Interplay of competing bond-order and loop-current fluctuations as a possible mechanism for superconductivity in kagome metals

Abstract

The pairing symmetry and underlying mechanism for superconducting state of AV${}_3$Sb${}_5$ (A=K, Rb, Cs) kagome metal has been a topic of intense investigation. In this work, we consider an 8-band minimal model, which includes V, and the two types of Sb, both within and above/below the kagome plane. This model captures the Fermi surface pocket with significant in-plane Sb contribution near the zone center, and also has the two types of van Hove singularities (VHS), one of which has a strong out of plane Sb weight. By including V-V and V-planar Sb nearest-neighbor Coulomb interactions, we obtain the susceptibilities for fluctuating bond-order and loop-current in both charge and spin channels, and examine the resulting superconducting instabilities. In particular, we find that the time-reversal odd (even) charge-loop-current (charge bond-order) fluctuations favor unconventional (conventional) pairing symmetry such as $s_{+-}$ and $d+id$ ($s_{++}$). Recent experimental works have highlighted the presence of $s$-wave pairing with two distinct gaps, one isotropic and one anisotropic. We discuss how this scenario may be compatible with either $s_{++}$ or $s_{+-}$ pairing, with an isotropic gap on the pocket dominated by in-plane Sb, but a highly anisotropic gap on V-dominated bands.

Figures (4)

• Figure 1: (a) The kagome plane, with V at $A,B,C$, in-plane-Sb at $S$, out-of-plane-Sb at the + positions within the grey unit cell. The green arrows $\Hat{\text{w}}_{A/B/C}$ show the different bond directions. (b) In the Brillouin zone reciprocal lattice vectors $Q_{AB}\equiv M_{C}$ and so on (same colour). (c) The electronic band structure of the 8-band model with the P- and M-type VHSs (blue and pink dots). (d) The 3-sheet Fermi surface for $\mu=3.88$ (dashed red line), the color scale denotes the weights of V-orbitals at sub-lattices $A,B,C$, and the in-plane-Sb orbital at $S$. We define, $\mathbf{x}_C=\mathbf{R}_1$, $\mathbf{x}_A=\mathbf{R}_2-\mathbf{R}_1$, $\mathbf{x}_B=-\mathbf{R}_2$.
• Figure 2: The RPA corrected susceptibilities (at $\mu=3.88$) for: bond-orders $\chi_{\mathcal{BB}}(\mathbf{q})$ ((a) between V-V, (e) between V-Sbip), and loop-currents $\chi_{\mathcal{JJ}}(\mathbf{q})$ ((c) between V-V, (g) between V-Sbip). Columns 1-5 show the susceptibilities in the charge and spin channels (different color scales) with increasing interaction strengths (shades of colors) as the relative strength of the charge and spin vertex is tuned by $\alpha$. Only the $\chi$’s in the disordered state are shown, some larger interaction strengths (or darker color shades) are omitted as those channels would condense. In (a,c) $\chi_{\mathcal{BB}}^{\texttt{i},\text{vv}}$ exceeds $\chi_{\mathcal{JJ}}^{\texttt{i},\text{vv}}$, and in (e,g) $\chi_{\mathcal{JJ}}^{\texttt{i},\text{sv}}$ exceeds $\chi_{\mathcal{BB}}^{\texttt{i},\text{sv}}$, when compared at the same interaction strengths (having same color shade). The ordered state resulting from condensation at the three $\mathbf{q}=M_{c}$ (or $3\mathbf{Q}$ states) for each BO & LC channel is shown on the right in (b,d,f,h). (i) A charge (spin) current corresponds to a locally conserved density $\mathds{1}$ (spin $S^{\textsf{a}}$), with spin up/down (in $\textsf{a}$-basis) flowing in the same (opposite) directions. (j) Fluctuations computed in the disordered state (blue arrow) where r is pressure/doping axis.
• Figure 3: SC phase diagram: (a) as a function of the strength of the different fluctuating channels parameterized by $\alpha$ (charge $\leftrightarrow$ spin), $\beta$ (bond-order $\leftrightarrow$ loop-current), $\gamma$ (V-V $\leftrightarrow$ V-Sbip) for an interaction scale $\xi=4$. (b) Sub-classification of the $A_{1g}$ phase into the $s_{++}$ type and $s_{+-}$ type based on the value of $\Tilde{\Delta}_{\text{in}}/\Tilde{\Delta}_{\text{out}}$ (defined in text). (c) The leading and sub-leading SC instabilities for two different interaction scales $\xi=4$ (circles), $\xi=5$ (squares). The gap structure of the leading SC instability for fluctuations of: (d) V-V cBO ($\alpha=\beta=\gamma=0$), (e) V-V cLC ($\alpha=\gamma=0, \beta=1$), (f) V-Sbip cBO ($\alpha=\beta=0, \gamma=1$), (g) V-Sbip cLC ($\alpha=0, \beta=\gamma=1$).
• Figure 4: (a1-a3) The $s$-wave ($A_{1g}$) gaps $\Delta_{\phi}$ of Figs. \ref{['fig:Fig_3_main_text']} (d, f, g), as a function of angle on the three Fermi sheets (see inset). (b) Triplet p-wave ($E_{1u}$) disappears on increasing $\xi$.