Wei-Norman approach for non-Hermitian driven spin-$S$ systems and its application to defect freezing

In the theoretical study of nonequilibrium non-Hermitian systems, obtaining exact analytical solutions for their nonadiabatic dynamics is highly desirable yet often challenging.

In this work, we identify a class of non-Hermitian quantum systems where this difficulty can be substantially reduced.

Employing the Wei-Norman approach, we show that for a spin-$S$ subject to a general non-Hermitian time-dependent drive, the matrix elements of the evolution operator can be expressed in closed analytical forms (via Jacobi polynomials) in terms of the corresponding spin-$1/2$ model.

This approach is straightforward and accessible to nonspecialists in Lie algebra.

As an application, we investigate a specific nonequilibrium non-Hermitian phenomenon known as defect freezing, i.e., the existence of excitations in the adiabatic limit, in spin-$S$ extensions of the $\mathcal{PT}$-symmetric Su-Schrieffer-Heeger model under linear quenches.

We derive exact analytical expressions for the momentum-resolved excitation probabilities and the total excitation densities.

Our results reveal that defect freezing occurs exclusively in momentum sectors that traverse the $\mathcal{PT}$-symmetry-broken region -- and thus pass through a pair of higher-order exceptional points (EPs) -- during the quench; notably, the excitation density exhibits a singularity at a critical value of the non-Hermiticity parameter.

This work enriches the analytical toolkit for nonadiabatic dynamics in multi-level non-Hermitian systems and provides quantitative, testable predictions for defect freezing across higher-order EPs, possibly accessible on platforms such as electric circuit networks and photonic lattices.