Spin liquid ground state of the spin-$\frac{1}{2}$ square $J_1$-$J_2$ Heisenberg model

We perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin-$1/2$ antiferromagnetic square lattice Heisenberg $J_1$-$J_2$ model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\le J_2/J_1\le 0.62$, separating conventional Néel and striped antiferromagnetic states for smaller and larger $J_2/J_1$, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy $\gamma =0.70\pm 0.02$, extremely close to $\gamma =\ln \left(2\right)\approx 0.69$ (expected for a $Z_2$ quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.

Figure 1

The ground-state phase diagram for the spin-$\frac{1}{2}$ AFM Heisenberg $J_1$-$J_2$ model on the square lattice, as determined by accurate DMRG calculations on long cylinders with $L_y$ up to 14. Changing the coupling parameter $J_2/J_1$, three different phases are found: N$\acute{\mathrm{e}}$el antiferromagnet (AFM), topological quantum spin liquid (QSL), and stripe AFM phase. $m_s({\mathbf{k}}_0=(\pi ,\pi ))$ [$m_s({\mathbf{k}}_x=(\pi ,0))$] denotes the staggered magnetization in the N$\acute{\mathrm{e}}$el AFM phase [stripe AFM phase], whose saturation value is $1/2$. ${\Delta }_S$ and ${\Delta }_T$ denote the spin singlet and spin triplet gaps, respectively.

Figure 2

Finite-size extrapolations of the magnetic order parameters and spin excitation gaps. (a) The N$\acute{\mathrm{e}}$el AFM order parameter $m_s^2\left(\mathbf{k}\right)$ at wave vector ${\mathbf{k}}_0=(\pi ,\pi )$ and (b) stripe AFM order parameter $m_s^2\left(\mathbf{k}\right)$ at wave vector ${\mathbf{k}}_x=(\pi ,0)$ or ${\mathbf{k}}_y=(0,\pi )$, for various values of $J_2$, fitted using second-order polynomials in $1/L$. N$\acute{\mathrm{e}}$el AFM order disappears for $J_2>0.41$, while stripe AFM order develops for $J_2>0.62$, as seen in the corresponding insets. (c) Spin triplet gap ${\Delta }_T$ and (d) spin singlet gap ${\Delta }_S$ for different values of $J_2$, also fitted using second-order polynomials in $1/L$. The inset in (c) shows ${\Delta }_T$ for $L=4,6,8,10$, and the extrapolated values in the 2D limit, as functions of $J_2$. For the spin singlet gap, due to the numerical cost, we focus on several typical data points as shown in (d).

Figure 3

Order parameters for horizontal and vertical dimers. (a) The dimer order parameter $m_{d,x}^2$ at wave vector ${\mathbf{k}}_x=(\pi ,0)$ and (b) plaquette order parameter $m_{d,y}^2$ at ${\mathbf{k}}_y=(0,\pi )$, as a function of $J_2$ for different system sizes, and extrapolated to $L=\infty$.

Figure 4

Finite-size extrapolations of the dimer order parameter and plaquette order parameter. (a) The dimer order parameter $m_{d,y}^2$ at wave vector ${\mathbf{k}}_y=(0,\pi )$ and (b) plaquette order parameter $m_p^2$ for various values of $J_2$, fitted using second-order polynomials in $1/L$. The inset shows the plaquette order parameter for $L=4,6,8,10$ and the extrapolated values in the 2D limit as functions of $J_2$.

Figure 5

The entanglement entropy at $J_2=0.5$ and $0.56$. (a) The entanglement entropy $S\left(L_y\right)$ for $L_y=6\text{--}14$. By fitting $S\left(L_y\right)=a{L}_y-\gamma$, we obtain $\gamma \sim 0.70\pm 0.02$ for $J_2=0.5$ and $\gamma \sim 0.72\pm 0.04$ for $J_2=0.56$. (b) Length dependence of the entanglement entropy for $J_2=0.50$ and several system widths. One observes that the entropy is almost independent of $L_x$ for long systems (a small increase with $L_x$ can be observed for the smallest $L_x$ at $L_y=8$).

Figure 6

Even-odd effect. (a) Expectation value of horizontal bond operator $\langle B_i^x\rangle$ for $L_y=5$ and $L_x=32$. (b) The same expectation value for $L_y=6$ and $L_x=32$. (c) Dimer order parameter $D_{d,\widehat{x}}$ for odd $L_y$ at $L_x=\infty$. The red line denotes the exponentially decaying fitting function with the form of Eq. (4). (d) Modified boundary induced dimer order parameter for $L_y=6,8$, with $d$, the distance from the boundary. Here, the dimer order parameter is defined as the dimer density difference between two nearest-neighbor vertical dimer bonds. Inset shows the correlation length $\xi$ along the cylinder as a function of $L_y$.