Annular Kitaev Chain: Topology & Majorana Modes

• Annular Kitaev chains are one-dimensional p-wave superconductors with closed-loop geometries that reveal how boundary conditions, flux, and defects orchestrate Majorana mode behavior.
• The models use Bogoliubov–de Gennes analysis and real-space diagnostics to unravel the effects of movable bonds and parity on topological invariants and spectral properties.
• Assessments of flux periodicity and transport coefficients demonstrate parity-controlled switching between single-electron and Cooper-pair processes, informing the design of quantum devices.

Searching arXiv for relevant papers on annular/ring geometries of the Kitaev chain and closely related formulations. An annular Kitaev chain is a one-dimensional spinless $p$-wave superconducting system realized on a closed or effectively closed geometry, including a periodic ring, a ring threaded by magnetic flux, or a “legged-ring” obtained by adding a single extra bond to an otherwise open chain. In the literature, this geometry is used to study how boundary conditions, fermion parity, magnetic flux, local defects, and geometric frustration reorganize the Bogoliubov–de Gennes spectrum, alter topological invariants, and control the existence or suppression of Majorana zero modes. Closely related constructions also arise from exactly solvable spin chains mapped to quadratic Majorana problems and from synthetic dimensions built from annular lowest-Landau-level orbitals (Saket et al., 2013, Maiellaro et al., 2019, Wang et al., 15 Jul 2025, Ali et al., 6 May 2026).

1. Geometric definitions and model variants

The annular geometry appears in several distinct but related forms. In the flux-threaded Kitaev ring, one considers $N$ sites with periodic boundary conditions $c_{N+1}=c_1$, nearest-neighbor hopping $t$, $p$-wave pairing $\Delta$, on-site potential $\mu$, and a Peierls phase $\phi$ per link induced by a uniform flux $\Phi=N\phi$. The Hamiltonian is

$$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$

This formulation makes the annular character explicit through periodicity and flux threading (Wang et al., 15 Jul 2025).

A second realization is the “Kitaev tie,” described as a Kitaev chain in the shape of a legged-ring. The underlying system is an open chain with sites $N$0, to which a single extra hopping $N$1 is added between sites $N$2 and $N$3. The full Hamiltonian is

$N$4

with

$N$5

and $N$6. The extra bond acts as a movable “tie knot,” and the topological properties are determined by the knot position $N$7 (Maiellaro et al., 2019).

A third usage arises in the exactly solvable spin-chain context, where an “annular Kitaev chain” is obtained from a two-spin XY–Ising model with periodic boundary conditions in the spin variables. After a Jordan–Wigner transformation, the model becomes a quadratic Majorana chain with periodic, anti-periodic, or open boundary conditions depending on fermion parity and defect configuration. In this construction, the ring geometry is encoded in the boundary term $N$8 and in the defect-free sector the chain is closed and uniform (Saket et al., 2013).

A further annular realization appears in synthetic dimension. In symmetric gauge, the lowest-Landau-level orbital $N$9 has a radial probability peak at

$c_{N+1}=c_1$0

Each $c_{N+1}=c_1$1 labels a ring-shaped orbital of radius $c_{N+1}=c_1$2, and a finite disk or annulus defines open boundaries at $c_{N+1}=c_1$3 and $c_{N+1}=c_1$4, which serve as the two ends of a synthetic Kitaev chain (Ali et al., 6 May 2026). This suggests that “annular Kitaev chain” is best understood as a geometry class rather than a single Hamiltonian.

2. Quadratic Hamiltonians and Bogoliubov–de Gennes structure

Across these realizations, the common structure is a quadratic superconducting Hamiltonian that can be represented in Nambu space and diagonalized through a Bogoliubov–de Gennes procedure.

For the Kitaev tie, one defines

$c_{N+1}=c_1$5

and writes

$c_{N+1}=c_1$6

where $c_{N+1}=c_1$7 is a $c_{N+1}=c_1$8 real-symmetric matrix which can be diagonalized numerically (Maiellaro et al., 2019). Because the movable bond breaks translation invariance, the problem is intrinsically real-space.

For the flux-threaded ring, translation invariance permits a momentum-space decomposition. With

$c_{N+1}=c_1$9

the Nambu-space Hamiltonian $t$0 becomes block diagonal,

$t$1

with

$t$2

The spectrum is

$t$3

This exact $t$4-space structure underlies the parity-dependent flux periodicity and the transport resonances discussed later (Wang et al., 15 Jul 2025).

In the exactly solvable spin-chain realization, the quadratic Majorana problem takes the Bloch form

$t$5

with dispersion

$t$6

The allowed values of $t$7 depend explicitly on the global fermion-parity eigenvalue $t$8, giving periodic or anti-periodic quantization (Saket et al., 2013).

In the synthetic-dimension construction, the effective extended Kitaev-chain Hamiltonian is

$t$9

which reduces in the Kitaev limit $p$0 to the standard nearest-neighbor form (Ali et al., 6 May 2026). Although this model is not periodic in the synthetic coordinate, the physical orbitals are annular.

3. Topological characterization and invariants

The topological characterization depends strongly on whether translation invariance is preserved.

For the exactly solvable ring, one defines the phase $p$1 by

$p$2

The half-Brillouin-zone winding

$p$3

takes the values $p$4 for $p$5 and $p$6 for $p$7. An equivalent $p$8 index is

$p$9

which also changes only at $\Delta$0 (Saket et al., 2013). In this setting, the ring geometry does not eliminate topological distinction, but it does remove boundary Majoranas in the defect-free closed chain unless the gap closes.

For the synthetic chain, the bulk Bloch Hamiltonian is

$\Delta$1

The class-D invariant changes sign when $\Delta$2, i.e. $\Delta$3 for $\Delta$4 (Ali et al., 6 May 2026). This reproduces the familiar topological criterion in a geometry built from annular orbitals.

For the Kitaev tie, the extra bond breaks translation invariance, so one cannot simply go to momentum space and compute the usual winding number. The analysis instead proceeds by numerically diagonalizing the finite-size $\Delta$5, tracking exact zero-energy modes and their spatial structure, and computing the Majorana polarization order parameter

$\Delta$6

Integrating over half the chain,

$\Delta$7

gives $\Delta$8 for trivial states and $\Delta$9 for genuine Majorana states (Maiellaro et al., 2019). The authors also point out that one may adapt a real-space transfer-matrix approach to derive a $\mu$0 invariant by following sign changes of suitably defined Pfaffians as the knot moves. This suggests that the annular geometry forces a shift from Bloch-topological diagnostics to real-space diagnostics whenever a local bond destroys translational symmetry.

4. Majorana zero modes, defects, and topological frustration

The annular geometry is particularly useful for distinguishing between bulk topology and the existence of localized Majorana modes.

In the exactly solvable spin-chain formulation, defect-free sectors correspond to a closed uniform chain, while an $\mu$1-defect sector “cuts” the ring into $\mu$2 open chains. Each open chain of length $\mu$3 has quantized standing-wave momenta solving

$\mu$4

In the topological regime $\mu$5, one solution is a zero-energy mode whose wavefunction is pushed exponentially to the two ends of that chain. Each open chain contributes a twofold ground-state degeneracy, so an $\mu$6-defect sector has total degeneracy $\mu$7, whereas in the defect-free closed chain the boundary conditions forbid zero-modes unless $\mu$8 (Saket et al., 2013).

The Kitaev tie realizes a related mechanism through geometry rather than gauge defects. Inside each topological island in the $\mu$9–$\phi$0 plane, one finds a pair of nearly zero-energy modes whose wavefunctions are localized on the two “legs” and decay into the central ring. For parameters such as $\phi$1 and $\phi$2 or $\phi$3, the zero mode has Majorana polarization $\phi$4 pinned at the two legs, while in the trivial region the lowest mode is at finite energy and delocalized (Maiellaro et al., 2019).

The authors emphasize that this system is a minimal model of “topological frustration”: a single extra bond can switch the system from nontrivial to trivial or back, depending on where it sits (Maiellaro et al., 2019). A plausible implication is that annular connectivity is not merely a boundary-condition detail; it can act as a control parameter that redistributes effective endpoints and therefore the support of Majorana modes.

In the synthetic-dimension realization, the Majorana operators are introduced through

$\phi$5

leading to

$\phi$6

For $\phi$7, the left and right zero modes are

$\phi$8

with

$\phi$9

These modes are exponentially localized at the two ends of the synthetic chain (Ali et al., 6 May 2026).

5. Flux periodicity and parity-dependent transport on the ring

The flux-threaded annular Kitaev chain exhibits a pronounced odd-even effect tied to the discrete momentum set.

If $\Phi=N\phi$0 is even, the shift $\Phi=N\phi$1 can be absorbed by relabeling $\Phi=N\phi$2, because $\Phi=N\phi$3 is an integer mod $\Phi=N\phi$4. Hence the entire set of levels is invariant under $\Phi=N\phi$5, and the many-body spectrum and transport coefficients are $\Phi=N\phi$6-periodic. If $\Phi=N\phi$7 is odd, $\Phi=N\phi$8 is not integral, there is no exact relabeling among the discrete $\Phi=N\phi$9’s, and the only exact symmetry is $$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$0. Thus the spectrum is $$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$1-periodic for odd $$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$2 (Wang et al., 15 Jul 2025).

Transport is decomposed into direct transmission (DT), local Andreev reflection (LAR), and crossed Andreev reflection (CAR). Using retarded and advanced Green’s functions in Nambu space, the corresponding zero-bias transmission coefficients are

$$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$3

$$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$4

and

$$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$5

In a “symmetrical” configuration, where $$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$6 on an even-$$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$7 ring, a mirror symmetry of the two Majorana pair paths forces all LAR and CAR amplitudes to cancel exactly, leaving only DT. Asymmetric configurations break that cancellation, so LAR and CAR become finite and can even dominate near particular flux values (Wang et al., 15 Jul 2025).

The characteristic conductance structure is strongly parity dependent. For even $$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$8, DT shows two equal-height peaks at $$H \;=\; -\sum_{j=1}^{N} \Bigl[ t\,e^{\,i\phi}\,c_j^\dagger c_{j+1} + \Delta\,c_j\,c_{j+1} + \text{h.c.} \Bigr] \;-\;\mu\sum_{j=1}^{N}c_j^\dagger c_j,\quad c_{N+1}=c_1\,.$$9 and $N$00, symmetric about $N$01, while LAR and CAR vanish in symmetric coupling. For odd $N$02, the DT peaks at $N$03 and $N$04 become asymmetric: the $N$05 DT peak is strongly suppressed, the $N$06 peak remains robust, and LAR and CAR develop large peaks around $N$07 but remain essentially zero for $N$08; there is no LAR/CAR peak at $N$09 (Wang et al., 15 Jul 2025).

The paper connects these features directly to the BdG gap at $N$10: for even $N$11 the gap closes simultaneously at both $N$12 and $N$13, whereas for odd $N$14 only at $N$15 does the gap close. Around $N$16 the odd-$N$17 system is gapped, which strongly favors Andreev processes (Wang et al., 15 Jul 2025). This provides an annular analogue of a parity-controlled transport switch.

6. Phase diagrams, energetics, and stability

The annular geometry supports phase structures that are not present in either a simple open chain or a fully periodic ring.

For the Kitaev tie, the phase diagram in the $N$18–$N$19 plane is described as richly structured and “interstitial.” For an unperturbed open Kitaev chain with $N$20, the topological region is $N$21. In the ring limit, defined by $N$22 plus a bond between $N$23 and $N$24, the system is always trivial, with no edge Majoranas. In the legged-ring with one bond at $N$25, nontrivial islands appear only for certain $N$26 and $N$27. In numerics with $N$28 and $N$29, topological islands appear roughly for $N$30 and $N$31, but only in narrow windows of $N$32 that depend sensitively on $N$33. No simple closed-form $N$34 is given; the phase boundaries are determined numerically by locating when the lowest BdG eigenvalue crosses zero (Maiellaro et al., 2019).

The energetics of the tie are analyzed through the electronic free energy

$N$35

where $N$36 are the positive eigenvalues of $N$37. For many $N$38, the open chain has higher free energy than either the closed ring or the tie, so a physical chain will tend to form a knot. As a function of $N$39, $N$40 oscillates: some knot positions are local minima and others maxima. Near $N$41, a trivial ring with $N$42 will relax into a topological-tie configuration by lowering its free energy, exhibiting a spontaneous emergence of topological order (Maiellaro et al., 2019).

In the exactly solvable annular chain, the phase diagram is simpler. The defect-free sector is the global ground state for $N$43; its spectrum is gapped except at $N$44, where the gap closes. The resulting phase structure along the $N$45-axis consists of a trivial phase for $N$46 and a topological phase for $N$47, with protected zero modes appearing only in defect sectors that cut the ring into open chains (Saket et al., 2013).

These results collectively show that annular topology does not enforce a unique phase behavior. Instead, the combined action of flux, parity, defects, and local reconnection determines whether the system behaves as a trivial closed loop, a frustrated ring with emergent endpoints, or a transport-active superconducting interferometer.

7. Related realizations, observables, and broader significance

The annular Kitaev chain has been used as a platform for both conceptual and potentially experimental extensions.

In the exactly solvable spin-chain realization, a standard nonlocal order parameter for the topological phase is the Jordan–Wigner string

$N$48

which is nonzero precisely in the topological regime $N$49 and zero in the trivial regime $N$50 (Saket et al., 2013). This provides a ring-compatible diagnostic when local edge observables are unavailable.

The synthetic-dimension annular construction adds a nonlocal readout mechanism. The parity of the two end Majoranas is

$N$51

and when a readout resonator modulates a control parameter, the Hamiltonian acquires the dispersive form

$N$52

Measuring the shift of the cavity resonance directly measures $N$53 in a nonlocal, quantum-non-demolition fashion, without tunneling probes at the synthetic ends (Ali et al., 6 May 2026). This suggests that annular orbital structure can be combined with open synthetic boundaries to separate geometry from topological readout.

The Kitaev tie work explicitly discusses possible realizations in looped carbon nanotubes with proximity superconductivity and a single mobile impurity or molecule, with the knot position controlled by electrostatics or mechanical motion (Maiellaro et al., 2019). The flux-threaded ring study, by contrast, emphasizes how the odd-even parity of the ring acts as a switch between predominantly single-electron transport and Cooper-pair transport at selected flux values (Wang et al., 15 Jul 2025).

A common misconception is that a ring geometry necessarily eliminates topological structure because a closed loop has no physical ends. The literature does not support such a blanket statement. In a defect-free closed chain, zero modes can indeed be forbidden by boundary conditions (Saket et al., 2013), and a perfect ring limit can be trivial (Maiellaro et al., 2019). However, flux, defects, asymmetric contacts, or a single movable bond can reintroduce topological diagnostics, transport asymmetries, or effective endpoints with localized Majorana character (Maiellaro et al., 2019, Wang et al., 15 Jul 2025). The annular Kitaev chain is therefore best viewed as a family of closed-geometry superconducting models in which topology is redistributed from ordinary edge physics into parity sectors, defect sectors, interference effects, and geometric frustration.