Breakdown of Fermi liquid behavior at the $(\pi ,\pi )=2k_F$ spin-density wave quantum-critical point: The case of electron-doped cuprates

Many correlated materials display a quantum-critical point between a paramagnetic and a spin-density wave (SDW) state. The SDW wave vector connects points, so-called hot spots, on opposite sides of the Fermi surface. The Fermi velocities at these pairs of points are in general not parallel. Here, we consider the case where pairs of hot spots coalesce, and the wave vector $(\pi ,\pi )$ of the SDW connects hot spots with parallel Fermi velocities. Using the specific example of electron-doped cuprates, we first show that Kanamori screening and generic features of the Lindhard function make this case experimentally relevant. The temperature dependence of the correlation length, the spin susceptibility, and the self-energy at the hot spots are found using the two-particle self-consistent theory and specific numerical examples worked out for band and interaction parameters characteristic of the electron-doped cuprates. While the curvature of the Fermi surface at the hot spots leads to deviations from perfect nesting, the pseudonesting conditions lead to drastic modifications of the temperature dependence of these physical observables: Neglecting logarithmic corrections, the correlation length $\xi$ scales like $1/T$, namely, $z=1$ instead of the naive $z=2$, the $(\pi ,\pi )$ static spin susceptibility $\chi$ like $1/\sqrt{T}$, and the imaginary part of the self-energy at the hot spots like $T^{3/2}$. The correction $T_1^{-1}\sim T^{3/2}$ to the Korringa NMR relaxation rate is subdominant. We also consider this problem at zero temperature, or for frequencies larger than temperature, using a field-theoretical model of gapless collective bosonic modes (SDW fluctuations) interacting with fermions. The imaginary part of the retarded fermionic self-energy close to the hot spots scales as $-{\omega }^{3/2}\ln \omega$. This is less singular than earlier predictions of the form $-\omega \ln \omega$. The difference arises from the effects of umklapp terms that were not included in previous studies.

Figure 1

The dashed lines indicate the magnetic Brillouin zone with ordering vector $\mathbf{Q}=(\pi ,\pi )$. The arrows give examples of pseudonesting conditions, namely, of points such that $2{\mathbf{k}}_F$ is equal to the antiferromagnetic wave vector. The $(\pi ,\pi )$ ordering wave vector is defined with respect to the $(q_x,q_y)$ coordinate system. In the field-theory approach introduced later, we work in the rotated ($k_x,k_y$) coordinate system.

Figure 2

Lindhard function near the $2k_F$ point as a function of doping for $U=6$, $t^{\prime }=-0.175$, and $t^{\prime \prime }=0.05$, values that are appropriate for electron-doped cuprates. The rapid fall with filling larger than $1.201$, close to the critical filling, is apparent.

Figure 3

The Fermi surface at the critical filling $n_c$ touches the antiferromagnetic Brillouin zone. The important integration region is over a rectangle of thickness $T$ and width $\sqrt{T}$. The critical chemical potential ${\mu }_c$ where the antiferromagnetic zone boundary touches $(\pi /2,\pi /2)$ on the Fermi surface is the solution of $-2t^{\prime \prime }[\cos \left(\pi \right)+\cos \left(\pi \right)]-{\mu }_c=0$. This corresponds to a filling $n_c=1.2007$ for $t^{\prime }=-0.175$ and $t^{\prime \prime }=0.05$. For $U=6$, the critical filling that we find, $n_c=1.20096$, is slightly larger.

Figure 4

${\xi }_0$ and ${\Gamma }_0$ evaluated at $n_c=1.2007$ obtained from the condition that the Fermi surface is tangent to the antiferromagnetic zone boundary, as explained in the caption of Fig. 3.

Figure 5

Log-log plot for the temperature scaling of the correlation length. The scaling is estimated from the width measured along one of the reciprocal lattice wave vectors $(\pi -q_H)$ at various fractions of the maximum height. On (a), for $U=5.56$, the critical doping corresponds to $n_c=1.2007$, where the Fermi surface is tangent to the antiferromagnetic zone boundary. The temperature scale is too small to detect possible logarithmic corrections. The results are consistent with $z=1$. On (b), deviations from $1/T$ occur at low temperature because, for the chosen value $U=6$, the calculation is at the critical doping $1.20096$, slightly away from the point where the Fermi surface is tangent to the antiferromagnetic zone boundary. Also shown, ${\xi }_{AS}$ obtained from the Ornstein-Zernicke form (12) using $U_{sp}$ from the self-consistency relation (7).

Figure 6

On (a), log-log plot of the temperature dependence of the imaginary part of the self-energy at various color-coded points on the Fermi surface. The color code is in the inset. On (b), the local exponent is given as a function of angle and temperature. The points near the hot spot at $\theta =\pi /4$ behave as $T^{3/2}$ over the accessible temperature range. Calculations are done with $U=5.56$, $t^{\prime }=-0.175$, and $t^{\prime \prime }=0.05$ at the quantum critical filling $n_c=1.2007$. The lower figures are the corresponding results for $U=6$, $t^{\prime }=-0.175$, and $t^{\prime \prime }=0.05$ at the quantum critical point $n=1.20096$. Since in that case the Fermi surface is not exactly tangent to the antiferromagnetic zone boundary, the $T^{3/2}$ behavior near $\theta =\pi /4$ is recovered only if the temperature is high enough that details of the Fermi surface can not be resolved. The black lines on (b) and (d) are the curves defined by $T_{\mathrm{onset}}=v_F\delta k_{\perp }\left(\theta \right)/2$ where $\delta k_{\perp }\left(\theta \right)$ is the component of ${\mathbf{k}}_F-(\pi /2,\pi /2)$ parallel to ${\mathbf{v}}_F(\pi /2,\pi /2)$ at a given angle $\theta$.

Figure 7

On (a) and (c), log-log plot for the frequency dependence of $-{\Sigma }^{\prime \prime R}({\mathbf{k}}_F,\omega )$ at various angles along the Fermi surface. The result of a power-law fit is shown on (b) and (d). The dashed lines on (a) and (c) correspond to the fitted power laws. At the hot spot located at $\theta =\pi /4$, $-{\Sigma }^{\prime \prime R}$ scales as ${\omega }^{3/2}$. The frequency range is small because of the low temperature saturation shown on the next figure. We have verified that the crossover from ${\omega }^{3/2}$ to the Fermi liquid regime ${\omega }^2$ occurs on a broader angular scale when the temperature is higher, as expected from the results of Fig. 6. Calculations are done at $T=0.002$, $t^{\prime }=-0.175$, and $t^{\prime \prime }=0.05$ at $U=5.56$ and $n=n_c=1.2007$ for (a) and (b) and $U=6$, $n=1.20096$ for (c) and (d).

Figure 8

Log-log plot for the frequency dependence of $-{\Sigma }^{\prime \prime R}$ at the hot spot for two different temperatures. The saturation at low frequency occurs at higher frequency when the temperature is higher. Calculations are done with $U=6$, $t^{\prime }=-0.175$, and $t^{\prime \prime }=0.05$ at the quantum critical filling $n_c=1.201$ appropriate for electron-doped cuprates.

Figure 9

The polarization bubble for the two-patch theory in Eq. (56). The internal solid lines in the loop correspond to the free-fermion propagators (different patches denoted by $s=\pm$) and the external wavy lines correspond to the boson ${\phi }^a$.

Figure 10

The electron self-energy ${\Sigma }_{+}\left(p\right)$ at leading order in $1/N$. The $\vec{\phi }$ propagator includes the one-loop bubbles evaluated earlier.

Figure 11

Left: $\mathrm{Im}{\Sigma }_R(0,\omega )$, calculated numerically from the second line of Eq. (73), shown as a red solid line. The black dashed line is the asymptotic result Eq. (78). For comparison, the blue dashed-dotted line indicates ${\omega }^{3/2}$. Right: different contributions to $\mathrm{Im}{\Sigma }_R(0,\omega )$. The red line shows the dominant contribution, denoted by $B$ in the main text. Black dashed line: asymptotic result Eq. (78). The other three curves show the subleading contributions $A,C$, and $D$, which scale as $\sim {\omega }^{3/2}$.

Figure 12

The one-loop contribution to the boson-fermion vertex.