Continuous phase transition from Néel state to $Z_2$ spin-liquid state on a square lattice

Recent numerical studies of the $J_1$-$J_2$ model on a square lattice suggest a possible continuous phase transition between the Néel state and a gapped spin-liquid state with $Z_2$ topological order. We show that such a phase transition can be realized through two steps: First bring the Néel state to the U(1) deconfined quantum critical point, which has been studied in the context of Néel–valence bond solid (VBS) state phase transition. Then condense the spinon pair–skyrmion/antiskyrmion bound state, which carries both gauge charge and flux of the U(1) gauge field emerging at the deconfined quantum critical point. We also propose a Schwinger boson projective wave function to realize such a $Z_2$ spin liquid state and find that it has a relatively low variational energy ($-0.4893J_1$/site) for the $J_1$-$J_2$ model at $J_2=0.5J_1$. The spin liquid state we obtain breaks the fourfold rotational symmetry of the square lattice and therefore is a nematic spin liquid state. This direct continuous phase transition from the Néel state to a spin liquid state may be realized in the $J_1$-$J_2$ model, or the anisotropic $J_{1x}$-$J_{1y}$-$J_2$ model.

Figure 1

Conjunctured phase diagram of the $J_1$-$J_2$-$Q$ model. In the phase diagram we set $J_1=1$ and vary the other two frustration terms. At the origin $J_2=Q=0$ the model is in the Néel state. Along the $x$ axis $Q=0$ and the model reduces to the $J_1$-$J_2$ model, which has a continuous phase transition between Néel and $Z_2$ spin liquid states [10, 11]. Along the $y$ axis $J_2=0$ and the model reduces to the $J$-$Q$ model, which has a continuous phase transition between Néel and VBS order [17]. The solid lines show phase boundaries described by the deconfined quantum criticality [12, 13], and the dashed line shows the phase boundary that is the subject of this study, in which we propose that it can also be described by the deconfined quantum criticality.

Figure 2

Pattern of pairing order parameters $Q_{ij}$ in Eq. (15). The arrows show the direction along which $Q_{ij}$ is positive. The two patterns correspond to spin liquid states obtained by condensing $f_x$ and $g_x$ as defined in Eq. (8), respectively.