Hubbard Model on Decorated Lattices

We introduce a family of lattices for which the Hubbard model and its natural extensions can be quasiexactly solved, i.e., solved for the ground and low energy states. In particular, we show rigorously that the ground state of the Hubbard model with off-site Coulomb repulsions on a decorated Kagomè lattice is an ordered array of local currents. The low energy theory describing this chiral state is an $S=\frac{1}{2}$ $XY$ model, where each spin degree of freedom represents the two possible chiralities of each local current.

Figure 1

(a) 1D lattice made up of corner sharing “double-tetrahedra” aligned in a linear chain and the general scheme for this family of 1D lattices. The squares are the connectors (C) and the circles belong to the basis blocks (B). (b) A possible general scheme for 2D lattices and a particular case of this family: the decorated Kagomè lattice.

Figure 2

(a)Two possible chiral states that can occur on each block B and the connection between the chiral and the spin language. (b) The block B (circles) and the connector (square) of the lattices shown in Fig. 1. (c) Unit cell of the decorated Kagomè lattice shown in Fig. 1b.