PT-symmetric lattices with spatially extended gain/loss are generically unstable

We illustrate, through a series of prototypical examples, that linear parity-time (PT) symmetric lattices with extended gain/loss profiles are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact PT-symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical computations.

💡 Research Summary

The paper investigates the spectral stability of linear parity‑time (PT) symmetric lattices when the gain–loss distribution is spatially extended. PT symmetry requires the real part of the lattice potential to be even and the imaginary part (representing gain and loss) to be odd, which in principle allows non‑Hermitian systems to possess entirely real eigenvalues. The authors, however, demonstrate that this promise collapses as soon as the imaginary component is not confined to a finite region.

Four representative models are examined. The first combines a quadratic (parabolic) real potential V_R(n)=α n² with a linear imaginary part V_I(n)=γ n. By applying a discrete Fourier transform, the Hamiltonian in k‑space becomes H(k)=2cos k + α(−∂²/∂k²) − γ ∂/∂k. The term proportional to γ acts as a non‑Hermitian drift operator, shifting every eigenvalue off the real axis for any γ≠0. Consequently, wave amplitudes either grow or decay exponentially, indicating intrinsic instability.

The second class removes the real potential altogether (V_R=0) and studies two cases of pure imaginary profiles: a constant gain–loss V_I=γ and a linear profile V_I=γ n. In k‑space these reduce to H(k)=2cos k + iγ and H(k)=2cos k − γ ∂/∂k, respectively. The constant term adds the same imaginary offset to the entire Bloch band, while the derivative term tilts the band in the complex plane. In both situations the spectrum becomes fully complex for any non‑zero γ, confirming that a globally distributed gain–loss inevitably destroys PT‑symmetric real spectra.

In contrast, the authors consider localized gain–loss configurations: (i) a single‑site delta‑function V_I(n)=γ δ_{n,0} and (ii) a compact support where V_I is non‑zero only within a finite interval |n|≤N₀. The Fourier analysis shows that these profiles introduce only high‑frequency perturbations, leaving the bulk of the Bloch band untouched. Numerical diagonalization confirms that the eigenvalues remain real, and no complex conjugate pairs emerge. Thus, PT symmetry can protect the spectrum provided the gain–loss is spatially confined.

The theoretical arguments hinge on the observation that an extended imaginary potential translates into either a constant shift or a derivative operator in momentum space, both of which act globally on the eigenfunctions and generate a linear imaginary component in the dispersion relation. This mechanism forces the eigenvalues off the real axis for any γ≠0. Conversely, a compact imaginary potential contributes only localized modifications that do not alter the global analytic structure of the dispersion, allowing the PT‑balanced condition to hold.

Extensive numerical simulations corroborate the analytical predictions. The authors construct finite‑size lattice Hamiltonians, vary γ, and track the movement of eigenvalues in the complex plane. The results consistently show immediate emergence of complex eigenvalues when the gain–loss profile is extended, and preservation of a purely real spectrum when the profile is localized.

The study carries significant implications for the design of PT‑symmetric photonic structures, electronic superlattices, and other engineered non‑Hermitian systems. It warns that simply imposing PT symmetry on a lattice is insufficient for stability; the spatial distribution of gain and loss must be carefully engineered, preferably confined to compact regions, to retain real spectra and avoid uncontrolled amplification or attenuation. This insight refines the criteria for practical PT‑symmetric device implementation and suggests new directions for achieving robust, loss‑balanced functionality in complex media.