Observation of a gapless linear dispersion at quantum criticality in the Ising chain ferromagnet CoNb$_2$O$_6$ in transverse field*

The Ising chain in transverse field is one of the canonical paradigms for a continuous field-driven quantum phase transition between spontaneous magnetic order and a quantum paramagnet. The mechanism driving this phase transition has long been predicted to involve the closing of the spin gap, or minimum excitation energy, at the quantum critical point, where a characteristic linear dispersion is expected at low energies. We report single-crystal neutron diffraction and inelastic neutron scattering measurements that unveil how the magnetic order and excitations evolve in the very close proximity of the quantum critical point in the quasi-1D Ising chain ferromagnet CoNb$_2$O$_6$. Near criticality, we observe an essentially gapless spectrum with an almost perfectly-linear dispersion along the chain direction. Below the critical field, the frustrated interchain couplings stabilize 3D incommensurate spin-density-wave order, as observed through diffraction measurements. To our knowledge, this is the first time that essentially-gapless, linearly dispersive excitations have been observed in the very close proximity of a transverse field-tuned quantum critical point.