Fractional excitations in Kitaev quasi-one-dimensional chain

Abstract: 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.

• The paper demonstrates a novel quasi-1D Kitaev chain that inherits fractionalized Majorana excitations and topological phases from the 2D Kitaev honeycomb model.
• It employs an integrable mapping to Majorana fermions, revealing band structure evolution including gap closings at critical anisotropy values.
• Edge-localized zero modes and domain-wall induced localized states, evidenced in the dynamical spin response, highlight potential for topological quantum applications.

Fractional Excitations and Topological Phase Transitions in a Quasi-One-Dimensional Kitaev Chain

Introduction and Motivation

The paper "Fractional excitations in Kitaev quasi-one-dimensional chain" (2606.20309) introduces a quasi-one-dimensional spin-1/2 model that inherits key properties of the two-dimensional Kitaev honeycomb model, particularly fractionalized Majorana excitations and topological phases. Through a systematic truncation of the honeycomb lattice, the authors construct a chain architecture with hexagonal plaquettes interconnected via anisotropic spin couplings. The resulting system enables a rigorous Majorana fermion representation while maintaining an extensive set of local conserved quantities, yielding integrability and a fragmented Hilbert space.

The relevance of this construction is underscored by intense interest in Kitaev and related models for realizing quantum spin liquids (QSLs), nontrivial anyonic excitations, and possible applications in topological quantum computation. Reduced-dimensional architectures, such as ladders or chain fragments, provide promising routes for material synthesis and experimental detection of emergent physics, including fractionalization signatures observed in proximate Kitaev materials through neutron scattering, Raman spectroscopy, and thermal response measurements.

Model Architecture and Majorana Representation

The model derives from truncating the honeycomb lattice to form a chain comprised of hexagonal plaquettes, with anisotropic nearest-neighbor couplings ($J_x$, $J_y$, $J_z$) between specific spin components. Each unit cell contains six sublattice sites partitioned into two classes, and the interactions are schematically depicted with colored bonds denoting coupling directions.

Figure 1: Schematic of quasi-one-dimensional hexagonal chain embedded in honeycomb geometry, illustrating sublattice classes and anisotropic spin couplings.

The Hamiltonian is integrable via a mapping to Majorana fermions, with both dynamic and non-dynamic flavors per site. The transformation projects the spin-1/2 degrees of freedom onto an enlarged Majorana Hilbert space, where conserved $\mathbb{Z}_2$ bond variables serve as static gauge fields. The physical subspace is extracted using appropriate projectors, ensuring the underlying spin algebra.

Non-dynamic Majorana modes, unique to this geometry, further fragment the Hilbert space and enhance degeneracy, leading to a rich structure of many-body configurations and gauge sectors.

Band Structure and Topological Regimes

The Majorana band structure in the uniform plaquette sector presents both dispersive and flat bands. A critical anisotropy value ($J_x = \sqrt{2} J$ for $J_y = J_z = J$) marks a gap closing and an ensuing topological phase transition. The gapless regime exhibits linear dispersion at the Brillouin zone edge or center depending on gauge choices, and flat bands shift proximity to the chemical potential as anisotropy varies.

Figure 2: Majorana band structures for varying $J_x$, showing gap closing at the topological transition and characteristic evolution of flat and dispersive bands.

The ground state spectrum is invariant under random choices of inter-plaquette $\mathbb{Z}_2$ field values, a result of symmetry and plaquette redundancy specific to the lattice construction.

Boundary Modes and Topological Classification

Under open boundary conditions (OBC), the system possesses edge-localized zero-energy states in the topological regime ($J_x > \sqrt{2} J$), protected by chiral symmetry and characterized by the winding number $\nu$. The system's block-chiral Hamiltonian places it in the BDI symmetry class, analogous to the SSH chain. The topological transition manifests as a discontinuous jump in $J_y$0 and the emergence of edge modes pinned at the chemical potential.

Figure 3: Topological characterization: (a) band gap evolution under different boundary conditions, (b) emergence of boundary states in the topological regime, (c) winding number as topological invariant, and (d) edge localization probability.

Plaquette Domain Wall Excitations and Localized Modes

Excitations constructed by flipping plaquette values generate novel localized modes with robust zero-energy states at domain boundaries. For sequences of negative plaquettes embedded in a positive background, the zero modes are exponentially localized at the domain edges, and their number increases monotonically with domain multiplicity. These modes are strong markers of trivial phase topology; they disappear in the chiral topological regime, where bulk states dominate.

Figure 4: Spectral evolution and localization probability of zero modes as a function of domain wall configuration and anisotropy, demonstrating domain-boundary-induced mode translation and emergent degeneracy.

Fractionalized Excitations and Dynamical Spin Response

The dynamical spin structure factor, computed via DMRG, reveals broad continua across both trivial and topological regimes, indicative of the fractionalization of spin excitations into itinerant Majorana modes. In the trivial phase, the spectral weight concentrates at finite momenta and low energies; at the critical point and deeper in the topological phase, zero-frequency contributions from edge modes dominate, and spectral intensity becomes spread across momenta.

Figure 5: Dynamical correlation function $J_y$1 across parameter regimes, showing continuum response, gap closing at transition, and edge mode contributions.

The broad spectral continuum and redistributed weight contrast sharply with conventional magnon responses in Heisenberg chains and provide fingerprints of topological Majorana physics accessible via experimental probes.

Implications and Future Directions

The quasi-one-dimensional Kitaev chain architecture offers an integrable platform that embodies fractionalized excitations, chiral topological phases, and symmetry-protected edge modes in a minimal geometry. The interplay between anisotropy-driven transitions, domain-wall-induced localized modes, and persistent fractionalization demonstrates a bridge between 2D Kitaev spin liquids and Majorana quantum wires, with the potential for material realization in systems such as CuPzN and related structures.

The model's response properties suggest clear routes for experimental identification of topological excitations via spectroscopic and transport measurements, distinguishing it from conventional magnetic chains. The presence of robust, tunable zero modes and extensive gauge sector fragmentation hint at new paradigms for quantum information encoding and manipulation in condensed matter settings.

Theoretically, further explorations could investigate the interplay between non-dynamic Majorana modes, bulk-edge correspondence in extended geometries, and the coupling of this chain to external drives or disorder. Realistic simulation of candidate materials, inclusion of Heisenberg and off-diagonal perturbations, and the dynamics of domain-wall manipulations remain promising avenues for advancing both fundamental understanding and quantum technology applications.

Conclusion

The study establishes a rigorous quasi-one-dimensional Kitaev chain with a hexagonal plaquette structure that captures the essential physics of fractionalization, topological phase transitions, and edge-mode protection by chiral symmetry. The interplay between band topology, localized domain-wall modes, and dynamical spin response positions this model as a versatile platform for the exploration of Majorana physics, with both theoretical and experimental significance in the field of quantum magnetism and topological matter.