Non-Hermitian 1D Kitaev Systems

• 1D non-Hermitian Kitaev systems are quantum chains for spinless fermions incorporating imbalanced pairings that generalize the Hermitian Kitaev model.
• They exhibit rich phase diagrams with quantized extended Zak phases and exceptional points marking transitions between topological and gapless states.
• Robust Majorana edge modes emerge in the unbroken symmetry phase, confirming the preserved bulk–edge correspondence in these systems.

A one-dimensional (1D) non-Hermitian Kitaev system is a quantum chain model for spinless fermions that incorporates non-Hermitian terms—such as imbalanced pair creation and annihilation amplitudes—thus generalizing the Hermitian Kitaev chain to non-Hermitian regimes. These systems serve as a rich platform to paper the interplay between topology, symmetry, and non-Hermiticity, manifesting unique bulk-edge correspondence, exceptional points, extended topological invariants, and robust Majorana edge modes within unbroken symmetry domains.

1. Fundamental Model and Phase Diagram

A 1D non-Hermitian Kitaev chain is defined by the Hamiltonian

$$\mathcal{H} = -t \sum_{j=1}^N (c_j^\dagger c_{j+1} + \text{h.c.}) - \mu \sum_{j=1}^N (1-2n_j) - \sum_{j=1}^N (\Delta_a c_j^\dagger c_{j+1}^\dagger + \Delta_b c_{j+1} c_j)$$

where $t$ is the hopping amplitude, $\mu$ is the chemical potential, $\Delta_a$ and $\Delta_b$ are real, generally unequal (non-Hermitian) pairing amplitudes for pair creation and annihilation, giving rise to non-Hermiticity when $\Delta_a \neq \Delta_b$.

After Fourier transformation and diagonalization, the Bogoliubov-de Gennes spectrum is

$$\epsilon_k = 2\sqrt{(\mu - t\cos k)^2 + (\Delta_a\Delta_b)\sin^2 k}$$

The non-Hermiticity induces a phase ($\Delta_a\Delta_b < 0$ for some $k$) where the spectrum becomes complex and time-reversal symmetry is spontaneously broken.

The topological phase diagram features:

Regime	Phase Type	Definition
$\Delta_{a}/t, \Delta_{b}/t > 0$ & Topological	$|\mu/t|<1$, $\Delta_a\Delta_b>0$
$\Delta_{a}/t, \Delta_{b}/t < 0$ & Topological	$|\mu/t|<1$, $\Delta_a\Delta_b>0$
$|\mu/t| > 1,\ \mu^2 + \Delta_a\Delta_b > t^2$	Trivial	Gapped, extended Zak phase $\mathcal{Z}_\pm=0$
$\Delta_a\Delta_b < 0$ or $|\mu|, t, \Delta$ otherwise	Coalescing (gapless)	Complex spectrum, at exceptional points (EPs)

At EPs, eigenstates coalesce and a phase transition from gapped/topological to gapless occurs via gap closing.

2. Topological Invariants in the Non-Hermitian Regime

The haLLMark integer-valued topological invariants in Hermitian 1D systems (Berry/Zak phase, winding number) acquire essential modifications due to biorthogonal quantum mechanics. In the non-Hermitian Kitaev chain, the topological phase is characterized by the extended Zak phase: $$\mathcal{Z}_\pm = \int_0^{2\pi} \mathcal{A}_k dk = \int_0^{2\pi} \langle \eta_\pm^k | \partial_k | \psi_\pm^k \rangle dk$$ where $|\psi_\pm^k\rangle$ and $|\eta_\pm^k\rangle$ are the right and left eigenvectors, forming a biorthonormal set ($$\langle \eta_\lambda^k | \psi_{\lambda'}^k \rangle = \delta_{\lambda\lambda'}$$).

The extended Zak phase is quantized: $$\mathcal{Z}_\pm = \begin{cases} -\pi\ \mathrm{sgn}[(\Delta_a + \Delta_b)/t], & |\mu|<1 \ 0, & |\mu|>1 \end{cases}$$ This invariant is robust within the time-reversal symmetric, gapped region and signals the presence of edge modes (nonzero value) even in the non-Hermitian context.

3. Symmetry, Exceptional Points, and Ground State Coalescence

Despite non-Hermiticity, the Hamiltonian can exhibit time-reversal symmetry (TRS) defined by an anti-linear operator $\mathcal{T}$ with $\mathcal{T}i\mathcal{T}^{-1} = -i$ and $[\mathcal{T}, \mathcal{H}]=0$. In the unbroken region, $\mathcal{T}|G\rangle = |G\rangle$ for the ground state $|G\rangle$. When parameters cross into regimes where $(\mu-t\cos k)^2 + \Delta_a\Delta_b\sin^2 k < 0$ for some $k$, the system hits EPs resulting in complex energies and the spontaneous breaking of TRS in eigenstates, i.e., $\mathcal{T}|G\rangle \neq |G\rangle$.

At EPs, eigenstates coalesce and ground-state degeneracy arises, indicating a transition between topologically nontrivial and coalescing (gapless) phases.

4. Majorana Edge Modes and Bulk–Edge Correspondence

In the open-boundary non-Hermitian Kitaev chain and within the unbroken phase, exact analysis in the Majorana representation reveals the existence of zero-energy edge modes. Majorana operators are defined as: $$a_j = c_j^\dagger + c_j,\qquad b_j = -i(c_j^\dagger - c_j)$$ with standard anticommutation relations. At special parameter points (e.g., $\sqrt{\Delta_a\Delta_b}=t$), the Hamiltonian can be mapped onto two coupled non-Hermitian SSH models, showing clear bulk–edge correspondence.

Zero modes are obtained as combinations: $$f_+ \propto \sum_j \mu^{j-1} a_j,\qquad f_- \propto \sum_j \mu^{N-j} b_j$$ The physical fermionic zero mode $f_N = (f_+ - i f_-)/2$ commutes with $H$, realizing a robust Majorana zero mode at the edge when $|\mu|<1$.

The presence of these edge modes corresponds to the nontrivial extended Zak phase, confirming that the bulk–edge correspondence persists in the non-Hermitian generalization. Majorana zero modes are robust to moderate pair imbalance (i.e., non-Hermiticity) as long as the spectrum remains real.

5. Generalization: Non-Hermitian Topological Band Theory

Non-Hermitian extensions require the generalization of Bloch band theory and topological invariants. The relevant procedure (Yokomizo et al., 2019):

1. Replace the conventional $e^{ik}$ by a complex parameter $\beta$ in the generalized Bloch Hamiltonian;
2. Solve $\det[\mathcal{H}(\beta)-E]=0$, yielding roots $\beta_1$, $\beta_2$;
3. The "generalized Brillouin zone" $C_\beta$ is defined by $|\beta_1|=|\beta_2|$ (as opposed to $|\beta|=1$ in Hermitian case);
4. The topological invariant (e.g., winding number $w$) is computed over $C_\beta$: $w = \frac{1}{2\pi} \oint_{C_\beta} d\arg[q(\beta)]$ with $q(\beta)$ related to the off-diagonal component of the Nambu-space Hamiltonian.

This construction restores bulk–edge correspondence even under the non-Hermitian skin effect, which leads to the exponential accumulation of bulk eigenstates at one edge (for asymmetric hopping or gain/loss) (Ezawa, 2018).

6. Related Models, Physical Realizations, and Extensions

The non-Hermitian Kitaev chain is closely connected to:

• Non-Hermitian SSH models, where non-reciprocal hopping induces the skin effect and modifies the bulk-boundary correspondence, and the topological invariants are generalized to non-Bloch winding numbers (Ezawa, 2018).
• Realizations in cold atom systems and electric circuits, where non-reciprocal or dissipative elements translate to non-Hermitian terms in quantum chain simulators or Laplacian networks, enabling impedance-based detection of topological edge states.
• Bosonic analogues, wherein Hermitian bosonic Kitaev-Majorana chains exhibit effective non-Hermitian dynamics at the quadratic level due to pairing/squeezing, leading to chiral transport, directional amplification, and the skin effect even absent explicit loss (McDonald et al., 2018, Busnaina et al., 2023, Fortin et al., 12 Dec 2024, Bomantara et al., 21 May 2025).

The robust extended Zak phase and corresponding edge modes survive in a variety of these platforms, although disorder and interactions can significantly affect phase boundaries and the multiplicity of edge modes (e.g., fourfold degeneracy in some interacting non-Hermitian Kitaev–Hubbard chains (Sayyad et al., 2023)).

7. Mathematical and Theoretical Frameworks

A summary of key mathematical structures:

Concept	Formula/Description
Hamiltonian	$$\displaystyle -t \sum_j (c_j^\dagger c_{j+1} + \text{h.c.}) - \mu \sum_j (1-2n_j) - \sum_j (\Delta_a c_j^\dagger c_{j+1}^\dagger + \Delta_b c_{j+1} c_j)$$
Spectrum	$$\displaystyle \epsilon_k = 2\sqrt{(\mu - t\cos k)^2 + (\Delta_a\Delta_b)\sin^2 k}$$
Extended Zak phase	$$\displaystyle \mathcal{Z}_\pm = \int_0^{2\pi} \langle \eta_\pm^k | \partial_k | \psi_\pm^k \rangle dk$$
Generalized Brillouin zone	$|\beta_1|=|\beta_2|$ (non-Bloch condition)
Winding number	$$\displaystyle w = \frac{1}{2\pi} \oint_{C_\beta} d\arg[q(\beta)]$$
Majorana decomposition	$a_j = c_j^\dagger + c_j$, $b_j = -i(c_j^\dagger - c_j)$

This framework provides a comprehensive and predictive structure for understanding 1D non-Hermitian Kitaev systems: analytic solutions, numerical stability, and robustness of topology all stem from mapping to biorthogonal formalism and non-Bloch band theory.

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Non-Hermitian Kitaev systems thus represent a paradigmatic platform at the intersection of symmetry-breaking, topology, and nonunitary quantum mechanics. Their analytic tractability and experimental relevance (via engineered dissipation, non-reciprocal hopping, and parametric driving) position them as a key testbed for next-generation studies of non-Hermitian quantum materials and devices (Li et al., 2017, Gong et al., 2018, Ezawa, 2018, McDonald et al., 2018, Yokomizo et al., 2019, Sayyad et al., 2023, Shi et al., 2023, Busnaina et al., 2023, Fortin et al., 12 Dec 2024, Ardonne et al., 22 Nov 2024, Bomantara et al., 21 May 2025).