Zero-mode contribution and quantized first order phase transition in a droplet quark matter

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.