Non-Fermi Liquid and Pairing in Electron-Doped Cuprates

We study the normal state and pairing instability in electron-doped cuprates in a model with long-ranged antiferromagnetic spin fluctuations close to an antiferromagnetic quantum-critical point. We show that the fermionic self-energy has a non-Fermi-liquid form leading to peculiar frequency dependencies of the conductivity and the Raman response. We solve the pairing problem and demonstrate that $T_c$ is determined by the curvature of the Fermi surface, and the pairing gap $\Delta (\mathbf{k},\omega )$ is strongly nonmonotonic along the Fermi surface. The normal state frequency dependencies, the value of $T_c\sim 10\mathrm{K}$, and the $\mathbf{k}$ dependence of the gap agree with the experiment.

Figure 1

Left: Fermi surface at the antiferromagnetic QCP with $2k_F=(\pi ,\pi )$. Diamond-shaped dashed lines bound the magnetic Brillouin zone. The diagonal points of the FS (nodal points of the $d_{x^2-y^2}$-wave gap) now become “hot.” Right: Graphic explanation of the attraction in the $d_{x^2-y^2}$ channel: parts of the Fermi surface on the same side of the zone diagonals are on average closer to the $\mathbf{Q}$ separation than the parts on the opposite sides, leading to attraction in the $d_{x^2-y^2}$ channel (plus and minus are the signs of the $d_{x^2-y^2}$ gap).

Figure 2

Normal state conductivity as a function of frequency shows a scaling behavior for ${\omega }_0<\omega <40{\omega }_0$. Left: $|\sigma (\omega )|$, the behavior is indistinguishable from the $\sigma (\omega )\sim {\omega }^{-0.64}$ dependence. Right: $\arctan \mathrm{Im}\sigma (\omega )/\mathrm{Re}\sigma (\omega )$, an almost constant value means that both $\mathrm{Im}\sigma (\omega )$ and $\mathrm{Re}\sigma (\omega )$ scale as ${\omega }^{-0.64}$.

Figure 3

Pairing instability temperature $T_c$ in units of the coupling constant $\overline{g}$ vs the curvature parameter $r=\overline{g}{\beta }^2/\pi v_F^2$. The solid line is the numeric solution, which takes increasingly long to compute as $r\to 0$. The dashed line is the analytic small-$r$ form $T_c=0.95\overline{g}r/(1+5r^{2/5}{)}^4$.

Figure 4

The pairing vertex $\Phi (k_y,{\omega }_n)$ for $r=0.05$ vs ${\omega }_n/(2\pi T_c)$ and $k_y/q_0$ ($q_0$ is defined in the text). Observe that $\Phi (k_y,{\omega }_n)$ is nonmonotonic in $k_y$.