Topological spectral form factor reveals emergent non-Hermitian single-particle $\mathcal{PT}$ transitions from many-body quantum chaos
In equilibrium, topological defect insertions in quantum and classical partition functions provide non-perturbative probes of phase transitions beyond local observables. In non-equilibrium physics, the spectral form factor provides a minimal probe of universal quantum dynamics and is a product of two partition functions at imaginary inverse temperature. We define the topological spectral form factor (TopSFF) by inserting topological defects acting non-trivially on the doubled partition functions, producing mismatched spacetime world-sheet topologies. For the minimal $\mathbb{Z}_2$ spatially extended defect, implemented by the global swap operator, we derive an exact mapping of the TopSFF of a generic $(1+1)$D many-body chaotic system to an emergent $(3+1)$D non-Hermitian single-particle problem describing a temporal domain wall (tDW). We show analytically that the effective tDW dynamics undergoes a $\mathcal{PT}$ symmetry breaking transition at finite interaction strength $\epsilon_{\mathrm{EP}}$: below $\epsilon_{\mathrm{EP}}$, the leading modes are polarized into Gaussian or non-Gaussian tDW sectors and the TopSFF varies monotonically and exponentially with system size; above $\epsilon_{\mathrm{EP}}$, the tDW sectors hybridize and the TopSFF oscillates with system size; at the exceptional point $\epsilon_{\mathrm{EP}}$, Jordan non-diagonality produces a linear-in-system-size enhancement. For minimal temporally extended topological defects associated with global swap or time translation, we derive exact universal TopSFF free-energy scaling forms. We verify these signatures numerically in independent models.