Exact correlation functions in the cuprate pseudogap phase: Combined effects of charge order and pairing

There is a multiplicity of charge-ordered, pairing-based, or pair density wave theories of the cuprate pseudogap, albeit arising from different microscopic mechanisms. For mean-field schemes (of which there are many) we demonstrate here that they have precise implications for two-body physics in the same way that they are able to address the one-body physics of photoemission spectroscopy. This follows because the full vertex can be obtained exactly from the Ward-Takahashi identity. As an illustration, we present the spin response functions, finding that a recently proposed pair density wave (Amperean pairing) scheme is readily distinguishable from other related scenarios.

Figure 1

Comparison of the spin density correlation function $\mathrm{Im}{\chi }_0\left(Q\right)=-\mathrm{Im}{P}_S^{00}(\omega ,\mathbf{q})$ for three different values of $\mathbf{q}$ in the Amperean, $d$-wave, and YRZ pseudogap theories. In (a) we have labeled the van Hove peaks appearing in the $d$-wave theory, which appear as saddle points in the contour plot of Fig. 2. Here we use the band structure given in Ref. [3] with $T=0.01$ and a broadening of $\Gamma =0.01$. The doping $p=0.12$ and the chemical potential $\mu$ is fixed by the Luttinger sum rule. The band structure and frequency are all normalized by $t$, and the gap function has an amplitude of ${\Delta }_0=0.15$. For the Amperean theory we use a $k_x,k_y$-symmetrized Gaussian [6] gap function.

Figure 2

The equal energy contours $E_2\left(\mathbf{k}\right)\equiv E_{\mathbf{k}}+E_{\mathbf{k}+\mathbf{q}}$ which appear as the integrand in $\mathrm{Im}{\chi }_0(\omega ,\mathbf{q})$ for both (a) $\mathit{d}$-wave and (b) the Amperean pseudogap schemes. Here $\mathbf{q}=(\pi ,\pi )$. Note there are several energy scales, as indicated by the legend. The labels $A$ and $B$ indicate the location of the saddle points of $\mathrm{Im}{\chi }_0(\mathbf{q},\omega )$.