Abstract
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.
Highlights
The size effect attracts wide interest in different physical systems
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
Even though the finite size effect in quantum chromodynamics (QCD) physics has attracted lots of interest for more than three decades, normally, both the periodic and the antiperiodic spatial boundary condition are applied for fermions
Summary
The size effect attracts wide interest in different physical systems. For example, in a recent article [1], scientists realized that the most essential factor of making a grape plasma in a microwave oven is the grape size, which is comparable with the typical microwave length, so that the grape can “trap” microwaves. Periodic boundary condition (PBC) is normally applied for fermions or quarks, and the antiperiodic boundary condition (APBC) is applied in most cases for fermions in the spatial direction in order to keep the so-called permutation symmetry between the time and space directions [13,14] and to be consistent with the results of the volumedependent pion mass from the chiral perturbation theory [5]. We carefully investigate quark matter in a finite system with both the antiperiodic and periodic spatial boundary conditions applied for quarks and analyze the two different physical results. The results of catalysis of chiral symmetry breaking and pseudo–Nambu-Goldstone (NG) pions are obtained in Sec. III, and in Sec. IV we show the quantized firstorder phase transition in cold droplet quark matter. It is worth mentioning that actual phase transitions are possible only for infinite volumes; we use “apparent phase transition” for finite size systems in the whole paper
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