The study of CoNb_{2}O_{6} sits at the confluence of simplicity and complexity: on one hand, the model for Ising chains—the building blocks of CoNb_{2}O_{6}—in a transverse field can be exactly solved and, thus, serves as an archetype of quantum criticality; on the other hand, the weak but nonzero interchain coupling adds geometric frustration to the stage, substantially complicating the phase diagram. Here we utilize low-temperature specific heat and thermal conductivity measurements to study the low-lying magnetic excitations in CoNb_{2}O_{6} and its spin-1 analog NiNb_{2}O_{6}. The thermal conductivity is found to be suppressed around the quantum critical point, where the specific heat is enhanced due to gapless magnetic excitations, pointing to the localized nature of the latter. These results highlight the predominant role of frustration in determining the quantum critical magnetic excitations of spin chains, which may furthermore underlie the remarkable similarities between the phenomenology of these spin-1/2 and spin-1 systems.
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