Mean-field theory for symmetry-breaking Fermi surface deformations on a square lattice

We analyze a mean-field model of electrons with pure forward scattering interactions on a square lattice which exhibits spontaneous Fermi surface symmetry breaking with a $d$-wave order parameter, the surface expands along the $k_x$ axis and shrinks along the $k_y$ axis (or vice versa). The symmetry-broken phase is stabilized below a dome-shaped transition line $T_c\left(\mu \right)$, with a maximal $T_c$ near van Hove filling. The phase transition is usually first order at the edges of the transition line, and always second order around its center. The $d$-wave compressibility of the Fermi surface is however strongly enhanced even near the first order transition down to zero temperature. In the weak coupling limit the phase diagram is fully determined by a single nonuniversal energy scale, and hence dimensionless ratios of different characteristic quantities are universal. Adding a uniform repulsion to the forward scattering interaction, the two tricritical points at the ends of the second order transition line are shifted to lower temperatures. For a particularly favorable choice of hopping and interaction parameters one of the first order edges is replaced completely by a second order transition line, leading to a quantum critical point.