Theory of fluctuating charge ordering in the pseudogap phase of cuprates via a preformed pair approach

We study the static and dynamic behavior of charge ordering within a $d$-wave pair pseudogap (pg) scenario. This is addressed using a density-density correlation function derived from the standard pg self-energy $\Sigma$ and compatible with the longitudinal and transverse sum rules. The broadening factor $\gamma$ in $\Sigma$ reflects the breaking of pairs into constituent fermions. We apply this form for $\Sigma$ (derived elsewhere for high fields) to demonstrate the existence of quantum oscillations in a non-Fermi liquid pg state. Our conclusion is that the pseudogap-induced pair breaking, via $\gamma$, allows the underlying fermiology to be revealed; $d$-wave (as distinct from $s$-wave) pairing in YBCO, with finite $\omega$ and $\gamma$ enables antinodal fluctuations, despite their competition in the static and superconducting limits.

Figure 1

Fermi surfaces for YBCO and LSCO with arrows indicating the dominant nesting vectors. The lower panel shows quantum oscillations persist in a non-Fermi liquid pseudogap state, due to the finite pair lifetime reflected in ${\gamma }^{-1}$. Their amplitude is reduced by a factor of about 5 from the standard Lifshitz-Kosevich theory in the $d$-wave case. In contrast, $s$-wave pairing (not shown) reduces the amplitude by a factor of at least 100.

Figure 2

Static studies: Vertical and diagonal cuts plotting Re$P_{\rho ,\rho }(\omega =0)$ vs $\mathbf{q}$ for YBCO and LSCO. The black, red, and green lines are for normal gas, $d$-wave, and $s$-wave pseudogaps, respectively. The $d$-wave gap is compatible with nodal peaks but suppresses antinodal peaks, while there is a more severe suppression of all peaks in the $s$-wave case. Here $\gamma \approx 0$.

Figure 3

Dynamical studies: Left and central figures show diagonal cuts of Im$P_{\rho ,\rho }$ vs $\mathbf{q}$ for $d$ wave with YBCO and LSCO band structure, and $\gamma \approx 0$. The peaks located at $q_x=q_y\approx 0.75\pi$ and $q_x=q_y\approx 0.4\pi$ become visible between 10 and 35 meV, and correspond to the nodal (N) and antinodal (AN) peaks we observed in Re$P_{\rho ,\rho }$ of the (gapless) normal gas. There is also a small peak which appears at $q_x=q_y\approx 0.6\pi$ in the $d$-wave curves for $\omega \approx \Delta$. The plot on the right shows the effect (for YBCO) of varying $\gamma$ at fixed frequency where one sees that larger $\gamma$ enhances the antinodal $q_x\approx 0.4\pi$ peak, while decreasing the relative height of the nodal $q_x\approx 0.75\pi$ peak. An upper limit of $\gamma /{\Delta }_o\approx 1/2$ is assumed to match the Fermi arc size.[16]