Mean-field analysis of intra-unit-cell order in the Emery model of the CuO${}_2$ plane

Motivated by recent experiments on high-$T_c$ cuprate superconductors pointing toward intra-unit-cell (IUC) order in the pseudogap phase, we investigate three distinct intra-unit-cell-ordering possibilities: nematic, nematic-spin-nematic, and current-loop order. The first two are Fermi-surface instabilities involving a spontaneous charge and magnetization imbalance between the two oxygen sites in the unit cell, respectively, while the third describes circulating currents within the unit cell. We analyze the three-band Emery model of a single CuO${}_2$ layer including various on-site and nearest-neighbor interactions within a self-consistent mean-field approach. We show how these on-site and further-neighbor repulsions suppress or enhance particular IUC orders. In particular, we show that the attractive interactions necessary for nematic and nematic-spin-nematic orders in one-band models have their natural microscopic origin in the O-O on-site and nearest-neighbor repulsions in the three-band model. Finally, we find that while the nematic and nematic-spin-nematic orders cannot coexist in this framework, the loop-current order can coexist with nematic order.

Figure 1

The unit cell of the CuO${}_2$ plane with the copper $d_{x^2-y^2}$ in the middle surrounded by the oxygen $p_x$ and $p_y$ orbitals. Also shown are the different hopping as well as interaction parameters used in the Emery model.

Figure 2

Static Fermi-surface instabilities analyzed in this work: (a) nematic phase breaking $C_4$ symmetry, (b) nematic-spin-nematic and (c) ferromagnetic instability. In (b) and (c), the solid and dashed lines denote the up- and down-spin band.

Figure 3

The critical strength of the oxygen-oxygen interaction $V_{pp}^c$ needed in order to enter a nematic phase as a function of hole density $n$ for $t_{pp}=0$. For numerical reasons, the calculation has been carried out at $T=5\times {10}^{-4}\left[t_{pd}\right]$. The dashed lines denote the values of $V_{pp}$ used for Fig. 5.

Figure 4

Critical interaction strength for different values of the oxygen-oxygen hopping $t_{pp}$.

Figure 5

(a) Phase diagram for the different values of the O-O nearest-neighbor interaction $V_{pp}=2$, $1.75$, and $1.5$. At low temperature, there would be first-order transitions, only shown for $V_{pp}=2$ by the solid lines, before the normal state becomes unstable (dashed lines). Figs. (b)–(d) show the free energy as a function of $\eta$ for $n=1.05$, $n=1.095$, and $n=1.105$ at $T=0.001$, illustrating the first-order character of the low-temperature transition. (e) The nematic order parameter $\eta$ as a function of temperature for $U_d=9$ and $n=1.05$ showing the second-order transition at $T_c=0.027$.

Figure 6

Phase diagram for $V_{pp}=1.75$ and different Cu on-site interaction strengths $U_d=8,9,10$. Shown are again only “second-order” phase boundaries.

Figure 7

Critical oxygen interaction strength for a nematic-spin-nematic (${\eta }_s$) and a magnetic ($m$) instability on the oxygen sites. Here, $U_d=9$, $V_{pd}=V_{pp}=1$, and $t_{pp}=0$.

Figure 8

The two bubble diagrams involved in the linearized self-consistency equation, where we have used the short notation ${\langle \delta \mathcal{H}\rangle }_{\alpha \beta }=\langle \alpha \left|\delta \mathcal{H}\right|\beta \rangle$.

Figure 9

The different current patterns arising from the operators ${\mathcal{A}}_{1-4s}$ in Eqs. (42) and (43) and ${\mathcal{D}}_{1-4s}$ in Eqs. (46, 47, 48). Combining ${\mathcal{A}}_{2s}$ with ${\mathcal{D}}_{2s}$ leads to the loop-current phase ${\Theta }_{\mathrm{I}}$, while ${\mathcal{A}}_{3s}$ (${\mathcal{A}}_{4s}$) combined with ${\mathcal{D}}_{3s}$ (${\mathcal{D}}_{4s}$) leads to ${\Theta }_{\text{II}}$.

Figure 10

Critical interactions $(V_{pp}^c,V_{pd}^c)$ for $U_d=9$, $U_p=3$, $t_{pp}=0.1$, $\Delta =2.5$, and different hole densities. The dashed and dotted lines for $n=0.9$ illustrate the influence of the Cu on-site interaction and the charge-transfer gap.