Unexpected behavior of the local compressibility near the B=0 metal-insulator transition

We have measured the local electronic compressibility of a two-dimensional hole gas as it crosses the B = 0 metal-insulator transition. In the metalli c phase, the compressibility follows the mean-field Hartree-Fock (HF) theor y and is found to be spatially homogeneous. In the insulating phase it devi ates by more than an order of magnitude from the HF predictions and is spat ially inhomogeneous. The crossover density between the two types of behavio r agrees quantitatively with the transport critical density, suggesting tha t the system undergoes a thermodynamic change at the transition.