Controlled evidence for phase separation in the fermionic 2D Hubbard model

In this study we inspect the repulsive two-dimensional fermionic Hubbard model at doping values $0.6\le n \le 1$ and moderate interaction strength $U=4$ by means of the Bold Diagrammatic Monte Carlo \footnote{Y. Deng et al. Europhysics Letters {\bf 110(5)}, 57001, (2015) }(DiagMC) and Determinant Diagrammatic Monte Carlo \footnote{E. Burowski et al. New Journal of Physics {\bf 8}, 153, (2006) }(DDMC). We use a stability condition for the inverse compressibility to prove in a controlled and unbiased way the existence of a first order phase transition and intermediate phase separated region between the $d_{x^2-y^2}$-wave superfluid at densities $n<0.8$ and the anti-ferromagnetic phase at half filling ($n=1.0$). The critical density of $n_c \sim 0.82$ obtained via Maxwell construction is in good correspondence with the region of densities where magnetic fluctuations become significant at $U=4$.