The Fluctuation-Dissipation Theorem at Low Temperatures in a 2D Optical Lattice*

We calculate local density fluctuations and the local compressibility\footnote{Q. Zhou, et al., {\em Phys. Rev. Lett.} {\bf 103}, 085701 (2009).} of bosons in a two dimensional optical lattice as a function of temperature $T$ and the tuning parameter $U/t$, the on-site boson repulsion strength in units of hopping, using worldline Quantum Monte Carlo. Our numerical results, coupled with the quantum fluctuation-dissipation theorem applied locally, make significantly different predictions for direct simulations of lattice bosons in a harmonic trap versus simulations that treat the trap within a local density approximation, especially at low temperatures. We discuss implications of our results for local thermometry, equilibration and characterization of the quantum critical regime.