Microscopic origin of period-four stripe charge-density-wave in kagome metal CsV$_3$Sb$_5$

TL;DR

The paper addresses the microscopic origin of the $4a_0$ stripe CDW observed in kagome superconductors like CsV$_3$Sb$_5$ by embedding a static $2\times 2$ BO into a $12$-site extended kagome model and analyzing CDW instabilities with a linearized density-wave equation that includes MT and AL vertex corrections. It demonstrates that BO-induced Fermi-surface reconstruction creates a new nesting vector leading to a $4a_0$ CDW, whose real-space pattern features dominant long-range bond modulations and on-site potentials in qualitative agreement with STM measurements. The work suggests that paramagnon interference within this framework explains the stripe CDW and its interplay with nonreciprocal transport phenomena like eMChA and the superconducting diode effect, and it points to a quantitative refinement via a two-orbital model for material-specific behavior.

Abstract

The interplay between unconventional density waves and exotic superconductivity has attracted growing interest. Kagome superconductors $A\rm{V}_3\rm{Sb}_5$ ($A = \rm{K}, \rm{Rb}, \rm{Cs}$) offer a platform for studying quantum phase transitions and the resulting symmetry breaking. Among these quantum phases, the $4a_0$ stripe charge-density-wave (CDW) has been widely observed for $A=\rm{Rb}$ and $\rm{Cs}$ by scanning tunneling microscopy (STM) and nuclear magnetic resonance (NMR) measurements. However, the microscopic origin of the $4a_0$ stripe CDW remains elusive, and no theoretical studies addressing this phenomenon have been reported so far. In this paper, we propose a microscopic mechanism for the emergence of the $4a_0$ stripe CDW. We analyze the CDW instability in the 12-site kagome lattice Hubbard model with the $2\times2$ bond order driven by the paramagnon-interference mechanism by focusing on the short-range magnetic fluctuations due to the geometrical frustration of kagome lattice. We reveal that the nesting vector of the reconstructed Fermi surface, formed by the $2\times 2$ bond order, gives rise to a $4a_0$-period CDW. Remarkably, the obtained stripe CDW is composed of both the off-site hopping integral modulations and on-site potentials. The real-space structure of the stripe CDW obtained here is in good qualitative agreement with the experimentally observed stripe pattern.

Figures (8)

• Figure 1: Lattice structure, Fermi surface of 3-site and 12-site model, and band structure. (a) Kagome lattice structure of the vanadium layer and Star-of-David (SoD) BO pattern. The red and gray bonds mean the modulation at $+\phi$ and $-\phi$, respectively. The two arrows $t$ and $t'$ shown at the top of the unit cell represent the nearest- and third-nearest-neighbor hopping integrals, with strengths of $-0.5~\rm{eV}$ and $-0.08~\rm{eV}$, respectively. (b) Fermi surface of 3 sites mode at $n=2.8$. $\bm{q}_1,\bm{q}_2$ and $\bm{q}_3$ are the nesting vectors of the $3Q$ BO, connecting van-Hove singularity (vHS) points. (c) Fermi surface of 12-site model under BO at $n=11.2$ (corresponding to $n=2.8$ in the 3-site model), $\phi=0.08~\rm{eV}$. $\bm{Q}$ is the new nesting vector by folded fermi surface. (d) The colored solid lines show the band structure with SoD BO shown in (a), while the dashed lines correspond to the case with $\bm{\phi} =0$, both calculated at filling $n = 11.2$. The band structure near the $\Gamma$ point is split into a 2-to-1 configuration due to the SoD BO. (e) Band structures along the $\eta$ and $\xi$ paths with a saddle point at $S$ point in (c). By $3Q$ BO symmetry, saddle points also exist at the other corners of the hexagonal Fermi surface.
• Figure 2: Diagrammatic representation of linearized DW equation, the development of the eigenvalue corresponding to the $4a_0$ stripe CDW (a) Diagrammatic representation of Linearized DW equation. (b),(c) $\bm{q}$ dependence of the eigenvalue for $\phi=0$ and $\phi=0.08~\rm{eV}$, and the $\bm{q}$-space path. Both analyses are based on the 12-site kagome lattice model. The red, green, and blue lines represent the results for $U=1.00~\rm{eV}$, $U=1.02~\rm{eV}$, $U=1.04~\rm{eV}$ at $n = 11.2$. The $\bm{q}$-space path follows the folded Brillouin zone along the route $\rm{\Gamma} \rightarrow M'_1 \rightarrow K' \rightarrow \Gamma$. All calculations are performed at $T = 0.01~\rm{eV}$.
• Figure 3: Real-space representation of the form factor and the $4a_0$ stripe CDW structure (a) Real-space structure of the form factor. The colors of the bonds and sites represent the amplitude of modulation indicated by the color bar: blue is negative modulation, while red is positive modulation. (b)-(c) show the structures with the constant phase at $\varphi =0,-\pi/4$, respectively. The modulation amplitude is also indicated by the color bar shown in (a).
• Figure 4: Evolution of the irreducible susceptibility with BO (a) Irreducible susceptibility $\tilde{\chi} ^0 _{47,47}(\bm{q})$ of 12 sites model without BO ($n=11.2$). (b) $\tilde{\chi} ^0 _{47,47}(\bm{q})$ of 12 sites model with BO $\phi=0.08 ~\rm{eV}$ ($n=11.2$). The value at the $\rm{M'_1}$ point is obviously enhanced by incorporating BO. The expression of $\tilde{\chi}^0$ is given by Eq. (\ref{['eq:txi']}).
• Figure 5: Band structure, Fermi surface, and $\bm{q}$ dependence of the eigenvalues under the TrH BO. (a) Band dispersion under the TrH BO for $\phi = 0.05~\mathrm{eV}$ and $n = 11.2$. (b) Corresponding Fermi surface. $\bm{Q}$ denotes the nesting vector corresponding to the $4a_0$ periodicity. (c) $\bm{q}$ dependence of the eigenvalues under the TrH BO for $\phi = -0.05~\mathrm{eV}$ and $n = 11.2$. The local maximum at $\bm{q} = \bm{Q}_m$ correspond to the nesting vectors $\bm{Q}_m$ indicated in (b).