Charge density waves in kagome-lattice extended Hubbard models at the van Hove filling

The Hubbard model on the kagome lattice is presently often considered as a minimal model to describe the rich low-temperature behavior of $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ compounds (with $A=\mathrm{K},\mathrm{Rb},\mathrm{Cs}$), including charge density waves (CDWs), superconductivity, and possibly broken time-reversal symmetry. Here, we investigate, via variational Jastrow-Slater wave functions, the properties of its ground state when both on-site $U$ and nearest-neighbor $V$ Coulomb repulsions are considered at the van Hove filling. Our calculations reveal the presence of different interaction-driven CDWs and, contrary to previous renormalization-group studies, the absence of ferromagnetism and charge- or spin-bond order. No signatures of chiral phases are detected. Remarkably, the CDWs triggered by the nearest-neighbor repulsion possess charge disproportionations that are not compatible with the ones observed in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. As an alternative mechanism to stabilize charge-bond order, we consider the electron-phonon interaction, modeled by coupling the hopping amplitudes to quantum phonons, as in the Su-Schrieffer-Heeger model. Our results show the instability towards a trihexagonal distortion with $2\times 2$ periodicity, in closer agreement with experimental findings.

Figure 1

(a) Phase diagram of the Hubbard model of Eq. (1) at the van Hove filling $n_F=5/6$. The symbols indicate the values of $U/t$ and $V/t$ for which the calculations have been performed (on finite-sized lattices with $L_1=12$ and $L_2=10$ and 18 [54]). Overlapping symbols are used when the variational energies of two phases are equivalent within error bars. (b) Charge pattern in the $2\times 2$ CDW phase. The color of each site $i$ indicates the average number of electrons $\langle n_i\rangle$, while the width and the darkness of the lines connecting nearest neighbors $(i,j)$ are proportional to the modulus of the expectation value of the hopping operator along the bond, namely, $|\langle c_{i,\uparrow }^{†}{c}_{j,\uparrow }^{\phantom{†}}+c_{i,\downarrow }^{†}{c}_{j,\downarrow }^{\phantom{†}}\rangle |$. Results for $U/t=0$ and $V/t=3$ are shown. (c) Sketch of the electronic charge patterns of the CDW I, II, and III phases, fulfilling the triangle rule. Blue (red) circles denote depletion (accumulation) of electrons. Following Ref. [9], for each CDW we show the dimer configuration on the honeycomb lattice formed by connecting the centers of the corner-sharing triangles.

Figure 2

Variational energy landscape of a wave function reproducing the $2\times 2$ CBO of Ref. [15], for $U/t=8$ and $V/t=4$. The energy is plotted as a function of the ratio between the hopping parameters of ${\mathcal{H}}_0$ for the nearest-neighbor bonds inside ($T^{\prime }$) and outside ($T$) the star of David (shown in the inset). Results for the uncorrelated state $|{\mathrm{\Phi }}_0\rangle$ (a) and the correlated one $\mathcal{J}|{\mathrm{\Phi }}_0\rangle$ (b) are shown. The error bars are smaller than the size of the symbols. The calculations have been performed on a finite lattice with $L_1=12$ and $L_2=10$ [54].

Figure 3

(a) Mean-square displacement $\delta {\mathcal{X}}^2$ as a function of the electron-phonon coupling $\lambda$ in the trihexagonal distorted phase of ${\mathcal{H}}_{\mathrm{ep}}$. (b) Lattice distortion induced by the electron-phonon coupling at $\lambda \approx 0.14$. The color of the sites represents the difference of the local electron density with respect to the filling, $\langle n_i\rangle -n_F$. The width and the darkness of $(i,j)$ bonds are proportional to $|\langle c_{i,\uparrow }^{†}{c}_{j,\uparrow }^{\phantom{†}}+c_{i,\downarrow }^{†}{c}_{j,\downarrow }^{\phantom{†}}\rangle |$. The calculations have been performed on a finite lattice with $L_1=8$ and $L_2=6$ [54].