Desperately seeking spin liquids V

I just read a very interesting Nature paper, which appeared last month.

Quantum spin liquid emerging in two-dimensional correlated Dirac fermions,

by Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad & A. Muramatsu

The authors perform Quantum Monte Carlo simulations on the Hubbard model at half-filling on the honeycomb lattice. [This is the relevant lattice for graphene].

As U/t increases there is a phase transition from a semi-metal (SM) (which has gapless excitations at corners of the Brillouin zone, Dirac fermions) to a Mott insulating phase. But, they also find that there is a spin liquid (SL) phase with a spin gap before entering a phase with antiferromagnetic order (AFMI). The latter what one expects from a strong coupling expansion, i.e U >>t), which is described by an unfrustrated Heisenberg model. This is summarised in the figure below.

A few random observations:

The spin gap is very small Delta_{s ~ t/40~J/40.}

The single-particle charge gap Delta_sp(K) is quite small in the spin liquid state (about t/10 ~ U/40).

Although the honeycomb lattice is bi-partite and so not frustrated the authors, suggest that near the Mott transition effective frustrating interactions occur.

The spin liquid state has dimer-dimer correlations similar to that in a single hexagon which can be described by the RVB states of benzene. See the figure below.