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We construct a microscopic spin-exchange Hamiltonian for the quasi-1D Isingmagnet CoNb$_2$O$_6$ that captures detailed and hitherto-unexplained aspects ofits dynamic spin structure factor. We perform a symmetry analysis that recallsthat an individual Ising chain in this material is buckled, with two sites ineach unit cell related by a glide symmetry. Combining this with numericalsimulations benchmarked against neutron scattering experiments, we argue thatthe single-chain Hamiltonian contains a staggered spin-exchange term. Wefurther argue that the transverse-field-tuned quantum critical point inCoNb$_2$O$_6$ corresponds to breaking this glide symmetry, rather than anon-site Ising symmetry as previously believed. This gives a unified microscopicexplanation of the dispersion of confined states in the ordered phase and`quasiparticle breakdown' in the polarized phase at high transverse field.

Glide symmetry breaking and Ising criticality in the quasi-1D magnet CoNb$_2$O$_6$