Space–Time Crystals: Reciprocity Classes, Forced Doublets and Defects
- 1Independent Reseacer
Abstract
A sheet modulated in space and in time is a space–time crystal. Among its symmetries is the glide time reversal: a reflection of time combined with a shift by half a lattice vector. It acts on the response by transposition. This article states what it enforces in 2+1 dimensions. Every space–time group falls into one of four reciprocity classes. Plain reciprocity comes with a time mirror, glide reciprocity with a glide time reversal. A primed point operation gives a mapped relation instead, in which the sign of the magnetic dipole is measurable. On a sheet a threefold or sixfold axis rules out glide reciprocity and every forced Kramers-type doublet, and a fourfold axis allows the glide only in fixed half-vector classes. In a bulk crystal a threefold axis still admits the class along the axis. In the band structure the glide forces diabolic doublets, with loss and gain. At a momentum-gap edge it doubles the exceptional point. A horizontal cone of second-order media is protected by a symplectic square root of the period map. The energy–momentum zone becomes a Klein bottle under a projective symmetry; the glide time reversal alone does not make it one. In scattering the glide forbids the Littrow orders with phase −1 and, without real loss, the partner channel of a time slab. It gives twin pair-creation channels, a frequency-weighted Kirchhoff law and an effective medium without drag. Around a pump vortex of odd charge the parametric response leaves the half-integer sector. A sector rule fixes which defects lock, and above threshold walls turn at a speed that falls as N^(−2) and grows with ℓ. Möbius closure forces a defect at the seam, a temporal one for a standing wave, and edges must run along the glide. The results hold for linear sheets with a real, causal kernel. They are measured on lattice and sheet models, not on a fabricated metasurface.
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BibTeX
@misc{mark2026spacetime,
title = {{Space–Time Crystals: Reciprocity Classes, Forced Doublets and Defects}},
author = {László Márk},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23136229},
url = {https://doi.org/10.5281/zenodo.23136229},
note = {CC BY},
}
RIS
TY - UNPB
TI - Space–Time Crystals: Reciprocity Classes, Forced Doublets and Defects
AU - László Márk
PY - 2026
DA - 2026/10/04
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23136229
UR - https://doi.org/10.5281/zenodo.23136229
AB - A sheet modulated in space and in time is a space–time crystal. Among its symmetries is the glide time reversal: a reflection of time combined with a shift by half a lattice vector. It acts on the response by transposition. This article states what it enforces in 2+1 dimensions. Every space–time group falls into one of four reciprocity classes. Plain reciprocity comes with a time mirror, glide reciprocity with a glide time reversal. A primed point operation gives a mapped relation instead, in which the sign of the magnetic dipole is measurable. On a sheet a threefold or sixfold axis rules out glide reciprocity and every forced Kramers-type doublet, and a fourfold axis allows the glide only in fixed half-vector classes. In a bulk crystal a threefold axis still admits the class along the axis. In the band structure the glide forces diabolic doublets, with loss and gain. At a momentum-gap edge it doubles the exceptional point. A horizontal cone of second-order media is protected by a symplectic square root of the period map. The energy–momentum zone becomes a Klein bottle under a projective symmetry; the glide time reversal alone does not make it one. In scattering the glide forbids the Littrow orders with phase −1 and, without real loss, the partner channel of a time slab. It gives twin pair-creation channels, a frequency-weighted Kirchhoff law and an effective medium without drag. Around a pump vortex of odd charge the parametric response leaves the half-integer sector. A sector rule fixes which defects lock, and above threshold walls turn at a speed that falls as N^(−2) and grows with ℓ. Möbius closure forces a defect at the seam, a temporal one for a standing wave, and edges must run along the glide. The results hold for linear sheets with a real, causal kernel. They are measured on lattice and sheet models, not on a fabricated metasurface.
ER -