Plaquette-type valence bond solid state in the $J_1$-$J_2$ square-lattice Heisenberg model

TL;DR

This study employs advanced tensor network methods to identify the VBS phase in the $J_1$-$J_2$ square-lattice Heisenberg model as a plaquette type without spontaneous rotational symmetry breaking, resolving a long-standing debate.

Contribution

It demonstrates the effectiveness of FAMPS and DMRG in characterizing VBS order, establishing that the VBS phase is of plaquette type in the $J_1$-$J_2$ model.

Findings

VBS phase identified as PVBS with no rotational symmetry breaking

FAMPS successfully used for large-scale 2D quantum systems

Resolved the long-standing debate on VBS order in the model

Abstract

We utilize Density Matrix Renormalization Group (DMRG) and Fully Augmented Matrix Product States (FAMPS) methods to investigate the Valence Bond Solid (VBS) phase in the $J_1$-$J_2$ square lattice Heisenberg model. To differentiate between the Columnar Valence Bond Solid (CVBS) and Plaquette Valence Bond Solid (PVBS) phases, we introduce an anisotropy $\Delta_y$ in the nearest neighboring coupling in the $y$-direction, aiming at detecting the possible spontaneous rotational symmetry breaking in the VBS phase. In the calculations, we push the bond dimension to as large as $D = 25000$ in FAMPS, simulating systems at a maximum size of $14 \times 14$. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite $\Delta_y$ scaling analysis of the VBS dimer order parameters, we identify the VBS phase as a PVBS type, meaning there is no spontaneous rotational symmetry breaking in the VBS phase. This study not only resolves the long-standing issue of the characterization of the VBS order in the $J_1$-$J_2$ square lattice Heisenberg model but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.