  The finite size effect on hadron physics and quark matter has attracted much
interest for more than three decades, normally both the periodic (with
zero-momentum mode) and the anti-periodic (without zero-momentum mode) spatial
boundary condition are applied for fermions. By comparing the thermodynamical
potential, it is found that if there is no other physical constraint, the
droplet quark matter is always more stable when the periodic spatial boundary
condition is applied, and the catalysis of chiral symmetry breaking is observed
with the decrease of the system size, while the pions excited from the droplet
vacuum keep as pseudo Nambu-Goldstone bosons. Furthermore, it is found that the
zero-momentum mode contribution brings significant change of the chiral
apparent phase transition in a droplet of cold dense quark matter: the
1st-order chiral apparent phase transition becomes quantized, i.e., the
1st-order apparent phase transition is completed in two steps, which is a
brand-new quantum phenomena. It is expected that the catalysis of chiral
symmetry breaking and the quantized 1st-order apparent phase transition are
common features for fermionic systems with quantized momentum spectrum with
zero-mode contribution, which also show up in quark matter under magnetic
field.
