PT-Symmetric Non-Hermitian AAH Model

• The PT-symmetric non-Hermitian AAH model is a framework for exploring non-Hermitian quantum lattices with balanced gain/loss, periodic driving, and spin-orbit coupling.
• It employs static and Floquet Hamiltonians, introducing complex perturbations and robust spectral transitions that reveal critical PT-breaking thresholds and localization effects.
• The model uncovers the interplay between topological invariants, bulk-boundary correspondence, and edge mode protection, advancing our understanding of non-Hermitian quantum phenomena.

The PT-symmetric non-Hermitian Aubry-André-Harper (AAH) model represents a central paradigm for the interplay of non-Hermiticity, parity-time symmetry (PT), and topological phases in quantum lattices. Its variants with static and driven non-Hermitian perturbations, as well as those incorporating spin-orbit coupling, have yielded a comprehensive and technically rich landscape of real-complex spectral transitions, topological invariants, and robust edge states. The model is of paramount relevance to the study of non-Hermitian topological matter, localization phenomena, and the extension of bulk-boundary correspondence beyond Hermitian settings.

1. Model Hamiltonians and Symmetry Structures

The non-Hermitian PT-symmetric AAH system manifests in two primary forms: static (time-independent), and periodically driven (Floquet), with further complexity introduced via spin-orbit coupling.

• Static off-diagonal AAH variant:

$$H = -t \sum_{n=1}^{N-1}[\,1+\lambda\cos(2\pi\beta n+\Phi)\,]\,a_n^\dagger a_{n+1} + \text{h.c.} + i\gamma(a^\dagger_j a_j - a^\dagger_{N-j+1}a_{N-j+1}),$$

incorporating two balanced gain/loss impurities at parity-symmetric sites $j$ and $N-j+1$ and a spatially modulated hopping reminiscent of the AAH (or Harper) model (Yuce, 2015).

• Driven (Floquet) PT-symmetric generalization:

$$H(t) = H_0 + V(t), \qquad V(t) = i\,\gamma\,\cos(\omega t)\,[a_{m_0}^\dagger a_{m_0} - a_{\bar m_0}^\dagger a_{\bar m_0}],$$

where the non-Hermitian terms are periodically modulated, introducing fundamentally new symmetry and stability properties (Blose, 2019).

• Inclusion of Rashba Spin-Orbit Coupling:

$H = H_A + H_R,$

where $H_A$ is a PT-symmetric non-Hermitian AAH Hamiltonian (on-site complex modulation), and $H_R$ encodes Rashba spin-orbit hopping with coefficients $\alpha_{y,z}$ and Pauli matrices acting in spin space (Acharya et al., 2021).

In all cases, PT symmetry is achieved via parity ($\mathcal{P}$: site inversion) and time-reversal ($\mathcal{T}$: complex conjugation) operators, acting as $\mathcal{P}a_n\mathcal{P}^{-1} = a_{N-n+1}$ and $\mathcal{T}i\mathcal{T}^{-1} = -i$.

2. Spectral Phases: PT Symmetry Breaking and Criticality

A hallmark feature of PT-symmetric non-Hermitian lattices is the existence of a finite threshold for the non-Hermitian parameter (e.g., $\gamma$ or $h$) below which all energies (or quasienergies) remain real (“unbroken PT phase”), and above which eigenvalues become complex (“broken PT phase”). In the static AAH case, this threshold $\gamma_c$ depends on system commensurability, impurity position, and chain length; for irrational $\beta$ (quasi-periodic case), the spectrum becomes complex as soon as $\gamma \ne 0$ (Yuce, 2015). For models with time-periodic driving, the critical threshold $\gamma_c$ persists for all defect positions as long as the drive frequency $\omega > 0$, and saturates at high frequencies—indicative of an effective Hermitianization and stabilization of real spectra (Blose, 2019).

When Rashba spin-orbit is present, the PT-breaking threshold $h_c$ (for the gain-loss parameter $h$ in the complex potential) can be computed analytically when only one Rashba channel is nonzero: $h_c = \ln \frac{2\alpha_{y \text{\,or}\,z}}{V}$; otherwise, it is found numerically, $h_c(\alpha_y, \alpha_z, t, V)$ (Acharya et al., 2021).

3. Topological Invariants and Bulk-Edge Correspondence

In the Hermitian limit, for rational $\beta = 1/2$, the AAH model reduces to the Su-Schrieffer-Heeger (SSH) chain, yielding a $\mathbb{Z}_2$ topological invariant: $$\nu = \begin{cases} 1 & |\Delta_+| > |\Delta_-| \ 0 & |\Delta_+| < |\Delta_-| \end{cases}, \quad \Delta_\pm = -2it(1 \mp \cos\Phi)$$ (Yuce, 2015). This topological classification persists into the PT-unbroken non-Hermitian regime as long as the bulk spectrum remains real and no gap closes off the real axis.

In the Floquet-driven case, the bulk-edge correspondence extends naturally to quasienergy spectra. The driven system exhibits a particle-hole (Majorana) symmetry, with a symmetric operator $\mathcal{C}$ enforcing $\mathcal{C} H_F \mathcal{C}^{-1} = -H_F^*$. Quasienergies thus occur in $\pm\epsilon^*$ pairs, allowing for robust Majorana zero or $\pi$-modes localized at chain ends (Blose, 2019). The topological invariant can again be computed through the sign of the Pfaffian or the winding of the effective Bloch vector.

With non-Hermitian on-site modulation and Rashba coupling, a flux-winding number invariant is constructed as: $$w(h) = \lim_{L\to\infty}\frac{1}{2\pi i}\int_0^{2\pi} d\theta\, \partial_\theta \ln\det[H(\theta, h) - E_b],$$ with an analytic topological transition at $$h_t = \ln \frac{2\sqrt{t^2 + (\alpha_y + \alpha_z)^2}}{V}$$ (Acharya et al., 2021).

4. Mechanisms and Relations Among Transitions: PT Breaking, Localization, and Topology

A central insight from both static and driven cases is the nuanced relationship between PT-symmetry breaking, localization transitions, and topological transitions.

• In the simple PT-symmetric AAH model without spin-orbit, the localization (metal-insulator) and PT-breaking transitions coincide at a critical parameter (Acharya et al., 2021). The topological transition, however, can occur at a higher threshold, particularly when both Rashba amplitudes are present, giving rise to a regime where states are localized and PT symmetry is broken, yet the winding number/topological index has not yet changed.
• For a single Rashba channel, $h_c = h_t$, and all transitions coincide. For two nonzero amplitudes, one invariably observes $h_c = h_c^{\text{(loc)}} < h_t$ (Acharya et al., 2021).
• In the presence of periodic driving, the driven model generically supports a real quasienergy spectrum and robust end-localized Floquet Majorana modes for all parameter regimes with $\gamma < \gamma_c(\omega, \cdots)$, newly enabled compared to the static model (Blose, 2019).

5. Analytical Methods and Floquet Framework

Analysis of the driven PT-symmetric non-Hermitian AAH model adopts the Floquet–Sambe formalism, requiring diagonalization of an infinite-dimensional matrix in the joint site and photon (Fourier) space: $$\mathcal{H}_{n,m}^{q,r} = - J[1+\lambda\cos(2\pi\beta n+\Phi)]\delta_{q,r}(\delta_{n,m+1} - \delta_{n,m-1}) - i q\omega \delta_{n,m}\delta_{q,r} + i(\gamma/2)\delta_{n,m}(\delta_{n,m_0} - \delta_{n,\bar{m}_0})(\delta_{q,r+1}+\delta_{q,r-1})$$ (Blose, 2019). Numerically, this is truncated to a finite number of photon sectors and diagonalized.

In the high-frequency limit, the periodically driven chain is well described by an effective time-averaged Hermitian Hamiltonian, with non-Hermitian terms vanishing to leading order—the mechanism by which the drive restores a real spectrum in parameter regimes where the static model's spectrum is inherently complex.

6. Edge Modes, Robustness, and Parameter Dependencies

The non-Hermitian PT-symmetric AAH family supports robust zero-energy (or zero-quasienergy) edge states, exponentially localized at chain boundaries, whenever the system is in the topologically nontrivial and unbroken-PT regime. Their localization length is

$\xi \approx [\ln(|\Delta_+/\Delta_-|)]^{-1}$

and the edge mode survives as long as the gain/loss impurities do not directly couple the end sites or exceed the critical threshold (Yuce, 2015, Blose, 2019). In periodically driven setups, the presence of protected Floquet Majorana modes is ensured by the intact antiunitary and particle-hole symmetries. Addition of local perturbations, such as small next-nearest-neighbor hopping, does not delocalize these modes until a comparable scale is reached.

7. Summary Table of Transitions and Regimes

Below, the distinct transitions and their interrelations in the PT-symmetric non-Hermitian AAH model are summarized (notations as above):

Regime / Parameter Range	PT-breaking $h_c$	Localization $h_c^{(\text{loc})}$	Topological $h_t$
Static, single Rashba channel	$=\ln\frac{2\alpha_{y/z}}{V}$	Same	Same
Static, both Rashba nonzero	Numerical, $<h_t$	Same	$h_t=\ln\frac{2\sqrt{t^2+(\alpha_y+\alpha_z)^2}}{V}$
Floquet (driven, any position)	$\gamma_c(\omega, \cdots)$	Not applicable	Real spectrum persists below $\gamma_c$

The spatial arrangement of gain/loss sites, commensurability parameter $\beta$, and drive frequency $\omega$ crucially control spectral and topological properties. For rational $\beta$, periodicities emerge in the critical threshold, whereas for irrational $\beta$, PT symmetry is generically broken for static non-Hermitian terms.

• "Floquet topological phase in a generalized PT-symmetric lattice" (Blose, 2019)
• "Localization, $\mathcal{PT}$-Symmetry Breaking and Topological Transitions in non-Hermitian Quasicrystals" (Acharya et al., 2021)
• "Topological phase in a non-Hermitian PT symmetric system" (Yuce, 2015)