A general, path-independent expression is derived for the excess chemical potential of integral equation closure approximations that contain a bridge function which depends on a renormalized indirect correlation function. Closed-form results are obtained for various cases, among them a partial series expansion of the hypernetted chain closure.

We address the timely issue of the presence of charge ordering at the hot-spots in the pseudo-gap phase of cuprate superconductors in the context of an emergent SU(2)-symmetry which relates the charge and pairing sectors. Performing the Hubbard-Stratonovich decoupling such that the free energy stays always real and physically meaningful we exhibit three solutions of the spin-fermion model at the hot spots. A careful examination of their stability and free energy shows that, at low temperature, the system tends towards a co-existence of charge density wave (CDW) and the composite order parameter made of diagonal quadrupolar density wave and pairing fluctuations of Ref. [Nat. Phys. 9, 1745].The CDW is sensitive to the shape of the Fermi surface in contrast to the diagonal quadrupolar order, which is immune to it. SU(2) symmetry within the pseudo-gap phase also applies to the CDW state, which therefore admits a pairing density pave counterpart breaking time reversal symmetry.

This paper exposes in a detailed manner the recent findings about the SU(2) scenario for the underdoped phase of the Cuprate superconductors. The SU(2) symmetry is formulated as a rotation between the d-wave SC phase and a d-wave charge order. We define the operators responsible for the SU(2) rotations and we derive the non-linear σ-model associated with it. In this framework, we demonstrate that SU(2) fluctuations are massless in finite portions of the Brillouin Zone corresponding to the anti-nodal regions (0,π), (π, 0). We argue that the presence of SU(2) fluctuations in the anti-nodal region leads to the opening of Fermi arcs around the Fermi surface and to the formation of the pseudo-gap. Moreover, we show that SU(2) fluctuations lead, in turn, to the emergence of a finite momentum SC order -or Pair Density Wave (PDW)-and more importantly to a new kind of excitonic particle-hole pairs liquid, the Resonant Excitonic State (RES), which is made of patches of preformed particle-hole pairs with multiple momenta. When the RES liquid becomes critical, we demonstrate that electronic scattering through the critical modes leads to anomalous transport properties. This new finding can account for the Strange Metal (SM) phase at finite temperature, on the right hand side of the SC dome, shedding light on another notoriously mysterious part of the phase diagram of the Cuprates. CONTENTS