A tensor product state approach to spin-1/2 square $J_1$-$J_2$ antiferromagnetic Heisenberg model: evidence for deconfined quantum criticality

TL;DR

The paper applies a cluster-update tensor-product-state approach with a four-sublattice $2\times 2$ unit cell to the spin-1/2 $J_1$-$J_2$ Heisenberg model on a square lattice, achieving ground-state energetics in agreement with exact diagonalization and other benchmarks. Finite-size scaling of spin correlations identifies a continuous transition at $J_2^{c_1}=0.572(5)$ with critical exponents $\nu=0.50(8)$ and $\eta_s=0.28(6)$, while in the paramagnetic regime spin correlations decay exponentially and dimer/plaquette correlations exhibit power-law decays with $\eta_p=0.24(1)$ and $\eta_c=0.28(1)$, consistent with a gapless $U(1)$ spin liquid and a deconfined quantum critical point (DQCP) scenario, potentially stabilized into a small VBS phase by instanton effects. The observed exponents closely match those of the $J$-$Q$ model’s DQCP, suggesting a Landau-forbidden transition from Néel order to VBS order at the critical point. This work demonstrates the viability of TPS with cluster updates for capturing subtle quantum critical behavior in frustrated spin systems and highlights a path toward understanding DQCP-like transitions in two dimensions.

Abstract

The ground state phase of spin-1/2 $J_1$-$J_2$ antiferromagnetic Heisenberg model on square lattice around the maximally frustrated regime ($J_2\sim 0.5J_1$) 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 $J_2^{c_1}=0.572(5)J_1$ and critical exponents $ν=0.50(8)$, $η_s=0.28(6)$. In the range of $0.572 < J_2/J_1 \leqslant 0.6 $ we find a paramagnetic ground state with exponentially decaying spin-spin correlation. Up to $24\times 24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent $η_p=0.24(1)$ and an anomalous columnar scaling exponent $η_c=0.28(1)$ at $J_2/J_1=0.6$. These results are consistent with a potential gapless $U(1)$ spin liquid phase. However, since the $U(1)$ spin liquid is unstable due to the instanton effect, a VBS order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point (DQCP) at $J_2^{c_1}$. Remarkably, all the observed critical exponents are consistent with the $J-Q$ model.

Figures (7)

• Figure 1: The finite size ground state energies using the largest available bond dimension $D=9$ measured at various $D_c=8,10,12,16,20,24,28$ in a VMC-tensor renormalization algorithm ling. The finite size energies are extrapolated to $D_c\to \infty$ limit by fitting to second order polynomials, the fitted results are in dashed lines.
• Figure 2: A benchmark of ground state energy with the SU(2) symmetric DMRG results ShengJ1J2 on tori and the VMC calculation with one Lanczos projection step HuJ1J2 on tori at $J_2=0.5,0.55$, 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.