Unified treatment of Fermi pockets and arcs scenarios for the cuprates: Sum-rule-consistent response functions of the pseudogap

Essential to understanding the cuprate pseudogap phase is a study of the charge (and spin) response functions, which we address here via a consistent approach to the Fermi arcs and the Fermi pockets scenario of Yang, Rice, and Zhang (YRZ). The two schemes are demonstrated to be formally similar, and to share a common physics platform; we use this consolidation to address the inclusion of vertex corrections which have been omitted in YRZ applications. We show vertex corrections can be easily implemented in a fashion analytically consistent with sum rules and that they yield important contributions to most observables. A study of the charge ordering susceptibility of the YRZ scenario makes their simple physics evident: They represent the inclusion of charged bosonic, spin singlet degrees of freedom, and are found to lead to a double peak structure.

Figure 1

(a) Normal state $\omega =0$, $q_y=0$ charge susceptibility $P_{\rho \rho }$ with (solid) and without (dotted) vertex corrections. The arrows indicate that a second peak is present in the former case. Here we follow the band structure used in Ref. [4], and use $T=0.01$ and broadening $\eta =0.01$ [4] to study a low-temperature system. The doping $p=0.12$, and chemical potential $\mu$ is fixed by the Luttinger sum rule [3]. These values are normalized to $t$, the primary single-particle dispersion parameter [3, 4]. The inset shows the contribution of the vertex term ($F_{\mathrm{pg}}{F}_{\mathrm{pg}}$) to $P_{\rho \rho }$. (b),(c) Plots of the momentum phase space contributions to $P_{\rho \rho }(\omega =0,\mathbf{q}=(0.3\pi ,0))$ for $\Delta =0.15$, overlaid on contour plots of the spectral function $A_{\mathrm{YRZ}}(\omega =0,\mathbf{k})$. Shown are green $\mathbf{k}$ (“origin”) regions and yellow $\mathbf{k}+\mathbf{q}$ (“destination”) regions for which the integrand magnitude is greater than a set threshold. (b) shows contributions from the vertex term greater than a threshold of $0.008$, while (c) shows single fermionic bubble ($GG$) contributions greater than a threshold of $0.02$. The plus and minus signs in (b) and (c) indicate the sign of the phase space contribution.