Numerical analysis of three-band models for CuO planes as candidates for a spontaneous T-violating orbital current phase

Recently, we have numerically evaluated the current-current correlation function for the ground states of three-band models for the CuO planes of high-$T_{\mathrm{c}}$ superconductors at hole doping $x=1∕8$ using systems with 24 sites and periodic boundary conditions. In this paper, the numerical analysis is extended to a wider range of parameters. Our results show no evidence for the time-reversal symmetry violating current patterns recently proposed by Varma [C. M. Varma, Phys. Rev. B 73, 155113 (2006)]. If such current patterns exist, our results indicate that the energy associated with the loop currents must be smaller than $5\mathrm{meV}$ per link even if the on-site chemical potential on the oxygen sites, which is generally assumed to be of the order of ${ϵ}_{\mathrm{p}}-{ϵ}_{\mathrm{d}}=3.6\mathrm{eV}$, is taken to 1.8, 0.9, 0.4, and finally $0\mathrm{eV}$. The current-current correlations remain virtually unaffected as we increase the interatomic Coulomb repulsion $V_{\mathrm{pd}}$, the term driving the system into the current carrying phase in Varma’s analysis, from $1.2\text{to}2.4\mathrm{eV}$. Assuming that the three-band models are adequate, quantum critical fluctuations of such patterns hence cannot be responsible for phenomena occurring at significantly higher energies, such as superconductivity or the anomalous properties of the pseudogap phase. We further derive an upper bound for the magnetic moment per unit cell from the upper bound we obtain for spontaneous currents, and find it smaller than the magnetic moment measured in a recent neutron scattering experiment. In this context, we observe that if the observed magnetic moments were due to a current pattern, the magnitude of these currents would be insufficient to determine the phase diagram. Finally, we discuss the role of finite size effects in our numerical experiments. In particular, we show that the net spin $1∕2$ of our finite size ground states does not infringe on the validity of our conclusions.

Figure 1

Orbital current pattern proposed by Varma.

Figure 2

We define the phases $q_1$ and $q_2$ acquired by translations shown in (a). For a system of eight unit cells, i.e., 8 Cu and correspondingly 16 O, the phases can have the values $q_1=n_1\pi ∕2$ and $q_2=n_2\pi$ with $n_1∊\{0,1,2,3\}$ and $n_2∊\{0,1\}$. For the Brillouin zone depicted in (b), $q_1$ and $q_2$ translate into $x$ and $y$ momenta by $q_x=q_1$ and $q_y=q_1-q_2$. Thus, the phase description $(q_1,q_2)$ of the $\Gamma$ and $M$ point is (0, 0) and $(\pi ,0)$, respectively.

Figure 3

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ and in units of $\frac{eV}{\hslash }$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=3.6$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster (8 $\mathrm{Cu}=\mathrm{open}$ circles, 16 $\mathrm{O}=\text{filled}$ circles) with periodic boundary conditions (PBCs). The reference link is indicated in the top and (due to the PBCs) bottom left corner. Except for the vertical lines, positive numbers indicate alignment with the pattern shown in Fig. 1.

Figure 4

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=3.6$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster (8 $\mathrm{Cu}=\mathrm{open}$ circles, 16 $\mathrm{O}=\text{filled}$ circles) with PBCs. The Cu-O reference link is indicated in the top corner.

Figure 5

Circulating current $j_{\mathrm{pp}}$ on a neighboring O triangle according to the pattern in Fig. 1. Lengths are given in units of the Cu-Cu distance $a_0$.

Figure 6

An orbital current pattern proposed earlier by Varma, which has subsequently been ruled out by experiment.

Figure 7

Spin-spin correlations multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=3.6$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster (8 $\mathrm{Cu}=\mathrm{open}$ circles, 16 $\mathrm{O}=\text{filled}$ circles). Due to PBCs the O reference site is in all corners of the plot and indicated by a big black circle.

Figure 8

Spin-spin correlations multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=3.6$, $V_{\mathrm{pd}}=1.2$ with PBCs. The Cu reference site to the upper left is indicated by a big black circle.

Figure 9

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=1.8$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster with PBCs (reference link indicated in the top and bottom left corners).

Figure 10

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=0.9$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster with PBCs.

Figure 11

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=0.4$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster with PBCs.

Figure 12

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=0.4$, $V_{\mathrm{pd}}=2.4$ on a 24 site cluster with PBCs.

Figure 13

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=0.0$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster with PBCs.

Figure 14

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=0.0$, $V_{\mathrm{pd}}=1.2$ on a 24 site cluster with PBCs. The Cu-O reference link is indicated by a black arrow in the upper left corner.

Figure 15

Current-current correlations $⟨j_{k,k+\widehat{x}}{j}_{l,m}⟩$ multiplied by ${10}^2$ for the ground state of Eq. (2) with ${ϵ}_{\mathrm{p}}=0.0$, $V_{\mathrm{pd}}=2.4$ on a 24 site cluster with PBCs.