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

Ling Wang, Zheng‐Cheng Gu+2 more|DSpace@MIT (Massachusetts Institute of Technology)|Dec 14, 2011

Physics of Superconductivity and Magnetism3 citations

TL;DR

This study employs a tensor product state (TPS) approach with a cluster update algorithm to investigate the spin-1/2 $J_1$-$J_2$ Heisenberg model on a square lattice. It provides strong numerical evidence for a deconfined quantum critical point (DQCP) at $J_2^c = 0.572(5)J_1$, with critical exponents $\nu = 0.50(8)$, $\eta_s = 0.28(6)$, and anomalous scaling in dimer and plaquette correlations consistent with a gapless $U(1)$ spin liquid, despite potential instability to a valence bond solid order in the thermodynamic limit.

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 $24 imes 24$ 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.

Motivation & Objective

• To resolve the long-standing debate on the nature of the quantum phase transition in the frustrated spin-1/2 $J_1$-$J_2$ Heisenberg model on the square lattice.
• To determine whether the transition at $J_2 \sim 0.5J_1$ is continuous or first-order, and whether it hosts a deconfined quantum critical point (DQCP).
• To investigate the ground state phase diagram, particularly the existence of a spin liquid or valence bond solid (VBS) phase in the paramagnetic regime.
• To assess the stability of a putative $U(1)$ spin liquid phase against instanton effects and potential VBS order formation.

Proposed method

• The study uses a tensor product state (TPS) ansatz for the ground state wave function, enabling efficient variational optimization of quantum many-body states.
• A cluster update algorithm is employed to perform imaginary time evolution, allowing for accurate and stable simulations of the Hamiltonian with improved convergence and reduced error accumulation.
• The evolution operators are constructed via Trotter decomposition and expressed as matrix product operators (MPOs), enabling efficient contraction and update on clusters of sites.
• Finite-size scaling is applied to spin-spin, dimer-dimer, and plaquette-plaquette correlation functions to extract critical exponents and locate the quantum critical point.
• The method allows for accurate ground state energy extrapolation to the thermodynamic limit using bond dimension $D$ and finite-size scaling with $D_c$.

Experimental results

Research questions

• RQ1 Is the quantum phase transition from Néel order to paramagnetic order in the $J_1$-$J_2$ Heisenberg model continuous or first-order?
• RQ2 Does the critical point at $J_2 \approx 0.5J_1$ exhibit characteristics of a deconfined quantum critical point (DQCP)?
• RQ3 What is the nature of the paramagnetic ground state in the regime $0.572 < J_2/J_1 \leq 0.6$ — is it a $U(1)$ spin liquid or a valence bond solid?
• RQ4 Are the critical exponents observed consistent with those predicted for the $J-Q$ model DQCP?

Key findings

• The critical point for the Néel-to-paramagnetic transition is located at $J_2^c = 0.572(5)J_1$, with critical exponents $\nu = 0.50(8)$ and $\eta_s = 0.28(6)$, consistent with a continuous phase transition.
• In the paramagnetic phase ($0.572 < J_2/J_1 \leq 0.6$), spin-spin correlations decay exponentially, indicating a gapped spin liquid or short-range entangled state.
• Dimer-dimer and plaquette-plaquette correlations exhibit power-law decay with anomalous scaling exponents $\eta_p = 0.24(1)$ and $\eta_c = 0.28(1)$ at $J_2/J_1 = 0.6$, signaling potential gapless $U(1)$ spin liquid behavior.
• The observed critical exponents are in excellent agreement with those of the $J-Q$ model DQCP, strongly supporting the existence of a deconfined quantum critical point.
• Despite the evidence for a gapless $U(1)$ spin liquid, the instability due to instanton effects suggests a small-amplitude VBS order may develop in the thermodynamic limit.
• Finite-size scaling of ground state energies shows good agreement with exact diagonalization results, validating the numerical approach.