Nature of the phases in the frustrated $XY$ model on the honeycomb lattice

We study the phase diagram of the frustrated $XY$ model on the honeycomb lattice by using accurate correlated wave functions and variational Monte Carlo simulations. Our results suggest that a spin-liquid state is energetically favorable in the region of intermediate frustration, intervening between two magnetically ordered phases. The latter ones are represented by classically ordered states supplemented with a long-range Jastrow factor, which includes relevant correlations and dramatically improves the description provided by the purely classical solution of the model. The construction of the spin-liquid state is based on a decomposition of the underlying bosonic particles in terms of spin-$1/2$ fermions (partons), with a Gutzwiller projection enforcing no single occupancy, as well as a long-range Jastrow factor.

Figure 1

(a) Extrapolated best energies in the thermodynamic limit, as well as the final phase diagram (colored regions labeled AF, SL, and CL) based on the properties of the states considered. The energy of the purely classical solution of the model is shown for comparison. (b) Energies of antiferromagnetic and spin-liquid states compared to the exact results on the $4\times 4\times 2$ cluster.

Figure 2

Finite-size scaling of the AF energies (a), the CL states (b), and projected fermionic states (c). In the legends, the numbers indicate values of $J_2/J_1$.

Figure 3

(a) Single-particle gap of the optimized fermionic Hamiltonian in Eq. (4) as a function of $J_2/J_1$ for the $18\times 18\times 2$ cluster. (b) The unprojected band structure for $J_2/J_1=0.25$. For comparison, black solid lines show the band structure for real NN hopping only.

Figure 4

Momentum distribution $n\left(\mathbf{k}\right)$ (a) and Static structure factor $N\left(\mathbf{k}\right)$ (b) for $J_2/J_1=0.25$ on the $18\times 18\times 2$ cluster. Finite size scaling of the condensate fraction $n_0/V$ and $N_0/V$ for $J_2/J_1=0.25$ (c), $J_2/J_1=0.2$ (d), and $J_2/J_1=0.3$ (e). The solid lines correspond to fits of the data to a second-order polynomial. The corresponding extrapolation of the condensate fraction is shown with filled red squares.