The Kitaev honeycomb model has attracted significant interest due to its quantum spin liquid ground state, fractionalized Majorana excitations, and topological properties. Motivated by these features, we introduce a quasi-one-dimensional Kitaev-like spin chain derived from a truncated honeycomb geometry. The resulting Majorana band structure contains both dispersive and flat bands, with a gap closing at a critical anisotropy that separates trivial and chiral topological phases. Under open boundary conditions, the topological regime hosts zero-energy edge modes protected by chiral symmetry, placing the system in the BDI symmetry class with a quantized winding number. In the excited spectrum, localized plaquette modes emerge near zero energy and can be tuned by introducing domains of negative plaquettes. The dynamical spin correlation function reveals broad continua associated with fractionalized excitations, together with characteristic low-energy spectral weight from edge Majorana modes. These features distinguish the present system from conventional Heisenberg spin chains and provide experimentally relevant fingerprints of chiral topological order. Our model thus establishes a conceptual bridge between the two-dimensional Kitaev spin liquid and Kitaev's one-dimensional quantum wire.
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 quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum---line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.
We study the gapped phase of Kitaev's honeycomb model (a
Z2 spin liquid) on a lattice with topological defects. We find that some dislocations and string defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a
Z2 gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a
Z2 vortex around one of the dislocations. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes. To facilitate these theoretical developments, we recast the degenerate perturbation theory for spins in the language of Majorana fermions.
In this article, we address the possibility of topological quantum spin liquids with only short-range (spin) correlations in the whole phase diagram in a ladder system, through an exactly solvable model which represents a generalization of the two-dimensional Kitaev honeycomb spin model to a ribbon geometry. This system gives rise to
Z2 gauge fields emerging in a static way in the ground state through Majorana fermions which then produce either
0 or
π flux. Through a quantum field theory perspective, mathematical and Density Matrix Renormalization Group results, we present our progress on the geometrical, topological and physical classification of this model through observables navigating on the whole phase diagram presenting three spin-liquid phases with a spin gap and a metallic phase in the sense of Majorana fermions. We propose a bulk-edge correspondence revealing the presence of protected zero-energy Majorana fermions at the edge in the phases with a gap and revealing also the presence of itinerant Majorana fermions in the bulk in the topological liquid phase. The proposed approach allows us to distinguish between properties of the brickwall ladder giving rise to a special line and those of the square ladder allowing for the gapless Majorana liquid phase. This model gives some hope on finding solutions for topological quantum spin models, e.g. with
π flux, on a square lattice geometry. This approach also allows for a correspondence onto models of topological superconductivity.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped
Z2 fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (
Kx=Ky=Kz). We explore the phase diagram along two axes, an external magnetic field
h, applied out of plane of the honeycomb, and anisotropic interactions
Kz larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and
h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped
Z2 toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (
e) and magnetic (
m) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower
Kz in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.