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The two-loop renormalization group (RG) calculation is considerably extendedhere for the two-dimensional (2D) fermionic effective field theory model, whichincludes only the so-called "hot spots" that are connected by thespin-density-wave (SDW) ordering wavevector on a Fermi surface generated by the2D $t-t'$ Hubbard model at low hole doping. We compute the Callan-Symanzik RGequation up to two loops describing the flow of the single-particle Green'sfunction, the corresponding spectral function, the Fermi velocity, and some ofthe most important order-parameter susceptibilities in the model at lowerenergies. As a result, we establish that -- in addition to clearly dominant SDWcorrelations -- an approximate (pseudospin) symmetry relating a short-range\emph{incommensurate} $d$-wave charge order to the $d$-wave superconductingorder indeed emerges at lower energy scales, which is in agreement with recentworks available in the literature addressing the 2D spin-fermion model. Wederive implications of this possible electronic phase in the ongoing attempt todescribe the phenomenology of the pseudogap regime in underdoped cuprates.

annals of physics

Evidence of a short-range incommensurate $d$-wave charge order from a  fermionic two-loop renormalization group calculation of a 2D model with hot  spots