The energy-momentum tensor plays an important role in QCD thermodynamics. Its
expectation value contains information of the pressure and the energy density
as its diagonal part. Further properties like viscosity and specific heat can
be extracted from its correlation function. Recently a new method based on the
gradient flow was introduced to calculate the energy-momentum tensor on the
lattice, and has been successfully applied to quenched QCD.
In this paper, we apply the gradient flow method to calculate the
energy-momentum tensor in (2+1)-flavor QCD. As the first application of the
method with dynamical quarks, we study at a single but fine lattice spacing
a=0.07 fm with heavy u and d quarks ($m_\pi/m_\rho=0.63$) and approximately
physical s quark. Performing simulations on lattices with Nt=16 to 4, the
temperature range of T=174-697 MeV is covered. We find that the results of the
pressure and the energy density by the gradient flow method are consistent with
the previous results using the T-integration method at T<280 MeV, while the
results show disagreement at T>350 MeV (Nt<8), presumably due to the small-Nt
lattice artifact of $O((aT)^2)=O(1/N_t^2)$.
We also apply the gradient flow method to evaluate the chiral condensate
taking advantage of the gradient flow method that renormalized quantities can
be directly computed avoiding the difficulty of explicit chiral violation with
lattice quarks. We compute the renormalized chiral condensate in the MS-bar
scheme at renormalization scale $\mu=2$ GeV with a high precision to study the
temperature dependence of the chiral condensate and its disconnected
susceptibility. Even with the Wilson-type quark action, we obtain the chiral
condensate and its disconnected susceptibility showing a clear signal of
pseudocritical temperature at T~190 MeV related to the chiral restoration
crossover.