  The nature of the zero-temperature phase diagram of the spin-$1/2$
$J_1$-$J_2$ Heisenberg model on a square lattice has been debated in the past
three decades, which may hold the key to understand high temperature
superconductivity. By using the state-of-the-art tensor network method,
specifically, the finite projected entangled pair state (PEPS) algorithm, to
simulate the global phase diagram the $J_1$-$J_2$ Heisenberg model up to
$24\times 24$ sites, we provide very solid evidences to show that the nature of
the intermediate nonmagnetic phase is a gapless quantum spin liquid (QSL),
whose spin-spin and dimer-dimer correlations both decay with a power law
behavior. There also exists a valence-bond solid (VBS) phase in a very narrow
region $0.56\lesssim J_2/J_1\leq0.61$ before the system enters the well known
collinear antiferromagnetic phase. The physical nature of the discovered
gapless QSL and potential experimental implications are also addressed. We
stress that we make the first detailed comparison between the results of PEPS
and the well-established density matrix renormalization group (DMRG) method
through one-to-one direct benchmark for small system sizes, and thus give rise
to a very solid PEPS calculation beyond DMRG. Our numerical evidences
explicitly demonstrate the huge power of PEPS for precisely capturing
long-range physcis for highly frustrated systems, and also demonstrate the
finite PEPS method is a very powerful approach to study strongly corrleated
quantum many-body problems.
