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

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.