Sublattice interference promotes pair density wave order in kagome metals

Motivated by the observation of a pair density wave (PDW) in the kagome metal $\mathrm{CsV}{}_3\mathrm{Sb}{}_5$, we consider the fate of electrons near a $p$-type Van Hove singularity (vHS) of the kagome lattice with on-site Hubbard $U$ and the nearest-neighbor repulsion $V$. We study the effect of such interactions on Fermi surface “patches” at the vHS. We show how a feature unique to the kagome lattice known as sublattice interference crucially affects the form of the interactions among the patches. The renormalization group (RG) flow of such interactions results in a regime where the nearest-neighbor interaction $V$ exceeds the on-site repulsion $U$. We identify this condition as being favorable for the formation of charge-density-wave (CDW) and PDW orders. In the weak-coupling limit, we find a complex CDW order as the leading instability, which breaks time-reversal symmetry. Beyond RG, we perform a Hartree-Fock study to a $V$-only model and find the pair-density-wave order indeed sets in at some intermediate coupling.

Figure 1

Sublattice interference in the kagome lattice. (a) Each unit cell has $\alpha =\mathrm{A}$, B, C different sublattice sites. (b) Band structure of the tight-binding model. We focus on the middle band, highlighted with red color, which exhibits a $p$-type Van Hove singularity. (c) For this band, the transformation matrix $u_{\alpha }\left(\mathit{k}\right)$, defined as $c_{\alpha }\left(\mathit{k}\right)=u_{\alpha }\left(\mathit{k}\right){\psi }_{\alpha }\left(\mathit{k}\right)$ where $c_{\alpha }$ and ${\psi }_{\alpha }$ are lattice and band fermions respectively, has a modulated distribution in the Brillouin zone.

Figure 2

(a) All symmetry allowed interactions in the six-patch model. (b) RG flow for the interactions and the order parameter vertex $\mathrm{\Gamma }$ with the SI effect. We set $U\left(0\right)=0.8t$ and $V\left(0\right)=0.3t$ and starting from some intermediate energy scale we have $V\gg U$. The resulting weak-coupling instabilities are degenerate complex ${\mathrm{CDW}}_{-}$ and imaginary ${\mathrm{SDW}}_{+/-}$. (c) The same RG flow but without SI effect. In this case all the $g_{ij,0}$ are set by the largest Hubbard $U$, and there does not exist a constant map $U,V$, and $g_{ij}\left(y\right)$. The leading weak-coupling instability is the $d$-wave SC [35, 36].

Figure 3

(a) Comparison between ${\mathrm{\Pi }}_{pp}^{\left(0\right)}\left(\mathit{Q}\right)$ and ${\mathrm{\Pi }}_{ph}^{\left(0\right)}\left(\mathit{Q}\right)$ as a function of temperature $T$. The fact that ${\mathrm{\Pi }}_{pp}^{\left(0\right)}\left(\mathit{Q}\right)>{\mathrm{\Pi }}_{ph}^{\left(0\right)}\left(\mathit{Q}\right)$ at larger $T$ indicates that PDW order may win over CDW order when interaction is sufficiently strong. (b) The weak-coupling instabilities (at perfect nesting) when other interactions such as $J_0$ and $K_0$ are taken into account. (c) RG flows for the lattice interactions for $U\left(0\right)=0.8t$ and $V\left(0\right)=0.3t$ and for various $J_0$ and $K_0$. In all cases there is a wide range in energy scale where $V$ is the largest.