  Using lattice QCD we study the spectrum of low-lying fermion eigenmodes.
According to the Banks-Casher relation, accumulation of the low-mode is
responsible for the spontaneous breaking of chiral symmetry in the QCD vacuum.
On the lattice we use the overlap fermion formulation that preserves exact
chiral symmetry. This is essential for the study of low-lying eigenmode
distributions. Through a detailed comparison with the expectations from chiral
perturbation theory beyond the leading order, we confirm the senario of the
spontaneous symmetry breaking and determine some of the low energy constants.
We also discuss on other related physical quantities, which can be studied on
the lattice with exact chiral symmetry.
