Charge and current orders in the spin-fermion model with overlapping hot spots

Experiments carried over the last years on the underdoped cuprates have revealed a variety of symmetry-breaking phenomena in the pseudogap state. Charge-density waves, breaking of $C_4$ rotational symmetry as well as time-reversal symmetry breaking have all been observed in several cuprate families. In this regard, theoretical models where multiple nonsuperconducting orders emerge are of particular interest. We consider the recently introduced [Volkov and Efetov, Phys. Rev. B 93, 085131 (2016)] spin-fermion model with overlapping ‘hot spots’ on the Fermi surface. Focusing on the particle-hole instabilities we obtain a rich phase diagram with the chemical potential relative to the dispersion at $(0,\pi );(\pi ,0)$ and the Fermi surface curvature in the antinodal regions being the control parameters. We find evidence for d-wave Pomeranchuk instability, d-form factor charge density waves, as well as commensurate and incommensurate staggered bond current phases similar to the d-density wave state. The current orders are found to be promoted by the curvature. Considering the appropriate parameter range for the hole-doped cuprates, we discuss the relation of our results to the pseudogap state and incommensurate magnetic phases of the cuprates.

Figure 1

Overlapping hot spots on the Fermi surface typical for the hole-doped cuprates.

Figure 2

Illustration of charge modulations for (a) d-form factor CDW with $\mathbf{Q}$ along the diagonal (b) d-wave Pomeranchuk phase.

Figure 3

Illustration of current modulations for (a) DDW (b) IDDW with $\mathbf{Q}$ along the diagonal. Only the currents between nearest neighbors are depicted.

Figure 4

Phase diagram of the model (3.1). ‘P’ is for the d-wave Pomeranchuk instability, ‘CDW(diag)’ is the d-form factor charge density wave with wave vector along the BZ diagonal, and ‘DDW’ (‘IDDW') stand for (incommensurate) d-density wave [$\mathbf{Q}\ne 0$ in (3.3)]. Axial CDW can emerge from Pomeranchuk or DDW phases at a lower temperature (see text).

Figure 5

Dependence of the IDDW incommensurability ${(\beta Q^2/T)}_{\max }$ on $\beta /\alpha$ for $\mu /T=0.1$. Dashed line is ${(\beta Q^2/T)}_{\max }$ for $\mathbf{Q}$ along the diagonal. Vertical lines mark transitions between different phases. In the leftmost region (D) $\mathbf{Q}$ has diagonal orientation, in the middle (A) axial, and in the rightmost DDW is commensurate.

Figure 6

Diagrammatic structure of approximations used; $a$ and $b$ correspond to different regions (1 or 2).

Figure 7

Leading particle-hole instability temperatures for the SF model. (a) $a=0.05$, $v_s^2/\alpha =0.05$, $\mu =0.02$, (b) $a=0.05$, $v_s^2/\alpha =0.05$, $\mu =0.05$. Gray dashed line is a guide to the eye.

Figure 8

Self-energy diagrams of the same order (a) without (b) with vertex corrections.