Theory of non-Fermi liquid and pairing in electron-doped cuprates

We apply the spin-fermion model to study the normal state and pairing instability in electron-doped cuprates near the antiferromagnetic quantum-critical point. Peculiar frequency dependencies of the normal state properties are shown to emerge from the self-consistent equations on the fermionic and bosonic self-energies, and are in agreement with experimentally observed ones. We argue that the pairing instability is in the $d_{x^2-y^2}$ channel, as in hole-doped cuprates, but theoretical $T_c$ is much lower than in the hole-doped case. For the same hopping integrals and the interaction strength as in hole-doped materials, we obtain $T_c\sim 10\mathrm{K}$ at the end point of the antiferromagnetic phase. We argue that a strong reduction of $T_c$ in electron-doped cuprates compared to hole-doped ones is due to critical role of the Fermi surface curvature for electron-doped materials. The $d_{x^2-y^2}$-pairing gap $\Delta (\mathbf{k},\omega )$ is strongly nonmonotonic along the Fermi surface. The position of the gap maxima, however, does not coincide with the hot spots, as the nonmonotonic $d_{x^2-y^2}$ gap persists even at doping when the hot spots merge on the Brillouin zone diagonals.

Figure 1

(Color online) 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 Fermi surface (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: $\mid \sigma \left(\omega \right)\mid$, the behavior is indistinguishable from the $\sigma \left(\omega \right)\sim {\omega }^{-0.64}$ dependence. Right: $\arctan \mathfrak{I}\sigma \left(\omega \right)∕\mathfrak{R}\sigma \left(\omega \right)$, an almost constant value means that both $\mathfrak{I}\sigma \left(\omega \right)$ and $\mathfrak{R}\sigma \left(\omega \right)$ scale as ${\omega }^{-0.64}$. (Figure reproduced from Ref. 25).

Figure 3

Diagrammatic equation on the anomalous vertex.

Figure 4

(Color online) Pairing instability temperature $T_c$ in units of the coupling constant $\overline{g}$ vs the curvature parameter $r=\overline{g}{\beta }^2∕\pi v_F^2$. Solid line gives the result of the numeric solution, which requires increasingly long computational time as $r\to 0$. Dashed line is the analytic small-$r$ form $T_c=0.95\overline{g}r∕(1+5r^{2∕5}{)}^4$. [Figure reproduced from Ref. 25].

Figure 5

(Color online) The pairing vertex $\Phi (k_y,{\omega }_n)$ for $r=0.05$ vs ${\omega }_n∕\left(2\pi T_c\right)$ and $k_y∕q_0$ ($q_0$ is defined in the text). Observe that $\Phi (k_y,{\omega }_n)$ is nonmonotonic in $k_y$. (Figure reproduced from Ref. 25).

Figure 6

(Color online) Scaling of the “critical temperature” $x\left(r\right)$ (solid lines) and the position of the maximum $k_{\mathrm{m}}\left(r\right)$ (dashed lines) with the parameter $r$ in the kernel (36) of the one-dimensional “toy model.” Numerical solution [red (gray) curves] closely follow the scaling laws obtained analytically (black straight lines): $3∕4x-1\approx 2r^{2∕5}$, and $k_{\mathrm{m}}\sim r^{-2∕5}$.

Figure 7

(Color online) Solutions $f\left(k\right)$ of the one-dimensional “toy model” for several values of the parameter $r$ ($r$ decreases for the curves from left to right). Dependence of the position of the maximum $k_{\mathrm{m}}\left(r\right)$ is plotted in Fig. 6.

Figure 8

Graphical solution to Eq. (B3). Straight line and the curve represent the left- and right-hand side of Eq. (B3). Crossing is at about $\gamma \approx 0.85$, or $\alpha =1-\gamma \approx 0.15$.

Figure 9

Critical temperatures $T_c$ found numerically vs the power law exponent $\gamma$ (left panel). Right panel represents the dependence of $T_c$ on the matrix size $N$ extrapolated to $1∕N\to 0$ for $\gamma =1∕4$; behavior for other $\gamma$ is similar; left panel shows $T_c$ already extrapolated to $1∕N\to 0.$