Wilson Fermions in Lattice Quantum Chromodynamics

We have studied the Wilson fermion matrix in the lattice QCD in the quenched approximation. Having implemented the Lanczos algorithm to study Wilson fermion spectrum on finite volumes we have presented results to confirm the existence of a phase transition accompanied by a massless mode from a phase where parity is restored to a phase where this discrete symmetry is violated in accordance with Aoki's lattice QCD phase diagram. In an effort to set up the most suitable algorithm to investigate hadron spectrum for Wilson fermions, we have also studied different versions of the Lanczos and conjugate gradient algorithms and have found that the block Lanczos algorithm is really superior for inverting large sparse matrices. In particular we have shown that the rate of convergence of the block Lanczos algorithm becomes effectively independent of the details of the fermion matrix such as gauge coupling constant and hopping parameter. The application of the block Lanczos algorithm to investigate scalar and pseudoscalar meson propagators shows that the massless mode associated with the transition from parity-restoring to parity-violating phase in the above phase structure is indeed where the pion becomes massless