Emergent $\PT$ Symmetry and Exceptional Points in a Driven Dirac Impurity

Abstract

Periodic driving can generate passive non-Hermitian impurity dynamics without microscopic gain. We derive this mechanism for an inversion-asymmetric spin--orbit-coupled Dirac impurity: off-shell angular harmonics produce a real spin-odd shift that the retarded bath converts into relative spin-dependent decay. After common damping is removed, the kernel contains the parity--time ($\mathcal{PT}$) core $\Delta_{\rm eff}\sigma_x+i\Gamma_{\rm PT}\sigma_z$ and its causal Kramers--Kronig detuning. Full finite-band frequency dependence turns the constant-core benchmark into an avoided coalescence. In contrast, a stationary rotating drive, evaluated with the full momentum integral and both helicity cuts, supports a family of nonlinear causal pole exceptional points (EPs), certified by a double-zero condition, local winding, and pole exchange. Zero-temperature complete-basis calculations on four $z$-shifted $N=10$ and $12$ matrix block-Wilson chains continue a representative EP to $U=0.0025$. On a common contour both chains have the same winding and pole exchange, while a Rouch'e ratio $0.678<1$ preserves the enclosed zero count. This is a controlled finite-chain result, not a thermodynamic-limit numerical renormalization group or strong-coupling theorem. Divergent biorthogonal projectors cancel in complete propagators and in the particle--hole-symmetric finite-$U$ charge resolvent, precluding universal screening enhancement. On an equal-velocity branchwise-linearized submanifold, the exact finite-$U$ Anderson contact matrix is rational in a dressed rapidity and $GL(2,\mathbb{C})$ covariant; its Yang--Baxter structure survives the non-unitary similarity as a unipotent EP boundary twist. This does not extend to the full curved, frequency-dependent driven kernel.