Particle-Time Duality in the Kicked Ising Chain I: The Dual Operator

Abstract: We demonstrate that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for $N$ spins at time $T$ is related to one of a non-unitary evolution operator for $T$ spins at time $N$. We investigate the spectrum of this dual operator with a focus on the different parameter regimes (chaotic, regular) of the spin chain. We present applications of this duality relation to spectral statistics in an accompanying paper.