Functional-Renormalization-Group Analysis on Electron Nematic State in Cuprate Superconductors

To elucidate the nematic phase transition recently observed at the pseudogap temperature $T^*$ in cuprate superconductors, we study the charge susceptibilities by utilizing the improved functional-renormalization-group method [1] to the $d$-$p$ Hubbard model. We reveal that the most dominant charge fluctuation is the uniform ($q=0$) charge modulation on the $p_x$ and $p_y$ orbitals with antiphase ($d$-symmetry) form factor. The spontaneous symmetry breaking with respect to the occupation of $p_x$ and $p_y$ orbitals with the wavevector $q=0$ accounts for the electronic nematic phase transition at $T^*$ in cuprates. In addition, we find that the $p$-orbital density wave instability at the wavevector $Q_{\rm{a}} \approx (0.3\pi, 0)$ is further enhanced by the presence of the $q=0$ nematic ordering, consistently with experimental observations of the density-wave states inside the pseudogap region. \\ \ [1] M. Tsuchiizu, Y. Yamakawa, H. Kontani, Phys. Rev. B \textbf{93}, 155148 (2016).