Renormalization group analysis of the pair-density-wave and charge order within the fermionic hot-spot model for cuprate superconductors

In light of the new experimental and theoretical important developments in high-$T_c$ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a $d$-wave form factor with the experimentally observed modulation $(\pm Q_0,0)$ and $(0,\pm Q_0)$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the “hidden” order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, $d$-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\pm Q_0,\pm Q_0)$, and $d$-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.

Figure 1

The present 2D fermionic model consisting of eight hot spots (with bare Fermi velocities ${\mathbf{v}}_{\mathbf{F}}$ and angle $\theta$) on the Fermi surface that are directly connected by the commensurate spin density wave (SDW) ordering wave vector $\mathbf{Q}=(\pi ,\pi )$. The incommensurate wave vectors ${\mathbf{Q}}_x=(Q_0,0),{\mathbf{Q}}_y=(0,Q_0),\tilde{\mathbf{Q}}=(Q_0,Q_0),{\mathbf{Q}}_x^{{}^{\prime }}$ and ${\mathbf{Q}}_y^{{}^{\prime }}$ are also displayed.

Figure 2

The relevant couplings in the present fermionic hot-spot model. For the $g_{1s},g_{3t}$, and $g_{3v}$ couplings, we show two possible scattering processes (one by a solid arrow and the other by a dashed arrow), which are always equal in our two-loop RG results.

Figure 3

A possible renormalized Fermi surface suggested by the present two-loop RG calculation of the fermionic hot-spot model. Note the resulting nesting condition of the hot spots connected by $\mathbf{Q}=(\pi ,\pi )$.

Figure 4

Feynman diagrams for (a) SC and (b) DW response functions of the present fermionic hot-spot model. Solid lines represent noninteracting fermionic single-particle Green's function, while the wavy lines stand for the renormalized coupling constants. Depending on the channel analyzed, the vector $\mathbf{q}$ can be $\mathbf{Q}=(\pi ,\pi ),\tilde{\mathbf{Q}}=(Q_0,Q_0),{\mathbf{Q}}_x=(Q_0,0),{\mathbf{Q}}_y=(0,Q_0),{\mathbf{Q}}_x^{\prime }$ and ${\mathbf{Q}}_y^{\prime }$.

Figure 5

The signs of the order parameters at all eight hot spots for a $d$-wave form factor. Instead, for an $s$-wave form factor, all signs become positive.

Figure 6

Relevant Feynman diagrams for the vertex corrections up to two-loop order in the present model. Solid lines denote noninteracting fermionic single-particle Green's function, while the wavy lines correspond to the renormalized coupling constants.