Finite-size effects on the phase structure of the Walecka model

In this work we investigate the finite-size effects on the phase structure of the Walecka model within the framework of a generalized $\zeta$-function, focusing on the influence of temperature as well as the number and length of compactified spatial dimensions. Here we concentrate on the situation of larger values of the coupling between the scalar and fermion fields, in which a phase transition of first order takes place. The phase transitions are analyzed and compared with the system in the situations of one, two, and three compactified spatial dimensions. Our findings suggest that the thermodynamic behavior of the system depends on the length and number of spatial dimensions, with the symmetric phase being favored as the size of the system diminishes.

Figure 1

Plot of values of $m_{\text{eff}}$ that are solutions of the gap equation in Eq. (22) for $d=0$ as a function of temperature at chemical equilibrium. We fix $m_{\sigma }=0.53$. Solid, dashed, and dotted lines represent, respectively, the cases for $g_{\sigma }=8.00,g_{\sigma }=16.00$, and $g_{\sigma }=26.00$.

Figure 2

Same as in Fig. 1, but with solid and dashed lines representing, respectively, the cases for $g_{\sigma }=16.00$ and $g_{\sigma }=26.00$, and with the temperature axis in the range near the transition point.

Figure 3

Plot of normalized thermodynamic potential density in Eq. (30) for $d=0$ as a function of effective mass and at chemical equilibrium. We fix $m_{\sigma }=0.53$ and $g_{\sigma }=26.00$. Solid, dashed, and dotted lines represent the cases for $T=0.120,T=0.125$, and $T=0.130$, respectively.

Figure 4

Same as in Fig. 3, but with $g_{\sigma }=16.00$. Solid, dashed, and dotted lines represent the cases for $T=0.153,T=0.158$, and $T=0.163$, respectively.

Figure 5

Plot of values of $m_{\text{eff}}$ that are solutions of the gap equation in Eq. (22) for $d=1$ as a function of temperature, at chemical equilibrium. We fix $m_{\sigma }=0.53$ and $g_{\sigma }=26.00$. Solid, dashed, and dotted lines represent, respectively, $L=15.00,L=10.00$, and $L=9.00$.

Figure 6

Plot of normalized thermodynamic potential density in Eq. (30) for $d=1$ as a function of effective mass and at chemical equilibrium. We fix $m_{\sigma }=0.53,g_{\sigma }=26.00$, and $L=15.00$. Solid, dashed, and dotted lines represent the cases for $T=0.1220,T=0.1254$, and $T=0.1293$, respectively.

Figure 7

Same as in Fig. 6, but with $L=10.00$. Solid, dashed, and dotted lines represent the cases for $T=0.1150,T=0.1200$, and $T=0.1250$, respectively.

Figure 8

Same as in Fig. 6, but with $L=9.00$. Solid, dashed, and dotted lines represent the cases for $T=0.1050,T=0.1127$, and $T=0.1190$, respectively.

Figure 9

Plot of values of $m_{\text{eff}}$ that are solutions of the gap equation in Eq. (22) for $d=1$ as a function of inverse of length ($x=1/L$) at chemical equilibrium. We fix $m_{\sigma }=0.53$ and $g_{\sigma }=26.00$. Solid, dashed, and dotted lines represent, respectively, $T=0.050,T=0.095$, and $T=0.110$.

Figure 10

Plot of normalized thermodynamic potential density in Eq. (30) for $d=1$ as a function of effective mass and at chemical equilibrium. We fix $m_{\sigma }=0.53,g_{\sigma }=26.00$, and $T=0.120$. Solid, dashed, and dotted lines represent the cases for $L=13.00,L=10.00$, and $L=9.00$, respectively.

Figure 11

Plot of values of $m_{\text{eff}}$ that are solutions of the gap equation in Eq. (22) as a function of temperature, at chemical equilibrium. We fix $m_{\sigma }=0.53,g_{\sigma }=26.00$, and $L=12.00$. Solid, dashed, and dotted lines represent, respectively, the case of $d=1,d=2$, and $d=3$ compactified spatial dimensions.

Figure 12

Plot of normalized thermodynamic potential density in Eq. (30) as a function of effective mass and at chemical equilibrium. We fix $m_{\sigma }=0.53,g_{\sigma }=26.00,L=12.00$, and $T=0.123$. Dotted, dashed, and solid lines represent the cases for $d=3,d=2$, and $d=1$, respectively.