Entangling electron-electron and electron-hole pairs in the pseudo-gap phase of cuprate superconductors

The mysterious pseudo-gap (PG) phase of cuprate superconductors has been the subject of intense investigation over the last thirty years, but without a clear agreement about its origin. Owing to a recent observation in Raman spectroscopy, of a precursor in the charge channel, on top of the well known fact of a precursor in the superconducting channel, we present here a novel idea: the PG is formed through a Higgs mechanism, where two kinds of preformed pairs, in the particle-particle and particle-hole channels, become entangled through a freezing of their global phase. We detail the $U(1)\times U(1)$ gauge theory at the origin of this phenomenon for underdoped cuprates. At $T^{*}$, the freezing of the global phase sets the unique PG energy scale $E^{*}$ which also opens a gap in the Fermi surface. Below $T^{*}$ we observe the formation of quasi long-range charge modulations at $T_{co}$ and a fluctuation regime of the Cooper pairs below $T_{c}^{\prime}$. A second condensation mechanism occurs at a lower temperature $T_{c}$, where the relative phase of the doublet of preformed pairs gets quenched. This defines the superconducting state, which here has the form of a super-solid with a finite superposition of condensed Cooper pairs and modulated particle-hole pairs. Through the idea of "locking" of phases between the charge and superconducting modes, the theory gives a unique explanation for the unusual global phase coherence of short-range charge modulations, observed below $T_{c}$ on phase sensitive scanning tunneling microscopy (STM). A simple microscopic model enables us to estimate the mean-field values of the precursor gaps in each channel and the PG energy scale, and to compare them to the values observed in Raman scattering spectroscopy. We also discuss the possibility of a multiplicity of orders in the PG phase.