Bond Order in Two-Dimensional Metals with Antiferromagnetic Exchange Interactions

We present an unrestricted Hartree-Fock computation of charge-ordering instabilities of two-dimensional metals with antiferromagnetic exchange interactions, allowing for arbitrary ordering wave vectors and internal wave functions of the particle-hole pair condensate. We find that the ordering has a dominant $d$ symmetry of rotations about lattice points for a range of ordering wave vectors, including those observed in recent experiments at low temperatures on ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$. This $d$ symmetry implies the charge ordering is primarily on the bonds of the Cu lattice, and we propose incommensurate bond order parameters for the underdoped cuprates. The field theory for the onset of Néel order in a metal has an emergent pseudospin symmetry which “rotates” $d$-wave Cooper pairs to particle-hole pairs [M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010)]; our results show that this symmetry has consequences even when the spin correlations are short ranged and incommensurate.

Figure 1

Fermi surface of the hole-doped cuprates, showing the Brillouin zone boundary to antiferromagnetism at $\mathit{K}=(\pi ,\pi )$ (dashed lines), the hot spots (filled circles), and the wave vectors ($Q_0$, 0) and ($Q_0$, $Q_0$).

Figure 2

Plot of ${\lambda }_{\mathit{Q}}/A$, where ${\lambda }_{\mathit{Q}}$ is the smallest charge order eigenvalue, as a function of $Q_x$ and $Q_y$. We used $\mu =-1.11856$, $\xi =2$, $T=0.06$, $\delta =1/4$, and $L=64$. Charge order appears when ${\lambda }_{\mathit{Q}}<-1$, which happens when $A$ is large enough. The global minimum is at ($Q_m$, $Q_m$), and $Q_m$ is plotted in Fig. 3 as a function of $\mu$. Notice also the valleys extending from ($Q_m$, $Q_m$) to ($Q_m$, 0) and (0, $Q_m$). The region with time reversal $\mathcal{T}$ preserved has the eigenfunctions ${\Delta }_{\mathit{Q}}(-\mathit{k})={\Delta }_{\mathit{Q}}(\mathit{k})$ which are predominantly $d$, while the region with $\mathcal{T}$ broken has ${\Delta }_{\mathit{Q}}(-\mathit{k})=-{\Delta }_{\mathit{Q}}(\mathit{k})$, as shown for some values of $\mathit{Q}$ in Table .

Figure 3

Plot of $Q_m$ (circles), where the minimum of ${\lambda }_{\mathit{Q}}$ occurs at $\mathit{Q}=(\pm Q_m,\pm Q_m)$. Also shown are the corresponding values of $Q_0$ (squares), as defined by the hot spots on the Fermi surface in Fig. 1. The near equality of $Q_m$ and $Q_0$ is evidence for the pseudospin symmetry; note that this holds even though ${\chi }_0(\mathit{q})$ in Eq. (7) is peaked at the wave vectors $\mathit{K}=(\pi ,\pm 3\pi /4),(\pm 3\pi /4,\pi )$, as is the case in many hole-doped cuprates.