---
title: Non-Hermitian 1D Kitaev Systems
url: https://www.emergentmind.com/topics/1d-non-hermitian-kitaev-systems
type: topic
---

# Non-Hermitian 1D Kitaev 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 study 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 [1902.10958]:

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) [1810.04527].

## 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 [1810.04527].
- 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 [1805.12557, 2309.06178, 2412.09744, 2505.15908].

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 [2302.13561]).

## 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 [1707.04718, 1802.07964, 1810.04527, 1805.12557, 1902.10958, 2302.13561, 2305.00496, 2309.06178, 2412.09744, 2411.14776, 2505.15908].

Source: https://www.emergentmind.com/topics/1d-non-hermitian-kitaev-systems