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  • Letter
  • Open Access

Space-time Floquet operator: Nonreciprocity and fractional topology of space-time crystals

Abhijeet Melkani1,2,* and Jayson Paulose1,†

  • *Contact author: abhijeet.melkani@ens-lyon.fr
  • †Contact author: jpaulose@uoregon.edu
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Phys. Rev. Research 8, L042009 – Published 23 October, 2026

DOI: https://doi.org/10.1103/xcls-6nzd

Abstract

We introduce a space-time Floquet operator for crystals whose spatial and temporal translations are intertwined. Unlike the conventional Floquet operator, which describes evolution over a full time period, the space-time Floquet operator evolves the system over a fraction of the period paired with a spatial shift that generates the primitive space-time unit cell. Its eigenmode spectrum defines a space-time band structure that unfolds conventional Floquet bands to reveal the consequences of the intertwined crystal symmetry in reciprocal frequency–wavevector space. We relate the topology of these space-time bands to quantized transport phenomena, such as Bloch oscillations and adiabatic charge transport, and uncover an intrinsically fractional charge pump enabled by the primitive space-time cycle. Numerical simulations quantitatively support these predictions with small deviations controlled by the speed of the modulation. We also demonstrate how nonreciprocal parametric resonances are naturally anticipated and accurately predicted by our framework. Our approach applies broadly to both classical and quantum systems with space-time symmetry, including non-Hermitian crystals.

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Supplemental Material

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Phys. Rev. Research 8, L042009– Published 23 October, 2026
  • Received 5 March 2026
  • Accepted 2 October 2026
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