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

TL;DR

The paper addresses the origin and 3D structure of density-wave order in kagome AV3Sb5 metals. It extends the paramagnon-interference mechanism, captured by Aslamazov-Larkin vertex corrections, to a realistic 3D multiorbital framework and solves the density-wave equation to predict a commensurate $2×2×2$ bond order with a transition temperature around $T_{\rm BO} \sim 100$ K. The 3D bond order pattern is set by the 3D Fermi surface and by the sign of the third-order Ginzburg-Landau coefficient $b_1$, yielding either a first-order transition to a nematic TrH/SoD $s$-BO or a second-order transition to an alternating vertical BO depending on $b_1$; hole doping tends to favor the TrH state. Overall, the PMI mechanism is confirmed as the essential origin of the 3D CDW in kagome metals and explains the experimentally observed diversity of BO states.

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.

Figures (11)

• Figure 1: (a) 2D wavevector $\bm{q}_m$$(m=1,2,3)$ in the original 2D Brillouin zone. (b) 2D TrH BO that corresponds to ${\bm\phi}\propto (1,1,1)$. In the BO state, the thick purple bond $(i,j)$ corresponds to a decrease in V-V bond length. (c) 2D SoD BO that corresponds to ${\bm\phi}\propto (-1,1,1)$.
• Figure 2: Possible $2\times2\times2$ 3D BO states: (a) "alternating v-BO" with $C_6$ symmetry. (b) "TrH s-BO" with nematic $C_2$ symmetry. The red arrow represents the nematic director.
• Figure 3: FS structure of the present kagome metal model for A=Cs. The FS at $n=31$ on the (a) $k_z=0$ plane, (b) $k_z=\pi$ plane, and (c) plane including $\Gamma$KHA points along the dotted line in (a) and (b). The red color denotes the $b_{3g}$-orbital weight. The FS at $n=30.8$ on the (d) $k_z=0$ plane, (e) $k_z=\pi$ plane, and (f) plane including $\Gamma$KHA points.
• Figure 4: FS structure of the present kagome metal models for A=Rb, K at $n=31$. The FS for A=Rb on the (a) $k_z=0$ plane, (b) $k_z=\pi$ plane, and (c) plane including $\Gamma$KHA points. The red color denotes the $b_{3g}$-orbital weight. The FS for A=K on the (d) $k_z=0$ plane, (e) $k_z=\pi$ plane, and (f) plane including $\Gamma$KHA points.
• Figure 5: (a) Charge-channel density-wave (DW) eigenvalue equation. The largest eigenvalue $\lambda_{ {\bm Q} }$ represents the instability of the charge-channel DW at wavevector ${ {\bm Q} }$, and the form factor $f$ gives the order parameter. The wavy lines represent the spin susceptibilities. Two AL terms describe the paramagnon interference mechanism that gives various unconventional DW states. (b-c) Obtained $\lambda_{ {\bm Q} }$ for A=Cs at $n=30.8,31$: (b) $\lambda_{ {\bm Q} }$ in the $q^z=0$ plane, where ${ {\bm q} }_K=\left(\frac{2\pi}{\sqrt{3}},\frac{2\pi}{3}\right)$. (c) Small $q_1^z$-dependence of $\lambda_{{ {\bm Q} }_1}$ for ${ {\bm Q} }_1=({ {\bm q} }_1,q_1^z)$. The BO solution is obtained for any $q_1^z$. (d) $\lambda_{{ {\bm Q} }}$ for $\alpha_S=0.95,0.98$ in $q^z=0$ at $n=31$ of A=Cs. (e) $\alpha_S$ dependence of $\lambda_{{ {\bm Q} }_1}$ for $q_1^z=0$ at $n=31$ of A=Cs.