Spin-fermion model with overlapping hot spots and charge modulation in cuprates

We study particle-hole instabilities in the framework of the spin-fermion (SF) model. In contrast to previous studies, we assume that adjacent hot spots can overlap due to a shallow dispersion of the electron spectrum in the antinodal region. In addition, we take into account effects of a remnant low energy and momentum Coulomb interaction. We demonstrate that at sufficiently small values $|\varepsilon (\pi ,0)-E_F|\lesssim \mathrm{\Gamma }$, where $E_F$ is the Fermi energy, $\varepsilon \left(\pi ,0\right)$ is the energy in the middle of the Brillouin zone edge, and $\mathrm{\Gamma }$ is a characteristic energy of the fermion-fermion interaction due to the antiferromagnetic fluctuations, the leading particle-hole instability is a $d$-form factor Fermi surface deformation (the Pomeranchuk instability) rather than the charge modulation along the Brillouin zone diagonals predicted within the standard SF model previously. At lower temperatures, we find that the deformed Fermi surface is further unstable to formation of a $d$-form factor charge density wave (CDW) with a wave vector along the Cu-O-Cu bonds (axes of the Brillouin zone). We show that the remnant Coulomb interaction enhances the $d$-form-factor symmetry of the CDW. These findings can explain the robustness of this order in the cuprates. The approximations made in the paper are justified by a small parameter that allows one to implement an Eliashberg-like treatment. Comparison with experiments suggests that in many cuprate compounds the prerequisites for the proposed scenario are indeed fulfilled and the results obtained may explain important features of the charge modulations observed recently.

Figure 1

The typical cuprate Fermi surface with eight hot spots. The arrows represent the direction of the Fermi velocities, while the dashed lines stand for possible CDW wave vectors (see text).

Figure 2

In order for the CDW wave vector $\mathbf{Q}$ to connect the hot spots, the CDW order parameter W, Eq. (2.3), should be localized at the points 1, 2, 3, and 4. Left part is for the order parameter of the modulation along horizontal axis, right is along vertical.

Figure 3

The CDW amplitude from Eq. (2.11) is maximal at the dots in the middle of the edges of BZ.

Figure 4

Definition of ${\mu }_0$. (a) Cut of the electron dispersion near the antinode (region shown by the line in the inset). (b) Dispersion at different temperatures obtained from ARPES experiment [63], with ${\mu }_0$ shown. Note that the pseudogap seen in the low-temperature dispersions is clearly larger than ${\mu }_0$.

Figure 5

BZ with regions 1 and 2.

Figure 6

Pictorial representation of two possible shapes of the Fermi surface below the Pomeranchuk transition and the corresponding intra-unit-cell charge redistributions.

Figure 7

Feynman diagrams for fermionic and bosonic propagators illustrating the approximations used.

Figure 8

$\mathrm{Im}\overline{f}$ as a function of the reduced Matsubara frequency ${\overline{\varepsilon }}_n=\pi \overline{T}(2n+1)$ for different temperatures. $\overline{a}=0.05, \sqrt{\frac{v_s^2/\alpha }{\mathrm{\Gamma }}}=0.5$, and $\overline{\mu }=0.05$.

Figure 9

${\overline{T}}_{\mathrm{Pom}}\left(\overline{\mu }\right)$ (dashed line) and ${\overline{T}}_{\mathrm{diag}}\left(\overline{\mu }\right)$ (dotted line) for $\overline{a}=0.05, \sqrt{\frac{v_s^2/\alpha }{\mathrm{\Gamma }}}=0.5\left(a\right),\text{and}0.1\left(b\right)$.

Figure 10

Dependence of the real (a) and imaginary (b) parts of the Pomeranchuk order parameter $\overline{P}$ on the reduced Matsubara frequency ${\overline{\varepsilon }}_n=\pi \overline{T}(2n+1), \overline{a}=0.05, \overline{\mu }=0.05, \sqrt{\frac{v_s^2/\alpha }{\mathrm{\Gamma }}}=0.5$, and $\overline{T}=0.01$.

Figure 11

Dependence of the Pomeranchuk order parameter $\overline{P}$ on temperature: (a) height of the peak in the real part $0.5(\overline{P}\left(\pi \overline{T}\right)+\overline{P}(-\pi \overline{T}))$ and (b) relative magnitude of the imaginary part given by ${max}_{{\varepsilon }_n}\left|\mathrm{Im}P\left({\varepsilon }_n\right)/\mathrm{Re}f\left({\varepsilon }_n\right)\right|$. The model parameters are $\overline{a}=0.05, \sqrt{\frac{v_s^2/\alpha }{\mathrm{\Gamma }}}=0.5$.

Figure 12

$T_{\mathrm{Pom}}\left(\overline{\mu }\right)$ (dashed line) and ${\overline{T}}_{\mathrm{CDW}}\left(\overline{\mu }\right)$ (dotted line) determined from Eq. (5.14) for $\overline{a}=0.05$ and $\sqrt{\frac{v_s^2/\alpha }{\mathrm{\Gamma }}}=0.5\text{(a)}\text{and}0.1\text{(b)}$.

Figure 13

Value of the Pomeranchuk order parameter at the CDW transition $0.5(\overline{P}\left(\pi \overline{T}\right)+\overline{P}(-\pi \overline{T}))$ for $\overline{a}=0.05, \sqrt{\frac{v_s^2/\alpha }{\mathrm{\Gamma }}}=0.5\left(\mathrm{a}\right),0.1\left(\mathrm{b}\right)$.