  The minimal model to describe many spin chain materials with ferroelectric
properties is the Heisenberg model with ferromagnetic nearest neighbor coupling
J1 and antiferromagnetic next-nearest neighbor coupling J2. Here we study the
thermodynamics of this model using a density-matrix algorithm applied to
transfer matrices. We find that the incommensurate spin-spin correlations -
crucial for the ferroelectric properties and the analogue of the classical
spiral pitch angle - depend not only on the ratio J2/|J1| but also strongly on
temperature. We study small easy-plane anisotropies which can stabilize a
vector chiral order as well as the finite-temperature signatures of multipolar
phases, stable at finite magnetic field. Furthermore, we fit the
susceptibilities of LiCuVO4, LiCu2O2, and Li2ZrCuO4. Contrary to the
literature, we find that for LiCuVO4 the best fit is obtained with J2 ~ 90 K
and J2/|J1| ~ 0.5 and show that these values are consistent with the observed
spin incommensurability. Finally, we discuss our findings concerning the
incommensurate spin-spin correlations and multipolar orders in relation to
future experiments on these compounds.
