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Equation of state of two--dimensional $^3$He at zero temperature

We have performed a Quantum Monte Carlo study of a two-dimensional bulk sample of interacting 1/2-spin structureless fermions, a model of $^3$He adsorbed on a variety of preplated graphite substrates. We have computed the equation of state and the polarization energy using both the standard fixed-node approximate technique and a formally exact methodology, relying on bosonic imaginary-time correlation functions of operators suitably chosen in order to extract fermionic energies. As the density increases, the fixed-node approximation predicts a transition to an itinerant ferromagnetic fluid, whereas the unbiased methodology indicates that the paramagnetic fluid is the stable phase until crystallization takes place. We find that two-dimensional $^3$He at zero temperature crystallizes from the paramagnetic fluid at a density of 0.061 Å$^{-2}$ with a narrow coexistence region of about 0.002 Å$^{-2}$. Remarkably, the spin susceptibility turns out in very good agreement with experiments.