Based on the generalized gauge transformation method, this paper investigates time-dependent non-Hermitian Hamiltonians described by the generators of the SU(1,1) and SU(2) Lie algebras, corresponding to the harmonic oscillator system under a periodically driven external field and the single-spin system with arbitrary parameters, respectively. By introducing a time-dependent Hermitian but non-unitary transformation operator Ȓ (t), the original time-dependent Hamiltonian is transformed into a time-independent effective Hamiltonian Ĥ'. Exact solutions of the dual eigenequations and the associated non-adiabatic Berry phases are derived. For the SU(2) system, an exceptional point occurs at G_c (ω) = Ω+ω/2, where the eigenvalues of the effective Hamiltonian diverge and the Berry phase becomes complex. Specifically, for G < G_c the Berry phase γ_n (T) is real, while for G > G_c it acquires an imaginary part, signaling the spontaneous breaking of pseudo-Hermitian phase. In the slowly varying external field limit (ω → 0), the non-adiabatic Berry phase reduces to the adiabatic counterpart. To more clearly reflect the variation of the Berry phase, we plot the phase diagram on the parameter plane of the driving frequency ω and the non-Hermitian coupling strength G using the magnitude of the imaginary part of the Berry phase. The result is shown in Fig.(a).
From the above results, we can clearly see that the Berry phase reflects the transition from the pseudo-Hermitian phase to the pseudo-Hermitian broken phase, and the analytical results agree well with the numerical results. In contrast, for the non-Hermitian SU(1,1) system, the eigenvalues of the effective Hamiltonian remain real for all parameters, and no exceptional point appears. As the non-Hermitian coupling G increases, the effective oscillator frequency Γ(ω) decreases monotonically without any singularity. The absence of exceptional points in the non-Hermitian SU(1,1) system is closely related to the noncompactness of the SU(1,1) group. The representation space of a non-compact group is infinite-dimensional and unbounded above, and the expectation values of the generators K±have no lower bound. This endows the system with greater robustness against non-Hermitian perturbations, preventing spectral degeneracy and the transition to complex eigenvalues at finite parameter values. In contrast, SU(2), as a compact group, has a finite-dimensional representation space and a bounded spectrum, making it more susceptible to spectral degeneracy at critical parameter values. These results provide a clear comparison of the geometric phase behavior between compact and non-compact Lie algebraic structures in non-Hermitian systems.