  Motivated by recent developments in conformal field theory (CFT), we devise a
Quantum Monte Carlo (QMC) method to calculate the moments of the partially
transposed reduced density matrix at finite temperature. These are used to
construct scale invariant combinations that are related to the negativity, a
true measure of entanglement for two intervals embedded in a chain. These
quantities can serve as witnesses of criticality. In particular, we study
several scale invariant combinations of the moments for the 1D hard-core boson
model. For two adjacent intervals unusual finite size corrections are present,
showing parity effects that oscillate with a filling dependent period. These
are more pronounced in the presence of boundaries. For large chains we find
perfect agreement with CFT. Oppositely, for disjoint intervals corrections are
more severe and CFT is recovered only asymptotically. Furthermore, we provide
evidence that their exponent is the same as that governing the corrections of
the mutual information. Additionally we study the 1D Bose-Hubbard model in the
superfluid phase. Remarkably, the finite-size effects are smaller and QMC data
are already in impressive agreement with CFT at moderate large sizes.
