Valence-bond crystal in the extended kagome spin-$\frac{1}{2}$ quantum Heisenberg antiferromagnet: A variational Monte Carlo approach

The highly frustrated spin-$\frac{1}{2}$ quantum Heisenberg model with both nearest ($J_1$) and next-nearest ($J_2$) neighbor exchange interactions is revisited by using an extended variational space of projected wave functions that are optimized with state-of-the-art methods. Competition between modulated valence-bond crystals (VBCs) proposed in the literature and the Dirac spin liquid (DSL) is investigated. We find that the addition of a small ferromagnetic next-nearest-neighbor exchange coupling $\left|J_2\right|>0.09J_1$ leads to stabilization of a 36-site unit cell VBC, although the DSL remains a local minimum of the variational parameter landscape. This implies that the VBC is not trivially connected to the DSL; instead it possesses a nontrivial flux pattern and large dimerization.

Figure 1

The $U(1)$ DSL ansatz, solid (dashed) bonds denote positive (negative) hoppings (a). The unit cell is doubled to accommodate $[0,\pi ]$ flux. Cases with n.n.n. hopping are also reported; the only possible (nonchiral) space group symmetric states built upon $[0,\pi ]$ state are $[0,\pi ;\pi ,0]$ (b) or $[0,\pi ;0,\pi ]$ (d) and those upon the uniform $[0,0]$ state (i.e., PFS) are $[0,0;\pi ,\pi ]$ (c) or $[0,0;0,0]$ (e).

Figure 2

(a) Optimized ${\chi }_{\mathrm{n}.\mathrm{n}.\mathrm{n}.}/{\chi }_{\mathrm{n}.\mathrm{n}.}$ versus $J_2$ for the extended PFS and DSL states of Fig. 1. A zoom around $J_2=0$ is shown in (b). The optimized n.n. (c) and n.n.n. (d) hopping parameters versus $J_2$ for the 36-site supercell VBC are also reported. ${\chi }_1=1$ is the reference bond. Only ${\chi }_{11}<0$, which implies $[0,0;\pi ,\pi ]$ flux in the P hexagons of Fig. 4.

Figure 3

Energy versus $J_2$ for spin liquid (see Fig. 1) and VBC states [see Figs. 4c and 4d].

Figure 4

Twelve-site supercell (a) with three different parameters for the hopping; 18-site supercell with two different parameters; 36-site supercell with six n.n. parameters (c), and with six n.n.n. parameters (d).