Tensor-product state approach to spin-$\frac{1}{2}$ square $J_1\text{−}{J}_2$ antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticality

The ground state phase of a spin-$\frac{1}{2} J_1\text{−}{J}_2$ antiferromagnetic Heisenberg model on a square lattice around the maximally frustrated regime $\left(J_2\sim 0.5J_1\right)$ has been debated for decades. Here we study this model using the cluster update algorithm for tensor-product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point $J_2^{c_1}=0.572\left(5\right)J_1$ and critical exponents $\nu =0.50\left(8\right),{\eta }_s=0.28\left(6\right)$. In the range of $0.572<J_2/J_1\le 0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\times 24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\eta }_p=0.24\left(1\right)$ and an anomalous columnar scaling exponent ${\eta }_c=0.28\left(1\right)$ at $J_2/J_1=0.6$. These results are consistent with a potential gapless $U\left(1\right)$ spin-liquid phase. However, since the $U\left(1\right)$ spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at $J_2^{c_1}$. Remarkably, all the observed critical exponents are consistent with the $J\text{−}Q$ model.

Figure 1

Finite size ground state energies using the largest available bond dimension $D=9$ measured at various $D_c=8$, 10, 12, 16, 20, 24, and 28 in a variational Monte Carlo (VMC)-tensor renormalization algorithm [35]. The finite size energies are extrapolated to $D_c\to \infty$ limit by fitting to second-order polynomials; the fitted results are shown in the dashed lines.

Figure 2

A benchmark of ground state energy with the SU(2) symmetric DMRG results [26] on tori and the VMC calculation with one Lanczos projection step [9] on tori at $J_2=0.5$ and 0.55, where $D_c$ is the Schmidt number kept in our VMC-tensor renormalization algorithm.

Figure 3

(a) The largest distance spin-spin correlation as a function of $J_2$ at $L=8$, 12, 16, 24. The same correlations $C(L/2,L/2)$ presented against $1/L$ in a regular plot (b) and in a log-log plot (c) for various $J_2$.

Figure 4

The finite size scaling function of $C(L/2,L/2)$.

Figure 5

The modified dimer-dimer correlation $C_{dx}^{*}(r,r)$ (a) and plaquette-plaquette correlation $C_{plq}^{*}(r,r)$ (b) as a function of separation $r$ at $J_2=0.6$ in log-log plots.

Figure 6

The valence bond solid order parameters $S_{\text{col}}^2$ and $S_{\text{plq}}^2$ at $J_2=0.6$ as functions of $1/L$ in a log-log plot. The power law decay behaviors are captured by decay exponents $1+{\eta }_p=1.24\left(1\right)$ and $1+{\eta }_c=1.28\left(1\right)$ for the plaquette and columnar VBS order, respectively.

Figure 7

(a) The simple update scheme. (b) The cluster update scheme with a cluster size $2\times 2$.