Quaternion-Charge Edge States

• Quaternion-charge edge states are boundary excitations defined by non-Abelian topological invariants, extending classification beyond conventional Chern numbers and Berry phases.
• Their construction employs techniques such as Floquet engineering and multi-channel tight-binding models, enabling precise control of exotic edge configurations.
• These states exhibit multifold bulk-edge correspondence and support phenomena like edge-of-edge modes, with significant implications for quantum devices and topological materials.

Quaternion-charge edge states are boundary or interface-localized excitations in topological systems whose properties—such as robustness, degeneracy, and transformation rules—are governed by topological invariants taking values in the quaternion group or in related non-Abelian (typically SU(2) or U(2)) structures. Unlike conventional edge states characterized by integer or Abelian invariants (Chern numbers, $\mathbb{Z}_2$ indices, Berry phases), quaternion-charge edge states display a rich interplay between multiple degrees of freedom (spin, valley, charge, or flavor), manifest non-Abelian bulk-edge correspondence, and can encode topological orders inaccessible to the standard classification paradigms. These edge states arise in settings ranging from Floquet-driven non-Abelian topological insulators and higher-dimensional quantum Hall systems to strongly interacting quantum wires and systems with synthetic gauge potentials, with the quaternionic (or more generally non-Abelian) charge playing a central role in their classification and dynamical response.

1. Quaternionic Topology in Edge-State Classification

Edge-state topology is conventionally described through bulk-boundary correspondence, where bulk invariants (Chern number, $\mathbb{Z}_2$ index) dictate the presence and nature of edge states. For quaternion-charge edge states, this correspondence generalizes to non-Abelian structures:

• In Floquet non-Abelian topological insulators (FNATIs), the bulk invariant is a quaternion charge $q$ belonging to the quaternion group $Q_8 \equiv \{1, -1, \pm i, \pm j, \pm k\}$, with multiplication rules $i^2 = j^2 = k^2 = ijk = -1$ and $ij = -ji$ (Li et al., 2023).
• The configuration and multiplicity of edge states are not determined solely by the value of $q$, but must also account for the nontrivial ordering and localization of “Dirac singularities” (phase-band crossings) in the momentum-time phase band structure.

A similar non-Abelian topological structure arises in higher-dimensional topological systems—for instance, in 4D class A insulators, where edge states are encoded by Berry connections with monopole structures and in certain boundary configurations realize the ’t Hooft–Polyakov monopole (an SU(2) object) rather than an ordinary Dirac monopole (Hashimoto et al., 2016).

2. Construction and Dynamics of Quaternion-Charge Edge States

The emergence and manipulation of quaternion-charge edge states require physical mechanisms that instantiate and probe their non-Abelian character:

• Floquet Engineering: Periodically driven systems (Floquet systems) can exhibit multiple quasi-energy band gaps. Topological characterization then relies on phase-band singularities, each carrying a local quaternion charge $\tilde{q}_m$. The global quaternion charge $q = \prod_m \tilde{q}_m$ (ordered product) fully determines the system's topology, and the specific arrangement of Dirac points sets the multifold edge-state structure (Li et al., 2023).
• Edge-of-Edge Localization: In higher-dimensional models, edge states at boundaries can carry nontrivial topological charges; when two such boundaries intersect (with appropriately chosen boundary conditions), exotic “edge-of-edge” modes form at the intersection. Compatibility conditions between boundary U(2) matrices $U_4$ and $U_5$—specifically, $\det[\mathbb{1} + U_4 - U_5 + U_5 U_4] = 0$—characterize the existence of these higher-codimension localized states, whose robustness is governed by non-Abelian Berry connections and fractional Chern–Simons forms in both momentum and boundary-parameter spaces (Hashimoto et al., 2017).
• Spin-Charge Transistor Devices: In two-dimensional topological insulators (e.g., HgTe quantum wells), device geometries such as constrictions bring spin-locked helical edge channels into spatial proximity, allowing for coherent charge and spin switching between edges. This system realizes a device whose output encodes “real” (charge) and “imaginary” (spin) components—a physical manifestation of quaternionic state control—enabling three-state transistor operation based on energy-dependent and gate-controlled overlap of edge-state wavefunctions (Krueckl et al., 2011).

3. Non-Abelian Bulk-Edge Correspondence and Anomalous Phases

Quaternion-charge systems exhibit a multifaceted bulk-edge correspondence and unique dynamical phenomena:

• The same global quaternion charge $q$ may correspond to distinct edge-state patterns, depending on how the constituent Dirac singularities distribute across energy gaps. This “multifold” correspondence is a direct consequence of the non-commutative group structure—e.g., for $q = j$, both $k \cdot i = j$ and $j$ alone can yield different edge-state configurations (Li et al., 2023).
• Anomalous Floquet phases exist where the global topological charge is trivial, but edge states persist in all energy gaps due to the fine structure of phase-band singularities. This phenomenon lacks a static analog and underscores the essential role of time-resolved (dynamical) topology (Li et al., 2023).
• At interfaces between regions driven with swapped time sequences, non-Abelian nature manifests in “swap effects”: even though subsystems have identical bulk spectra, their interface supports protected domain-wall modes due to a change in the local quaternion charge, related to nontrivial loops in SO(3) and the group homotopy $\pi_1(SO(3)) = \mathbb{Z}_2$ (Li et al., 2023).

4. Mathematical Structure and Monopole Realizations

Quaternion-charge edge states are deeply linked to non-Abelian gauge field structures in both real and momentum spaces:

• Berry Connection and Monopoles: Edge states in 4D topological systems can carry Berry connections whose field strengths realize either Dirac (U(1)) monopoles or ’t Hooft–Polyakov (SU(2)) monopoles, depending on boundary configuration. When two edge states coexist on parallel boundaries, the non-Abelian Berry connection and an adjoint “Higgs” scalar $\Phi$ satisfy BPS equations $D_i \Phi = \frac{1}{2}\epsilon_{ijk} F_{jk}$, yielding a non-Abelian topological charge analogous to the TKNN number (Hashimoto et al., 2016).
• Generalized Berry Connections: For edge-of-edge states, Berry connections are extended beyond momentum space to the space of boundary parameters (U(2) or SU(2)), with associated Chern–Simons invariants (e.g., $S_{CS} = \pi/4$ over S$^3$ parameter space) protecting the exotic localization (Hashimoto et al., 2017).

These structures admit realization and interpretation in string-theory-inspired D-brane setups: D1-branes (Dirac monopoles) ending on one or two D3-branes (t’Hooft–Polyakov monopoles), with the “position” in the bulk corresponding to the scalar field $\Phi$ (Hashimoto et al., 2016).

5. Quaternionic Analyticity and Multi-component Edge States

Quaternionic analyticity provides a unifying mathematical lens for constructing and understanding high-dimensional topological states:

• Landau Level Analyticity: The lowest Landau level wavefunctions in 3D and 4D, reorganized via SU(2) Aharonov–Casher gauge potentials, exhibit quaternionic analyticity by satisfying generalized Cauchy–Riemann–Fueter equations. For a two-component spinor $\Psi = (\psi_\uparrow, \psi_\downarrow)$ mapped to $f = \psi_\uparrow + j\psi_\downarrow$, analyticity conditions select wavefunctions with nontrivial topological properties and, by implication, edge states that can encode quaternionic structure (Wu, 2019).
• Coset CFT and Fractionalization: In 2+1D non-Abelian topological phases, chiral edge states described by constrained fermions and coset CFTs support SU(2)$_k$ sectors with central charge $c=3k/(k+2)$. These edge states, through their non-Abelian algebra, realize “fractionalized” (quaternion-charge) excitations. Anomalies arising from chiral singlet constraints are managed through chiral bosonization and are canceled by bulk Chern–Simons inflow (Hernaski et al., 2017).

6. Generalization to Multichannel Systems and Outlook

The full extension of quaternion-charge edge state concepts to practical and interacting systems remains an active area:

• In multi-channel (multi-band, multi-orbital) tight-binding models, the unique quantization of boundary charge and edge state energies present in single-channel systems generalizes to a non-Abelian setting, where the boundary invariant may take multiple values, and a non-Abelian version of the winding number becomes necessary. The gauge choices for multi-component Bloch states and the matrix-valued Zak–Berry phase play a critical role in setting up precise bulk-boundary correspondences. A plausible implication is that experimental systems, such as Floquet-engineered or cold atom platforms, can access these nontrivial invariants by engineering suitable multi-channel couplings and boundary configurations (Pletyukhov et al., 2019).

7. Experimental and Theoretical Outlook

The quaternion-charge framework connects high-dimensional or strongly-correlated quantum matter, Floquet systems, gauge-field engineering, and even D-brane models in string theory. Key challenges include:

• Realizing robust quaternionic invariants in finite, disordered, or interacting systems.
• Classifying and detecting edge-state patterns protected by non-Abelian topological charges under realistic experimental conditions.
• Developing a unified theory of “bulk–boundary–corner–hierarchy” for topological systems, allowing the systematic inclusion of exotic localized modes (edges, edge-of-edge, etc.) and their associated quaternionic or higher non-Abelian invariants.

Theoretical advances suggest that quaternionic analyticity and non-Abelian Berry connections will serve as guiding principles in the exploration of next-generation topological materials and devices, especially those targeting multi-functional quantum, spintronic, or information-processing applications where manipulation of intertwined spin, charge, and further internal degrees of freedom is essential.