Dispersion relations near quantum criticality in the quasi one-dimensional Ising chain CoNb$_2$O$_6$ in transverse magnetic field

The Ising chain in a transverse magnetic field is one of the canonical examples of a quantum phase transition. We have recently realized this model experimentally in the quasi-one-dimensional (1D) Ising-like ferromagnet CoNb$_2$O$_6$ [1]. Here, we present single-crystal inelastic neutron scattering measurements of the magnetic dispersion relations in the full three-dimensional (3D) Brillouin zone for magnetic fields near the critical point and in the high field paramagnetic phase. We explore the gap dependence as a function of field and quantify the cross-over to 3D physics at the lowest energies due to the finite interchain couplings. We parametrize the dispersion relations in the high-field paramagnetic phase to a spin wave model to quantify the sub-leading terms in the spin Hamiltonian beyond the dominant 1D Ising exchange. [1] R. Coldea, D.A. Tennant, E.M. Wheeler et al, Science 327 177-180 (2010).