Plaquette-type valence bond solid state in the J 1 − J 2 square-lattice Heisenberg model

We utilize density matrix renormalization group and fully augmented matrix product states (FAMPS) methods to investigate the valence bond solid (VBS) phase in the ${J}₁\text{\ensuremath{-}}{J}₂$ square lattice Heisenberg model. To differentiate between the columnar valence bond solid and plaquette valence bond solid (PVBS) phases, we introduce an anisotropy ${\mathrm{\ensuremath{\Delta}}}_y$ 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 $D=25\phantom{\rule{0.16em}{0ex}}000$ in FAMPS, simulating systems at a maximum size of $14\ifmmode\times\else\texttimes\fi{}14$. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite ${\mathrm{\ensuremath{\Delta}}}_y$ 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 paper not only supports the presence of PVBS order in the ${J}₁\text{\ensuremath{-}}{J}₂$ square lattice Heisenberg model, but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.