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. A nonperturbative evaluation of it on the lattice is called. 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 ( 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 scheme at renormalization scale µ = 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 which violates the chiral symmetry explicitly, 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.

We calculate the energy gap (latent heat) and pressure gap between the hot and cold phases of the SU(3) gauge theory at the first order deconfining phase transition point. We perform simulations around the phase transition point with the lattice size in the temporal direction N t = 6, 8 and 12 and extrapolate the results to the continuum limit. We also investigate the spatial volume dependence. The energy density and pressure are evaluated by the derivative method with non-perturabative anisotropy coefficients. We adopt a multi-point reweighting method to determine the anisotropy coefficients. We confirm that the anisotropy coefficients approach the perturbative values as N t increases. We find that the pressure gap vanishes at all values of N t when the non-perturbative anisotropy coefficients are used. The spatial volume dependence in the latent heat is found to be small on large lattices. Performing extrapolation to the continuum limit, we obtain ∆ǫ/T 4 = 0.75 ± 0.17 and ∆(ǫ − 3p)/T 4 = 0.623 ± 0.056.

We study the end point of the first-order deconfinement phase transition in two and 2+1 flavor QCD in the heavy quark region of the quark mass parameter space. We determine the location of critical point at which the first-order deconfinement phase transition changes to crossover, and calculate the pseudo-scalar meson mass at the critical point. Performing quenched QCD simulations on lattices with the temporal extents N t = 6 and 8, the effects of heavy quarks are determined using the reweighting method. We adopt the hopping parameter expansion to evaluate the quark determinants in the reweighting factor. We estimate the truncation error of the hopping parameter expansion by comparing the results of leading and next-toleading order calculations, and study the lattice spacing dependence as well as the spatial volume dependence of the result for the critical point. The overlap problem of the reweighting method is also examined. Our results for N t = 4 and 6 suggest that the critical quark mass decreases as the lattice spacing decreases and increases as the spatial volume increases.

The wrong expression is not used in our numerical simulations and any results are not affected by this typo.

We study latent heat and the pressure gap between the hot and cold phases at the first-order deconfining phase transition temperature of the SU(3) Yang–Mills theory. Performing simulations on lattices with various spatial volumes and lattice spacings, we calculate the gaps of the energy density and pressure using the small flow-time expansion (SF$t$X) method. We find that the latent heat $\Delta \epsilon$ in the continuum limit is $\Delta \epsilon /T^4 = 1.117 \pm 0.040$ for the aspect ratio $N_s/N_t=8$ and $1.349 \pm 0.038$ for $N_s/N_t=6$ at the transition temperature $T=T_c$. We also confirm that the pressure gap is consistent with zero, as expected from the dynamical balance of two phases at $T_c$. From hysteresis curves of the energy density near $T_c$, we show that the energy density in the (metastable) deconfined phase is sensitive to the spatial volume, while that in the confined phase is insensitive. Furthermore, we examine the effect of alternative procedures in the SF$t$X method—the order of the continuum and the vanishing flow-time extrapolations, and also the renormalization scale and higher-order corrections in the matching coefficients. We confirm that the final results are all very consistent with each other for these alternatives.