  There has been considerable recent progress in identifying candidate
materials for the transverse-field Ising chain (TFIC), a paradigmatic model for
quantum criticality. Here, we study the local spin dynamical structure factor
of different spin components in the quantum disordered region of the TFIC. We
show that the low-frequency local dynamics of the spins in the Ising- and
transverse-field directions have strikingly distinctive temperature
dependencies. This leads to the thermal-activation gap for the secular term of
the NMR $1/T_2^{\prime}$ relaxation rate to be half of that for the $1/T_1$
relaxation rate. Our findings reveal a new surprise in the nonzero-temperature
dynamics of the venerable TFIC model and uncover a means to evince the material
realization of the TFIC universality.
