Fractional topological invariants via ( q , τ )-Gamma deformation in the non-Hermitian SSH model

We introduce a (q, τ)-deformation framework for analyzing topological invariants in non-Hermitian systems, with a focus on the Su–Schrieffer–Heeger model exhibiting asymmetric hopping. By incorporating the generalized (q, τ)-Gamma function as a spectral weighting mechanism, we define a deformed winding number ν(q,τ) that captures nonlocal memory and energy-selective effects. This new invariant retains the classical quantized winding number in the Hermitian limit but becomes fractional and continuous under non-Hermitian deformation, particularly near exceptional points or in the presence of broken reciprocity. Numerical analysis reveals that ν(q,τ) can interpolate smoothly between integer topological phases, providing a tunable geometric response sensitive to the spectral structure of the system. Our results demonstrate that (q, τ)-deformations offer a powerful tool for probing non-Hermitian topology, skin effects, and generalized bulk-boundary correspondence. This approach lays the groundwork for studying fractional topological states in spectrally complex quantum systems.