Emergence of charge order in a staggered loop-current phase of cuprate high-temperature superconductors

We study the emergence of charge-ordered phases within a $\pi$-loop-current $\left(\pi \mathrm{LC}\right)$ model for the pseudogap based on a three-band model for underdoped cuprate superconductors. Loop currents and charge ordering are driven by distinct components of the short-range Coulomb interactions: loop currents result from the repulsion between nearest-neighbor copper and oxygen orbitals, while charge order results from repulsion between neighboring oxygen orbitals. We find that the leading $\pi \mathrm{LC}$ phase has an antiferromagnetic pattern similar to previously discovered staggered flux phases, and that it emerges abruptly at hole dopings $p$ below the Van Hove filling. Subsequent charge-ordering tendencies in the $\pi \mathrm{LC}$ phase reveal that diagonal $d$-charge density waves (dCDWs) are suppressed by the loop currents while axial order competes more weakly. In some cases we find a wide temperature range below the loop-current transition, over which the susceptibility towards an axial dCDW is large. In these cases, short-range axial charge order may be induced by doping-related disorder. A unique feature of the coexisting dCDW and $\pi \mathrm{LC}$ phases is the emergence of an incommensurate modulation of the loop currents. If the dCDW is biaxial (checkerboard) then the resulting incommensurate current pattern breaks all mirror and time-reversal symmetries, thereby allowing for a polar Kerr effect.

Figure 1

Unit supercell and Brillouin zone. (a) The supercell (shaded region) contains two ${\mathrm{CuO}}_2$ primitive unit cells, with orbitals numbered 1 through 6 as shown. The plus and minus signs indicate the sign convention for the lobes of the $\mathrm{Cu}{d}_{x^2-y^2},\mathrm{O}{p}_x$, and $\mathrm{O}{p}_y$ orbitals. The lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ lead to the folded Brillouin zone shown in (b) (shaded region). Backfolded Fermi surfaces are shown for hole filling $p=0.15$ (thick red lines).

Figure 2

Largest eigenvalue of the static susceptibility matrix $\tilde{\mathbf{X}}\left(\mathbf{q}\right)$ in the normal state at $T=0.010,V_{pd}=2.5$, and $p=0.10$. Throughout the paper, energies and temperatures are in units of $t_{pd}$.

Figure 3

Components of the induced generalized charge density $\delta {\rho }_{i\alpha ,j\beta }$ associated with the ${\mathbf{q}}_0$ eigenmode for the susceptibility shown in Fig. 2. Real part of $\delta {\rho }_{i\alpha ,j\beta }$ for $\alpha \ne \beta$ (left), induced currents $\delta J_{i\alpha ,j\beta }$ (middle), and induced charge modulations $\delta n_{i\alpha }$ (right). Currents with a component of their flow in the positive $x$ direction are deemed positive; currents flowing entirely along the $y$ axis are positive in the positive $y$ direction. $\delta {\rho }_{i\alpha ,j\beta }$ is calculated from Eq. (37) with the prefactor $\chi \left(\mathbf{q}\right){\phi }_{\mathbf{q}}$ set to 1; the color scale is therefore arbitrary.

Figure 4

As in Fig. 3, but for the peak in Fig. 2 at ${\mathbf{q}}_1=(0.84,0.84)$. This case corresponds to a diagonal dCDW with no circulating currents.

Figure 5

Effects of $V_{pd},V_{pp}$, and $U_p$ on the susceptibility eigenvalue $\chi \left(\mathbf{q}\right)$. Plots show cuts along the Brillouin zone diagonal, $\mathbf{q}=(q,q)$. Results are for (a) fixed $V_{pp}=1.0$ and varying $V_{pd}$; (b) fixed $V_{pd}=2.5$ and varying $V_{pp}$. $V_{pd}$ enhances the loop-current susceptibility at ${\mathbf{q}}_0$, while $V_{pp}$ enhances the charge susceptibility peak at ${\mathbf{q}}_1$. The inset shows that $U_p$ opposes charge order but has no effect on the $\pi \mathrm{LC}$ susceptibility. $U_d$ has no effect on either the $\pi \mathrm{LC}$ or the dCDW susceptibility (not shown). Results are for $T=0.025$ (in units of $t_{pd})$ and hole density $p=0.12$.

Figure 6

Phase diagram of the three-band model from linear response theory. The figure shows the doping dependence of the leading instability temperature of the generalized susceptibility $\tilde{\mathbf{X}}\left(\mathbf{q}\right)$. The Van Hove hole density $p_{\mathrm{VH}}$ is indicated by a vertical dashed line, and representative Fermi surfaces for $p<p_{\mathrm{VH}}$ and $p>p_{\mathrm{VH}}$ are shown in the insets. The leading instability is to a $\pi \mathrm{LC}$ for $p<0.154$ and to a charge-ordered state for $p>0.154$. There are two charge-ordered states shown: a diagonal dCDW below $T_{\mathrm{diag}}$ for $p<p_{\mathrm{VH}}$, and a nematic phase below $T_{\mathrm{nem}}$ for $p>p_{\mathrm{VH}}$. The nematic phase consists of a translationally invariant intra-unit-cell transfer of charge between adjacent oxygen orbitals [25]. Results are for $V_{pd}=3.0,V_{pp}=1.0$.

Figure 7

Self-consistent Hartree-Fock results for the orbital current as a function of hole filling $p$, temperature $T$, and interaction $V_{pd}$. The current $J_{pd}$ along the $p\text{−}d$ bond is shown as a function of (a) $T$ for $V_{pd}=3.0$ and (b) $p$ for various $V_{pd}$ at $T=0.001$. The ratio of the $p\text{−}p$ and $p\text{−}d$ bond currents is $J_{pp}/J_{pd}=0.32$, independently of $T,p$, and $V_{pd}$. It was previously found to depend on the ratio of $t_{pp}/t_{pd}$ [73]. The current is in units of $e{t}_{pd}/\hslash$, with $e$ the electron charge.

Figure 8

Temperature evolution of the peaks in $\chi \left(\mathbf{q}\right)$ at ${\mathbf{q}}_0,{\mathbf{q}}_1$, and ${\mathbf{q}}_2$ at four different fillings: (a) $p=0.10$, (b) $p=0.12$, (c) $p=0.14$, (d) $p=0.15$. The corresponding mean-field current along the $p\text{−}d$ bond is also shown in each figure (left scale). The three wave vectors are ${\mathbf{q}}_0=(0,0),{\mathbf{q}}_1=(q_1,q_1),{\mathbf{q}}_2=(q_2,0)$, where $q_1$ and $q_2$ are the peak positions along diagonal and axial directions. Fermi surfaces corresponding to the different hole fillings are shown in (e)–(h) for the normal state (red lines) and for the $\pi \mathrm{LC}$ state (black lines). As shown in (e), the axial wave vector connects tips of the hole pockets in the $\pi \mathrm{LC}$ phase.

Figure 9

Largest eigenvalue of the susceptibility matrix in the $\pi \mathrm{LC}$ phase. Results are at $p=0.15$ for $V_{pd}=3.0,V_{pp}=1.0$, and at $T=0.021$.

Figure 10

Left panel: Induced shift in the real part of $\langle c_{i\alpha }^{†}{c}_{j\beta }\rangle$, middle panel: the bond current, and right panel: the charge density. Parameters are as in Fig. 9. The charge modulations on the Cu sites are $\sim 15\%$ of those on the O sites.