Low Temperature Domain Wall Freezing and Non-Equilibrium Dynamics in the Transverse-Field Ising Model Material CoNb$_2$O$_6$

CoNb$_2$O$_6$ is a rare realization of the transverse field Ising model (TFIM), making it a useful tool for studying both equilibrium and non-equilibrium many-body quantum physics. Despite a large body of work dedicated to characterizing this material, details of the ordered states in the presence of relatively weak transverse fields have not been discussed in detail. Here, we present a detailed study of CoNb$_2$O$_6$ via ac susceptibility measurements in order to further characterize its low temperature behavior in the presence of a transverse field. Specifically, we call attention to an unconventional freezing transition in zero-field below T$_F$ = 1.2~K, occurring \emph{within} the well-known commensurate antiferromagnetic (CAFM) state that onsets at $T_{N2}$ = 1.9~K. We performed a series of transverse-field quenches into this frozen state, which resulted in a slowly relaxing susceptibility, $\chi^{\prime}(t)$, that followed a logarithmic decay within the time range measured. We discuss the frozen state in the context of the freezing of previously discussed "free" chains arising from domain walls between the four degenerate sublattices of the CAFM state. We also attempted to observe Kibble-Zurek scaling by quenching the transverse field into the frozen state at different rates. This produced a null result; the behavior can be fully explained by coarsening of domains over the timescale of the quenches. The absence of a clear Kibble-Zurek scaling is itself surprising, given the proposed ubiquity of the phenomenon for general second order phase transitions.