We construct four kinds of Z3-symmetric three-dimensional (3-d) Potts models, each with different number of states at each site on a 3-d lattice, by extending the 3-d three-state Potts model. Comparing the ordinary Potts model with the four Z3-symmetric Potts models, we investigate how Z3 symmetry affects the sign problem and see how the deconfinement transition line changes in the µ-κ plane as the number of states increases, where µ (κ) plays a role of chemical potential (temperature) in the models. We find that the sign problem is almost cured by imposing Z3 symmetry. This mechanism may happen in Z3-symmetric QCD-like theory. We also show that the deconfinement transition line has stronger µ-dependence with respect to increasing the number of states.

As an effective model corresponding to Z3-symmetric QCD (Z3-QCD), we construct a Z3-symmetric effective Polyakov-line model (Z3-EPLM) by using the logarithmic fermion effective action. Since Z3-QCD tends to QCD in the zero-temperature limit, Z3-EPLM also agrees with the ordinary effective Polyakov-line model (EPLM) there; note that (ordinary) EPLM does not possess Z3 symmetry. Our main purpose is to discuss a sign problem appearing in Z3-EPLM. The action of Z3-EPLM is real, when the Polyakov line is not only real but also its Z3 images. This suggests that the sign problem becomes milder in Z3-EPLM than in EPLM. In order to confirm this suggestion, we do lattice simulations for both EPLM and Z3-EPLM by using the reweighting method with the phase quenched approximation. In the low-temperature region, the sign problem is milder in Z3-EPLM than in EPLM. We also propose a new reweighting method. This makes the sign problem very weak in Z3-EPLM.

We study the equation of state of pure SU(2) gauge theory using Monte Carlo simulations. The scale-setting of lattice parameters has been carried by using the gradient flow. We propose a reference scale t 0 for the SU(2) gauge theory satisfying t 2 E | t=t0 = 0.1, which is fixed by a natural scaling-down of the standard t 0 -scale for the SU(3) case based on perturbative analyses. We also show the thermodynamic quantities as a function of T /T c , which are derived by the energy-momentum tensor using the small flow-time expansion of the gradient flow. consistent with the most perfect-liquid property rather than the gas [10]. A theoretical large-N c analysis based on AdS/CFT correspondence gives the lower bound for η/s [11], while the 1/N c correction terms to η/s have not yet been determined even for its sign in the finite N c [12]. Although the determination of transport coefficients using the lattice calculations has been developing [13][14][15][16], it is still a challenging subject. The measurement of the correlation function of the energy-momentum tensor (EMT) is the first step to obtain the viscosities, and there are at least three difficulties: (i) the small signal-to-noise ratio of the correlator of EMT, (ii) the definition of the "correct" renormalization of EMT as a conserved quantity on the lattice, (iii) solving an inverse-problem to obtain the spectral function from the correlation function. In fact, the recent study [15] has used more than 6-million configurations to obtain the shear viscosity for one set of lattice parameter in the pure SU(3) gauge theory.Based on these situations, we focus on the pure SU(2) gauge theory, which is a good testing ground for the SU(3) gauge theory since the numerical cost is lower than the one for the SU(3) because of the smallness of the matrix size, nevertheless it has almost the same properties as the SU(3) gauge theory. In addition, the study of SU(2) gauge theory will provide the larger signal of the correction term of 1/N c to η/s because of the smaller N c .Several works have obtained the thermodynamic quantities for the SU(2) gauge theory using the integral method [17-20] on the lattice. A recent work, which mainly focuses on T < T c , shows the consistency with the massive free glueball model [17]. Another one [20], which utilizes the improved gauge action, presents the thermodynamic quantities without taking continuum limit. It is reported that, in intermediate temperature (T c T 5T c ), the scaling law of the trace anomaly of the SU(2) gauge theory has the different behavior from the SU(N c ) gauge theories with N c ≥ 3 as shown in Fig. 1, where the N c -dependence of the trace anomaly (∆/T 4 ) normalized by the pressure in the Stefan-Boltzmann (SB) limit (P SB /T 4 = π 2 (N 2 c − 1)/45) for the SU(N c ) theories is plotted. The results for the SU(2) gauge theories present as the circle (red) symbols, which we precisely determine in this work, and the triangle (cyan) symbols obtained in Ref. [20]. The dashed lines present the interpolating functions of the...

The persistent homology analysis is applied to the effective Polyakov-line model on a rectangular lattice to investigate the confinement-deconfinement nature.The lattice data are mapped onto the complex Polyakov-line plane without taking the spatial average and then the plane is divided into three domains. This study is based on previous studies for the clusters and the percolation properties in lattice QCD, but the mathematical method of the analyses are different. The spatial distribution of the data in the individual domain is analyzed by using the persistent homology to obtain information of the multiscale structure of center clusters. In the confined phase, the data in the three domains show the same topological tendency characterized by the birth and death times of the holes which are estimated via the filtration of the alpha complexes in the data space, but do not in the deconfined phase. By considering the configuration averaged ratio of the birth and death times of holes, we can construct the nonlocal order-parameter of the confinement-deconfinement transition from the multiscale topological properties of center clusters.

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