Abstract:
We propose a method to probe the nature of phase transitions in lattice QCD at finite temperature and density, which is based on the investigation of an effective potential as a function of the average plaquette. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass $m/T = 0.4$, and find that a first order phase transition line appears in the high density regime for $\mu_q/T \simge 2.5$. We also discuss the difference between the phase structures of QCD with non-zero quark chemical potential and non-zero isospin chemical potential.

Abstract:
We propose a method to find the QCD critical point at finite density calculating the canonical partition function ${\cal Z}_{\rm C} (T,N)$ by Monte-Carlo simulations of lattice QCD, and analyze data obtained by a simulation with two-flavor p4-improved staggered quarks with pion mass $m_{\pi} \approx 770 {\rm MeV}$. It is found that the shape of an effective potential changes gradually as the temperature decreases and a first order phase transition appears in the low temperature and high density region. This result strongly suggests the existence of the critical point in the $(T, \mu_q)$ phase diagram.

Abstract:
We discuss the nature of the phase transition for lattice QCD at finite temperature and density. We propose a method to calculate the canonical partition function by fixing the total quark number introducing approximations allowed in the low density region. An effective potential as a function of the quark number density is discussed from the canonical partition function. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass $m/T = 0.4$ on a $16^3 \times 4$ lattice. The results suggest that the finite density phase transition at low temperature is of first order.

Abstract:
We investigated a behavior of monopole currents in the high temperature phase of abelian projected finite temperature SU(2) QCD in maximally abelian gauge. Wrapped monopole currents which are closed by periodic boundary play an important role for the spatial string tension. And the wrapped monopole current density seems to be non-vanishing in the continuum limit. These results may be related to Polyakov's analysis of the confinement mechanism using monopole gas in 3-dimensional SU(2) gauge theory with Higgs fields.

Abstract:
We propose a method to probe the nature of phase transitions in lattice QCD at finite temperature and density, which is based on the investigation of an effective potential as a function of the average plaquette. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass $m/T = 0.4$, and find that a first order phase transition line appears in the high density regime for $\mu_q/T \sim 2.5$. The effective potential as a function of the quark number density is also studied. We calculate the chemical potential as a function of the density from the canonical partition function and discuss the existence of the first order phase transition line.

Abstract:
We review recent progress in lattice QCD at finite density. The phase diagram of QCD and the equation of state at finite temperature and density are discussed. In particular, we focus on the critical point terminating a first order phase transition line in the high density region. The critical point is one of the most interesting features that may be discovered in heavy-ion collision experiments. We summarize the current discussion on the existence of a critical point in the QCD phase diagram and discuss some attempts to find the critical point by numerical simulations.

Abstract:
We study the relation between the abelian monopole condensation and the deconfinement phase transition of the finite-temperature pure QCD. The expectation value of the monopole contribution to the Polyakov loop becomes zero when a long monopole loop is distributed uniformly in the configuration of the confinement phase. On the other hand, it becomes non-zero when the long monopole loop disappears in the deconfinement phase. We also discuss the relation between the monopole behaviors and the usual interpretation of the spontaneous breaking of Z(N) symmetry in finite-temperature SU(N) QCD. It is found that the boundary condition of the space direction is important to understand the Z(N) symmetry in terms of the monopoles.

Abstract:
We studied a behavior of monopole currents in the high temperature (deconfinement) phase of abelian projected finite temperature SU(2) QCD in maximally abelian gauge. Wrapped monopole currents closed by periodic boundary play an important role for the spatial string tension which is a non-perturbative quantity in the deconfinement phase. The wrapped monopole current density seems to be non-vanishing in the continuum limit. These results may be related to Polyakov's analysis of the confinement mechanism using monopole gas in 3-dimensional SU(2) gauge theory with Higgs fields.

Abstract:
We comment on the reweighting method in the chemical potential $(\mu_{\rm q})$ direction. We study the fluctuation of the reweighting factor during Monte-Carlo steps. We find that it is the absolute value of the reweighting factor that mainly contributes to the shift of the phase transition line $(\beta_c)$ by the presence of $\mu_{\rm q}$. The phase fluctuation is a cause of the sign problem, but the effect on $\beta_c$ seems to be small. We also discuss $\beta_c$ for Iso-vector chemical potential and $\beta_c$ determined from simulations with imaginary chemical potential.

Abstract:
Recent developments in finite-temperature QCD with dynamical quarks are reviewed focusing on the topics of critical temperature, the equation of state, and critical behaviors around the chiral phase transition.