Topological spectral form factor reveals emergent non-Hermitian single-particle $\mathcal{PT}$ transitions from many-body quantum chaos

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Original abstract

In equilibrium physics, 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 admits a representation as 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 1D 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 a finite interaction strength $ε_{\mathrm{EP}}$: below $ε_{\mathrm{EP}}$, the leading modes are polarized into Gaussian or non-Gaussian tDW sectors and the TopSFF varies monotonically and exponentially with system size; above $ε_{\mathrm{EP}}$, the tDW sectors hybridize and the TopSFF oscillates with system size; at the exceptional point $ε_{\mathrm{EP}}$, Jordan non-diagonality produces a linear-in-system-size enhancement. For temporally extended topological defects, we derive exact universal scaling forms for the TopSFF free energy in systems with time reversal or time translation symmetry, and verify them numerically in independent models.

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