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The Compressibility in Strongly Correlated Superconductors and Superfluids: From BCS to BEC

We present a theoretical study of the compressibility, $κ$, in a Fermi gas with attractive contact interactions, providing predictions for the strongly-attractive regime and the superfluid phase. Our work emphasizes the compressibility sum rule and gauge invariance as constraints on $κ$ and we show how within a particular $t$-matrix approach, these can be satisfied in the normal phase when no approximations are made. For tractability, approximations must be introduced, and it is believed that thermodynamical approaches to $κ$ are more reliable, than correlation function based schemes. Contrasting with other studies in the literature, we present thermodynamic calculations of $κ$; these yield semi-quantitative agreement with experiment and provide physical insight into similar results obtained via quantum Monte Carlo simulations.