Nature of the Spin Liquid State of the Hubbard Model on a Honeycomb Lattice

Recent numerical work [Z. Y. Meng et al., Nature (London) 464, 847 (2010)] indicates the existence of a spin liquid (SL) phase that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum $J_1-J_2$ spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model. We find three phases—antiferromagnetic, an unusual $Z_2$ SL state, and a dimerized state with spontaneously broken rotational symmetry. We identify the $Z_2$ SL state as the likely candidate for the SL phase of the Hubbard model.

Figure 1

Phase diagram of the quantum $J_1-J_2$ model, obtained using the variational Monte Carlo method. As $J_2/J_1$ increases, the system undergoes a phase transition between an AFM state, and a gapped SL. We estimate the critical value of $J_2/J_1=x_1\approx 0.08$. The arrow indicates the location for this transition in the Hubbard model [4]. The SL state is best described variationally by SPS. At higher $J_2/J_1=x_2$ the SL gives way to a dimerized phase. Our variational study gives an estimate $x_2\approx 0.3$. The hashed area indicates values of $J_2/J_1$ that correspond to values of $U/t$ in the Hubbard model that are in the Néel or SL state [4].

Figure 2

Energies of AFM, ASL, SPS, and dimerized phases compared for a $10\times 10$ system. The AFM state is favorable at $J_2/J_1\lesssim 0.08$; ASL and SPS states have energies lower than the AFM state at $J_2/J_1\gtrsim 0.08$, but their energies are very close. Spontaneous breaking of the rotational symmetry occurs at $J_2/J_1\approx 0.3$, giving rise to the dimerized state.

Figure 3

Energies of the SPS state for different values of pairing amplitude $\Delta$ as a function of pairing phase $\theta$ compared to the ASL energy (for system size $14\times 14$). The SPS state is favored, and its energy is minimized for $\Delta \approx 0.1$, $\theta \approx 1.1$. * denotes energy gain in $J_1$ if SPS exactly obeyed the Marshall sign.

Figure 4

Energy difference between a SPS state with optimized (for $14\times 14$) pairing parameters ($\Delta \approx 0.1$, $\theta \approx 1.1$) and an ASL state as a function of system size. The energy difference extrapolates to a nonzero value in the thermodynamic limit $L\to \infty$. Nonmonotonicity of the points is a result of incommensurability effects with the lattice.

Figure 5

Dimer-dimer correlation function, defined as in 4 for a SPS state. Green (dark gray) is the reference slice. Red (light gray) indicates a positive correlation with the reference slice and blue (medium gray) a negative correlation. The SPS has the same positive and negative correlations as the dimer-dimer correlations found in 4.