  We present results of numerical simulations performed on one-dimensional spin
chains in order to extract the so-called relaxation rate $1/T_1$ accessible
through NMR experiments. Building on numerical tensor network methods using the
Matrix Product States (MPS) formalism, we can follow the non-trivial crossover
occurring in critical chains between the high-temperature diffusive classical
regime and the low-temperature response described by the Tomonaga-Luttinger
liquid (TLL) theory, for which analytical expressions are known. In order to
compare analytics and numerics, we focus on a generic spin-$1/2$ XXZ chain
which is a paradigm of gapless TLL, as well as a more realistic spin-$1$
anisotropic chain, modelling the DTN material, which can be either in a trivial
gapped phase or in a TLL regime induced by an external magnetic field. Thus, by
monitoring the finite temperature crossover, we provide quantitative limits on
the range of validity of TLL theory, that will be useful when interpreting
experiments on quasi one-dimensional materials.
