Paramagnon-Interference Mechanism for Three-Dimensional Bond Order in Kagome Metals AV$_3$Sb$_5$ (A=Cs, Rb, K): Analysis by the Density-Wave Equation

Abstract

The mechanism of CDW and its 3D structure are important fundamental issues in kagome metals. We have previously shown that, based on a 2D model, $2\times 2$ bond order (BO) emerges due to the paramagnon-interference (PMI) mechanism and that its fluctuations lead to $s$-wave superconductivity. This paper studies these issues based on realistic 3D models of kagome metals AV$_3$Sb$_5$ (A=Cs, Rb, K). We reveal that a commensurate 3D $2\times 2\times 2$ BO is caused by the PMI mechanism, by performing the 3D density-wave (DW) equation analysis for all A=Cs, Rb, K models in detail. Our results indicate a BO transition temperature $T_{\rm BO}\sim 100$K within the regime of moderate electron correlation. The 3D structure of BO is attributed to the three-dimensionality of the Fermi surface, while the 3D structure of BO is sensitively changed, since the Fermi surface is quasi-2D. Based on the analysis of the DW equation, by taking into account a finite third-order Ginzburg-Landau (GL) term, (i) shift stacking $2\times 2\times 2$ BO can be realized via a first-order transition below $T_{\rm BO}$. Here, the in-plane BO pattern (tri-hexagonal or star-of-David) is determined by the sign of the third-order GL term, with hole doping tending to favor the tri-hexagonal state. On the other hand, if the third-order GL term is very small, (ii) alternating vertical stacking BO may instead be realized via a second-order transition. The present study enhances our understanding of the rich variety of BOs observed experimentally. It is confirmed that the PMI mechanism is the essential origin of the 3D CDW of kagome metals.