  We perform an in-depth investigation of the phase diagram of the $J_1-J_2$
Heisenberg model on the square lattice. We take advantage of Density Matrix
Renormalization Group and Fully-Augmented Matrix Product States methods and
reach unprecedented accuracy with large bond dimensions. We utilize
excited-level crossing analysis to pinpoint the phase transition points. It was
believed before that there exists a narrow spin liquid phase sandwiched by the
N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through
careful finite size scaling of the level crossing points, we find a direct
phase transition between the N\'eel AFM and VBS phases at $J_2/J_1 = 0.535(3)$,
suggesting the absence of an intermediate spin liquid phase. We also provide
accurate results for ground state energies for a variety of sizes, from which
we find that the transition between the N\'eel AFM and VBS phases is
continuous. These results indicate the existence of a deconfined quantum
critical point at $J_2/J_1 = 0.535(3)$ in the model. From the crossing of the
first derivative of the energies with $J_2$ for different sizes, we also
determine the precise location of the first order phase transition between the
VBS and stripe AFM phases at $J_2/J_1=0.610(5)$.
