We study the correlation functions of quantum spin $1/2$ ladders at finite temperature, under a magnetic field, in the gapless phase at various relevant temperatures $T\neq 0$, momentum $q$ and frequencies $ω$. We compute those quantities using the time dependent density matrix renormalization group (T-DMRG) in some optimal numerical scheme. We compare these correlations with the ones of dimerized quantum spin chains and simple spin chains, that we compute by a similar technique. We analyze the intermediate energy modes and show that the effect of temperature lead to the formation of an essentially dispersive mode corresponding to the propagation of a triplet mode in an incoherent background, with a dispersion quite different from the one occurring at very low temperatures. We compare the low energy part of the spectrum with the predictions of the Tomonaga-Luttinger liquid field theory at finite temperature. We shows that the field theory describes in a remarkably robust way the low energy correlations for frequencies or temperatures up to the natural cutoff (the effective dispersion) of the system. We discuss how our results could be tested in e.g. neutron scattering experiments.

17 pages, 14 figures