SU(2) symmetry in a realistic spin-fermion model for cuprate superconductors
We consider the Pseudo-Gap (PG) state of high-$T_c$ superconductors in form of a composite order parameter fluctuating between 2$p_F$-charge ordering and superconducting (SC) pairing. In the limit of linear dispersion and at the hotspots, both order parameters are related by a SU(2) symmetry and the eight hotspot model of Efetov $\textit{et al.}$ [Nat. Phys. $\bf{9}$, 442 (2013)] is recovered. In the general case however, curvature terms of the dispersion will break this symmetry and the degeneracy between both states is lifted. Taking the full momentum dependence of the order parameter into account, we measure the strength of this SU(2) symmetry breaking over the full Brillouin zone. For realistic dispersion relations including curvature we find generically that the SU(2) symmetry breaking is small and robust to the fermiology and that the symmetric situation is restored in the large paramagnon mass and coupling limit. Comparing the level splitting for different materials we propose a scenario that could account for the competition between the PG and the SC states in the phase diagram of high-$T_c$ superconductors.