Finite-size effects on the phase structure of the Nambu–Jona-Lasinio model

Abstract

The Nambu-Jona-Lasinio (NJL) model is one of the most frequently used four-fermion models in the study of dynamical symmetry breaking. In particular, the NJL model is convenient for that analysis at finite temperature, chemical potential and size effects, as has been explored in the last decade. With this motivation, we investigate the finite-size effects on the phase structure of the NJL model in D = 3 Euclidean dimensions, in the situations that one, two and three dimensions are compactified. In this context, we employ the zeta-function and compactification methods to calculate the effective potential and gap equation. The critical lines that separate trivial and non-trivial fermion mass phases in a second order transition are obtained. We also analyze the system at finite temperature, considering the inverse of temperature as the size of one of the compactified dimensions.The last decades witnessed significant investigations on the phase structure of quantum field theories, in particular on the chiral symmetry phase transitions in Quantum Cromodynamics (QCD). However, due to its complex structure, effective models have been largely employed to simplify that analysis. Among them, one of the most frequently used is the four-fermion theory known as Nambu-Jona-Lasinio (NJL) model [1]. The NJL model is specially convenient for the investigation of dynamical symmetries when the system is under certain conditions, like finite temperature, finite chemical potential, external gauge field, gravitation field and others [2,3,4].Finite-size effects on the phase transitions of four-fermion models have also attracted a great interest [5,6,7]. This question emerges when the system has a finite size and it is not clear if it is large enough to apply the thermodynamic limit in a usual way; frequently it is necessary to take into account the fluctuations due to finite-size effects. In particular, Ref.[6] performed a numerical investigation of a three-dimensional four-fermion model in a finite-size scaling analysis, where the finite-size effects act as an external field. On the other hand, Ref.[7] studied the NJL model in the framework of the multiple reflection expansion, where in terms of a modified density of states finite-size effects are included (see also Ref. [8]). The critical temperature was suggested to decrease as the system is reduced.In this paper, we investigate finite-size effects on the dynamical symmetry breaking in a different way. We study the Euclidean three-dimensional NJL model in the framework of zeta-function and compactification methods [9]. This procedure in principle allows us to explore the mentioned model with one, two or three compactified dimensions with antiperiodic boundary conditions [17] and compare their effects in the phase diagram of the model. With the choice of all dimensions being spatial, the system is considered confined between two parallel planes a distance L apart, confined to a infinity cylinder having a square transversal section of area L 2 , and to a cubic box of volume L 3 , for ...

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On the phase structure of QCD in a finite volume

“…Examples include NJLtype models and quark-meson-type models, see e. g. Refs. [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], as well as Polyakov-loop extended versions thereof, see e. g. Refs. [39,40,41,42,43,44,45,46,47,48,49,50,51,52].…”

Finite-size effects on the phase diagram of difermion condensates in two-dimensional four-fermion interaction models

“…With this purpose, different approaches have been used to study various aspects of these effects [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In particular, in Ref.…”