Prediction of pseudogap formation due to $d\text{-wave}$ bond-order in organic superconductor $\kappa \text{−}{\left(\text{BEDT-TTF}\right)}_2\mathrm{X}$

Rich hidden unconventional orders with pseudogap formation, such as the intersite bond order (BO), attract increasing attention in condensed matter physics. Here, we investigate the hidden order formation in organic unconventional superconductor $\kappa \text{−}{\left(\text{BEDT-TTF}\right)}_2\mathrm{X}$. We predict the formation of $d$-wave BO at wavelength $\mathit{q}={\mathit{Q}}_{\mathrm{B}}=(\delta ,\delta )$ ($\delta =0.38\pi$) for the first time, based on both the functional renormalization group (fRG) and the density-wave equation theories. The origin of the BO is the quantum interference among antiferromagnetic spin fluctuations. This prediction leads to distinct pseudogap-like reduction in the NMR $1/T_1$ relaxation rate and in the density-of-states, consistently with essential experimental reports. The present theory would be applicable for other strongly correlated metals with pseudogap formation.

Figure 1

(a) Anisotropic triangular lattice for $\kappa \text{−}{\left(\text{BEDT-TTF}\right)}_2\mathrm{X}$. (b) Band dispersion and (c) the FS at $n=1$. The major nesting vector is ${\mathit{Q}}_{\mathrm{S}}\approx (\pi -\delta /2,\pi -\delta /2)$ and the minor nesting vector is ${\mathit{Q}}_{\mathrm{B}}\approx (\delta ,\delta )$ with $\delta \approx 3\pi /8$. (d) Differential equation of the fRG theory. The crossed line represents electron propagator inside of the energy cutoff ($|{\epsilon }_{{\mathit{k}}^{\prime }}|<{\mathrm{\Lambda }}_l$), while the slashed line denotes on-shell one ($|{\epsilon }_{\mathit{k}}|={\mathrm{\Lambda }}_l$).

Figure 2

The $\mathit{q}$ dependencies of (a) ${\chi }^{\mathrm{S}}\left(\mathit{q}\right)$ and (b) ${\chi }^{\mathrm{BO}}\left(\mathit{q}\right)$ obtained by the $\mathrm{RG}+\mathrm{cRPA}$ method at $U=3.5$. (c) The RG flow for spin and BO susceptibilities at $U=3.54$. (d) Obtained optimized SC gap function, which belongs to $d_{x^2-y^2}$-wave symmetry. (e) Schematic BO pattern at $\mathit{q}={\mathit{Q}}_{\mathrm{B}}$ in the real space, where $\mathit{\lambda }\approx (8/3,8/3)$ is the wave vector.

Figure 3

Obtained RG results in the case of ${\omega }_c^{\mathrm{pp}}=10T$ at $U=3.82$. (a) Obtained RG flow for ${\chi }^{\mathrm{BO}}\left({\mathit{Q}}_{\mathrm{B}}\right)$ and ${\chi }^{\mathrm{S}}\left({\mathit{Q}}_S\right)$. (b) Obtained pairing interaction $V^{\mathrm{SC}}(\mathit{p},{\mathit{p}}^{\prime })$ due to the spin-fluctuation-mediated repulsion [for $(\mathit{p},{\mathit{p}}^{\prime })=(33,16)]$ and the BO fluctuation-mediated attraction [for $(\mathit{p},{\mathit{p}}^{\prime })=(62,31)]$. The patch number $\mathit{p}$ and ${\mathit{p}}^{\prime }$ are shown in Fig. 1.

Figure 4

(a) The DW equation for wave vector $\mathit{q}$. ${\mathit{k}}_{\pm }\equiv \mathit{k}\pm \mathit{q}/2$ and ${\mathit{p}}_{\pm }\equiv \mathit{p}\pm \mathit{q}/2$. The kernel function is composed of the mean field, MT, and AL terms 1 and 2. These diagrams are also produced systematically by solving the RG equation. (b) Obtained eigenvalue of the BO ${\lambda }_{\mathit{q}}^{\mathrm{DW}}$ and (c) the form factor $f_{\mathit{q}}\left(\mathit{k}\right)$ at $\mathit{q}={\mathit{Q}}_{\mathrm{B}}$ and ${\epsilon }_n=\pi T$. (d) Obtained Fermi arc structure in the unfolded zone, and the pseudogap in the DOS with $f^{\max }=0.1$ is shown in the inset. (e) Obtained $1/T_{1}T$, where $T^{*}$ is the BO transition temperature.