The Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated $J_1$-$J_2$ Heisenberg model

TL;DR

This paper introduces a hybrid computational method combining Green's function Monte Carlo with projected entangled pair states to efficiently and accurately study the frustrated $J_1$-$J_2$ Heisenberg model, overcoming traditional sign problems.

Contribution

The paper presents a novel integration of GFMC with PEPS, improving simulation accuracy and efficiency for frustrated quantum spin systems, especially where sign problems hinder traditional methods.

Findings

Enhanced ground-state energy accuracy using PEPS-guided GFMC.

Identification of a possible columnar valence-bond phase.

Effective characterization of the phase diagram in the $J_1$-$J_2$ model.

Abstract

The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic $J_1$-$J_2$ Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of $0.52 < J_2/J_1 < 0.58$. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.