Analysis of Charge Order in the Kagome Metal $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K},\mathrm{Rb},\mathrm{Cs}$)

Motivated by the recent discovery of unconventional charge order, we develop a theory of electronically mediated charge density wave formation in the family of kagome metals $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K},\mathrm{Rb},\mathrm{Cs}$). The intertwining of van Hove filling and sublattice interference suggests a three-fold charge density wave instability at ${\mathrm{T}}_{\mathrm{CDW}}$. From there, the charge order forming below ${\mathrm{T}}_{\mathrm{CDW}}$ can unfold into a variety of phases capable of exhibiting orbital currents and nematicity. We develop a Ginzburg Landau formalism to stake out the parameter space of kagome charge order. We find a nematic chiral charge order to be energetically preferred, which shows tentative agreement with experimental evidence.

Figure 1

Electronic features of $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. (a) Kagome lattice with basis of three sublattices, occupied by vanadium $d_{\mathrm{xy}}$ (blue) and $d_{\mathrm{xz}/\mathrm{yz}}$ (red) orbitals. (b) Corresponding band structure induced by $d_{\mathrm{xy}}$ (blue) and $d_{\mathrm{xz}/\mathrm{yz}}$ (red) orbitals and corresponding nature of van Hove singularities. Higher-lying bands of $d_{\mathrm{xz}/\mathrm{yz}}$ are not displayed for simplicity. (c) Schematic of Fermi surfaces in the hexagonal Brillouin zone. The V $d_{\mathrm{xy}}$ Fermi surface, which is nested by the ordering wave vectors ${\mathit{Q}}_j$, $j=1$, 2, 3, has weight on distinct sublattices at each $M$ point, giving rise to the sublattice interference mechanism. Dirac cones slightly above the Fermi energy are indicated in red according to their orbital origin.

Figure 2

Interaction strength phase diagram and corresponding electronic order in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. Mean field critical temperatures of the charge density order introduced in Eq. (3) and the charge bond order introduced in Eq. (5) as a function of the electronic interaction strengths $U/V$ ($\mu =-35\mathrm{meV}$ [26]). For $U\lesssim 2.1\mathrm{V}$ a charge bond order dominates, which forms a complex star of David pattern of strong and weak bonds. The inset highlights the mean field parameters ${\mathrm{\Delta }}_j$, $j=1$, 2, 3, and their arrangement on the kagome lattice. For systems with interaction scales above $U=2.1\mathrm{V}$ a charge density order emerges, modulating the real space site occupation.

Figure 3

Ginzburg Landau free energy and resulting current pattern. (a) The second order coefficient ${\alpha }_1$ as a function of temperature $T$ signals the second order phase transition of the emerging charge bond order by a sign change at $k_B{T}_{\mathrm{CDW}}\approx 1.011\mathrm{eV}$, whereas the coefficient ${\alpha }_2$ is purely positive, requiring an imaginary order parameter ${\mathrm{\Delta }}_j$, breaking time-reversal symmetry ($V=1\mathrm{eV}$, $\mu =-35\mathrm{meV}$). (b) The bond correlation pattern $⟨{\psi }_{\mathrm{GS}}|c_{\mathit{r}}^{†}{c}_{{\mathit{r}}^{\prime }}|{\psi }_{\mathrm{GS}}⟩$ in this setting (${\mathrm{\Delta }}_j=0.1\mathrm{i}$) shows a star of David arrangement of strong and weak bonds, equipped with a nonzero current. The current is homogeneous on the lattice, however alternating in its flow, indicated with blue arrows. Additionally, the order parameter modulates the on-site occupation within and between the hexagons. (c) The resulting band structure in the reduced Brillouin zone shows a gap opening around the $M$ points, reducing the density of states (DOS) at the Fermi level (${\mathrm{\Delta }}_j=0.1\mathrm{i}$).