Ground-state phase diagram of the quantum $J_1-J_2$ model on the square lattice

We study the ground-state phase diagram of the quantum $J_1-J_2$ model on a square lattice by means of an entangled-plaquette variational Ansatz. In the range $0⩽J_2/J_1⩽1$, we find classical magnetic order of Néel and collinear type, for $J_2/J_1\lesssim 0.5$ and $J_2/J_1\gtrsim 0.6$, respectively. For intermediate values of $J_2/J_1$ the ground state is a spin liquid (i.e., paramagnetic with no valence-bond crystalline order). Our estimates of the entanglement entropy show that such a spin liquid is topological.

Figure 1

Left: GS energy per site of the $J_1-J_2$ model on the square lattice with PBCs, as a function of the system size. Estimates are obtained with $N$ plaquettes of $l=4$ (boxes), $l=9$ (triangles), and $l=16$ (circles) sites. Each dashed line is a fit to numerical data with the same $l$ value. Right: GS energy per site, extrapolated to the thermodynamic limit, as a function of the plaquette size. Each dashed line is a fitting function linear in $l^{-3/2}$ (see the text). The QMC estimate in the $J_2=0$ case is also shown for comparison (star) (Ref. 4).

Figure 2

Left: Néel SSM of the $J_1-J_2$ model on the square lattice with PBCs, as a function of the system size. Estimates are obtained with $N$ plaquettes of $l=4$ (boxes), $l=9$ (triangles), and $l=16$ (circles) sites. Each dashed line is a fit to numerical data with the same $l$ value (see text). Right: Néel SSM, extrapolated to the thermodynamic limit, as a function of the plaquette size. Lines are functions built to infer the $l$ dependence of our data (see the text). The QMC estimate in the $J_2=0$ case is also shown for comparison (star) (Ref. 4). Inset: pictorial representation of classical Néel order where “up” (“down”) spins reside on sublattice $\alpha$ ($\beta$).

Figure 3

As Fig. 2 where the SSM describes now magnetic order of collinear type (sublattices $\alpha$ and $\beta$ are chosen according to the inset).

Figure 4

Finite-size scaling of the plaquette VBC order parameter for two values of $J_2$ at which the system is found in a magnetically disordered GS. Dashed lines are fits to the numerical data with the same $J_2$. This order parameter clearly vanishes, for all the $l$ values considered in this work, in the thermodynamic limit. Data shown refer to the $l=16$ case.

Figure 5

Upper left: Dependence of the Monte Carlo average of the swap operator on the plaquette size for square subregion $A$ comprising $a\times a$ spins. Different symbols refer to different $a$ values ($a=4$ to 8 from top to bottom). Each dashed line is a quadratic (in $l^{-1/2})$ fit to numerical data with the same $a$. Upper right: Extrapolated (to the $l\to N$ limit) values of the entanglement entropy as a function of the boundary length $x$ of subregion $A$ (diamonds). Also shown, for comparison, are the QMC estimates (crosses) (Ref. 19). Lower panel: Entanglement entropy for two values of $J_2$ at which the GS is a spin liquid. Each dashed line is a linear fit (to numerical data with the same $J_2$) performed to extract the topological entropy $\gamma$ (see the text). Here $A$ is an $L/2\times L$ subregion of an $L\times L$ lattice with PBCs. Hence, $A$ has no corners.