions. The reduction of the degeneracy in the non-Abelian phase of the model is seen to be due to the vanishing of one of the corresponding candidate BCS ground states in that phase. This occurs in particular in the fully periodic vortex-free sector. The true ground state in this sector is exhibited and shown to be gapped away from the three partially antiperiodic ground states whenever the non-Abelian phase is gapped.
We develop an exact map of all states and operators from two-dimensional lattices of spins-1/21/21/2 into lattices of fermions and bosons with mutual semionic statistical interaction that goes beyond previous dualities of Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories because it does not rely on imposing local conservation laws and captures the motion of ``charges'' and ``fluxes'' on equal footing. This map allows to explicitly compute the Berry phases for the transport of fluxes in a large class of symmetry-enriched topologically ordered states with emergent Z2{\mathbb{Z}}_{2}Z2​ gauge fields that includes chiral, nonchiral, Abelian or non-Abelian, that can be perturbatively connected to models where the visons are static and the emergent fermionic spinons have a noninteracting dispersion. The numerical complexity of computing such vison phases reduces, therefore, to computing overlaps of ground states of free-fermion Hamiltonians. Among other results, we establish numerically the conditions under which the Majorana-carrying flux excitation in Ising-topologically ordered states enriched by translations acquires the 0 or $\ensuremath{\pi}$ phase when moving around a single plaquette.