$d$- and $p$-wave quantum liquid crystal orders in cuprate superconductors, $\kappa$-(BEDT-TTF)$_2$X, and coupled chain Hubbard models: functional-renormalization-group analysis

Unconventional symmetry breaking without spin order,such as the rotational symmetry breaking (=nematic or smectic) orders as well as the spontaneous loop-current orders, have been recently reported in cuprate superconductors and their related materials.They are theoretically represented by non-$A_{1g}$ symmetry breaking in self-energy, which we call the form factor $f_{k,q}$.In this paper, we analyze typical Hubbard models by applying the renormalization-group (RG) method, and find that various unconventional ordering emerges due to the quantum interference among spin fluctuations. Due to this mechanism,nematic ($q=0$) and smectic ($q \ne 0$)bond orders with $d$-wave form factor appear $f_{k,q}\propto \cos k_x - \cos k_y$ in both cuprates and $\kappa$-(BEDT-TTF)$_2$X. The derived bond orders naturally explain the pseudogap behaviors in these compounds. The quantum interference also induces various current orders with odd-parity form factor. For example, we find the emergence of the charge and spin loop-current orders with $p$-wave form factor in geometrically frustrated Hubbard models. Thus, rich quantum phase transitions with $d$- and $p$-wave form factors are driven by the paramagnon interference in many low-dimensional Hubbard models.