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

TL;DR

This paper reveals a duality in kicked spin chains where the trace of the evolution operator for N spins over T steps relates to a non-unitary operator for T spins over N steps, with implications for spectral analysis.

Contribution

It introduces a novel duality property in kicked spin chains connecting unitary and non-unitary operators across different dimensions.

Findings

Duality relates N-spin, T-step evolution to T-spin, N-step evolution.

Spectrum analysis varies across chaotic and regular regimes.

Applications to spectral statistics are discussed in follow-up work.

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.