Unveiling the Physics of the Doped Phase of the $t-J$ Model on the Kagome Lattice

We investigate the ground state properties of the kagome lattice $t-J$ model at low doping by variational Monte Carlo calculations. The resulting state possesses an interesting balance of spin exchange and kinetic exchange through the building blocks of stars which are linked by triangles and their internal hexagons. While the spin exchange is taking place mainly on the stars, hopping is favored on the hexagons. There is a density modulation, resulting in the holes having an effective static contribution. From this observation, how holes lead to dimerization in this model and why a particular valence bond crystal pattern is formed can be understood. Furthermore, we argue the optimal doping for this state. We discuss our result in connection with static impurities, and show the likely relevance to the diluted kagome lattice Heisenberg model, describing actual compounds.

Figure 1

The parametrization of variational VBC states considered. All fluxes are zero. Top left: HVBC, the blue thick bonds correspond to the VBC of Hastings and are controlled by ${\chi }_1$. Top right: DVBC, similar to HVBC, but additionally the inner hexagon is parametrized (red dashed bonds; controlled by ${\chi }_2$). Bottom left: IVBC, the up-pointing triangles (blue thick bonds) are controlled by a variational parameter ${\chi }_1$. Bottom right: YVBC, the thick blue bonds are the strongest bonds (parameter ${\chi }_1$), the “8-site loop” is indicated by thick dashed red lines (parameter ${\chi }_2$), the weakest bonds are depicted as thin dashed green lines (parameter ${\chi }_3$). In all of the above the remaining connecting bonds (black thin) are set to unity.

Figure 2

Local bond and site measurements in the optimized state for a $N=192$ sites system and a doping level of 9.4%. Top left: Bond spin exchange (strong bonds, $S_{ij}\approx -0.30$; medium bonds, $-0.14<S_{ij}<-0.11$; weakest bonds, $S_{ij}\approx -0.05$). Top right: Bond kinetic energy (strong bonds, $-0.14<T_{ij}<-0.15$; weak bonds, $T_{ij}\approx -0.09$). Bottom: Site densities (large density sites, $n_i\approx 0.92$; lower density sites, $n_i\approx 0.90$). A finite value has been subtracted from all bonds or sites for better visualization.

Figure 3

Top: Mechanism forming the Star of David: When a hole moves within the inner hexagon completing a route, the outer Star of David is formed. Bottom: Distribution of holes: Assuming one hole per star, we show the position of closest and longest distance.

Figure 4

Variational parameters for the DVBC, and the ratio between the two. Both parameters develop their maximum or minimum very close to our suggested optimal doping ($=1/12$) for this state (indicated by a dashed vertical line). The ratio between the two has the maximum exactly at this point.