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Phase diagram and critical properties of Yukawa bilayers

We study the ground-state Wigner bilayers of pointlike particles with Yukawa pairwise interactions, confined to the surface of two parallel hard walls at dimensionless distance $η$. The model involves as limiting cases the unscreened Coulomb potential and hard spheres. The phase diagram of Yukawa particles, studied numerically by Messina and Löwen [Phys. Rev. Lett. 91 (2003) 146101], exhibits five different staggered phases as $η$ varies from 0 to intermediate values. We present a lattice summation method using the generalized Misra functions which permits us to calculate the energy per particle of the phases with a precision much higher than usual in computer simulations. This allows us to address some tiny details of the phase diagram. Going from the hexagonal phase I to phase II is shown to occur at $η=0$, which resolves a longtime controversy. We find a tricritical point where Messina and Löwen suggested a coexistence domain of several phases which was suggested to divide the staggered rhombic phase into two separate regions. Our calculations reveal one continuous region for this rhombic phase with a very narrow connecting channel. Further we show that all second-order phase transitions are of mean-field type. We also derive the asymptotic shape of critical lines close to the Coulomb and hard-spheres limits. In and close to the hard-spheres limit, the dependence of the internal parameters of the present phases on $η$ is determined exactly.