The finite size effect on hadron physics and quark matter has attracted much interest for more than three decades; normally, both the periodic (with a zero-momentum mode) and the antiperiodic (without a 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 remain as pseudo–Nambu-Goldstone bosons. Furthermore, it is found that the zero-momentum-mode contribution brings a significant change of the chiral apparent phase transition in a droplet of cold dense quark matter: The first-order chiral apparent phase transition becomes quantized; i.e., the first-order apparent phase transition is completed in two steps, which is a brand-new quantum phenomenon. It is expected that the catalysis of chiral symmetry breaking and the quantized first-order phase transition are common features for fermionic systems with a quantized momentum spectrum with a zero-mode contribution, which also shows up in quark matter under a magnetic field.
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