We study the finite density phase transition in the lattice QCD at real chemical potential. We adopt a canonical approach and the canonical partition function is constructed for N f = 2 QCD. After derivation of the canonical partition function we calculate observables like the pressure, the quark number density, its second cumulant and the chiral condensate as a function of the real chemical potential. We covered a wide range of temperature region starting from the confining low to the deconfining high temperature; 0.65T c ≤ T ≤ 3.62T c . We observe a possible signal of the deconfinement and the chiral restoration phase transition at real chemical potential below T c starting from the confining phase. We give also the convergence range of the fugacity expansion.

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The canonical approach for finite density lattice QCD has a numerical instability. This instability makes it difficult to use the method reliably at the finite real chemical potential region. We studied this instability in detail and found that it is caused by the cancellation of significant digits. In order to reduce the effect of this cancellation, we adopt the multiple precision calculation for our discrete Fourier transformation (DFT) program, and we get the canonical partition function Z C (n, T ) with required accuracy. From the obtained Z C (n, T ), we calculate Lee-Yang zero distribution varying the number of significant digits. As a result, some curves surround the origin in the fugacity plane, but they are moved by varying the number of significant digits. Hence, we conclude that these curves are pseudo phase transition lines, and not real ones.

QCD is the fundamental theory which describes the dynamics of quarks and gluons. If we understand the dynamics at finite density and temperature, i.e. QCD phase diagram and equation of states, we can progress many studies such as the studies of unstable nucleus, nuclear fusion, early universe and neutron stars. The study of QCD phase diagram is very interesting, but we have not understood it well for a long time. This is because we face a problem in this thesis at finite density. The problem is called the sign problem. It causes a decrease of the calculation accuracy. That is why we can not calculate physical quantities with high accuracy at finite chemical potential.In this thesis, we try to beat the sign problem using the canonical approach of finite density lattice QCD. Although it is known that the canonical approach has several numerical problems, we can reduce them and calculate thermodynamic observables well at finite density. Concretely, in order to reduce the computation cost of the fermion determinant, we use the winding number expansion, in order to enhance the calculation accuracy of physical quantities, we adopt the multi precision calculation to our program based on the canonical approach. In this thesis, we will see how to improve the canonical approach and a result of thermodynamic observables which is related with the QCD phase transition at finite density.Our result shows that we do not observe the peak which represents the confinementdeconfinement phase transition in baryon number susceptibility. Therefore, we do not see the QCD phase transition yet. However, in this thesis, we find that canonical approach can explore the QCD phase diagram beyond µ B /T = 3 (µ B is the baryon chemical potential). That is, we explored the QCD phase structure beyond the validity range of Taylor expansion and reweighting method. This opens a bright window of study of QCD phase diagram at finite density. With our improvement, canonical approach has the possibility for investigation of thermodynamic observables at any chemical potential.

We report our recent project to study the QCD phase diagram by the canonical approach (CA), which is expected to solve the sign problem. First we briefly describe the sign problem and several approaches to fight against it. Then we argue that the CA may be a break-through if we combine it with the multiple precision calculation. We study the method using the hopping parameter expansion (HPE), in order to investigate how the CA works. We show explicitly how we calculate the winding number in HPE and obtain the fugacity expansion whose coefficients are the canonical partition functions. Finally we discuss dark sides of the current CA, which should be beaten for the method to become a real tool of QCD phase explorers.