Doping on the kagome lattice: A variational Monte Carlo study of the $t$-$J$ model

We study doping on the kagome lattice by exploring the $t$-$J$ model with variational Monte-Carlo. We use a number of Gutzwiller projected spin-liquid and valence-bond-crystal states and compare their energies at several system sizes. We find that introducing mobile holes drives the system away from the spin-liquid state proposed by Ran et al. [Y. Ran, M. Hermele, P. A. Lee, and X. G. Wen, Phys. Rev. Lett. 98, 117205 (2007)] for the undoped system, toward a uniform state with zero flux. On top of the uniform state a valence-bond crystal of the Hastings type is formed for low doping. The results are compared and agree well with exact diagonalization on small clusters.

Figure 1

The kagome lattice consisting of corner-sharing triangles. The three-site physical unit cell is indicated with dotted lines. The underlying Bravais lattice is triangular, leading to a hexagonal Brillouin zone as in the triangular lattice.

Figure 2

The flux states U, D, and C, being defined by the value of a flux when surrounding a plaquette, as explained in the text.

Figure 3

(Left) The pattern of the VBC-1 state. The dotted lines indicate the 12-site unit cell. (Right) The pattern of the VBC-2 state. The dotted lines indicate the 36-site unit cell. The variational parameter $\chi$ tunes the ratio between the weak and strong bonds.

Figure 4

Ground state energy for various states and system sizes in their optimum as a function of doping $x$ and the comparison with the ED[32] on small clusters of $N=21$. The error bar is smaller than the symbol size.

Figure 5

(Top) Difference between the U state energy compared to the U+VBC-1 state and U+VBC-2 state energies for various sizes. (Bottom) Variational parameter $\chi$ (as described in the text) for the U+VBC-1 and U+VBC-2 states for some sizes as a function of doping $x$. For clarity we show only a few typical error bars.

Figure 6

Contributions of the various terms in the $t$-$J$ model [kinetic energy, spin-exchange, and $\frac{n_i{n}_j}{4}$-term (compare Eq. (1))], for the D+VBC-1 state and the U+VBC-1 state with the best energy at each doping level $x$.

Figure 7

Phase diagram of the $t$-$J$ model on the kagome lattice.