Strongly Interacting Fermions and Phases of the Casimir Effect

With the intent of exploring how the interplay between boundary effects and chiral symmetry breaking may alter the thermodynamical behavior of a system of strongly interacting fermions, we study the Casimir effect for the setup of two parallel layers using a four-fermion effective field theory at zero density. This system reveals a number of interesting features. While for infinitely large separation (no boundaries), chiral symmetry is broken or restored via a second order phase transition, in the opposite case of small (and, in general, finite) separation the transition becomes first order, rendering effects of finite size, for the present setup, similar to those of a chemical potential. Appropriately moving on the separation-temperature plane, it is possible to generate a peculiar behavior in the temperature dependence of the thermodynamic potential and of the condensate, compensating thermal with geometrical variations. A behavior similar to what we find here has been predicted to occur in bilayer graphene. Chiral symmetry breaking induces different phases (massless and massive) in the Casimir force separated by critical lines.

Figure 1

Temperature dependence of the effective potential for fixed separation. The left-hand plot refers to the case of infinitely large separation for which the phase transition is second order. The curves refer (bottom to top) to the values of $T/T_{\mathrm{crit}}=0.00$, 0.60, 0.80, 1.00, 1.17. The right-hand plot refers to the case of small separation $a\times T_{\mathrm{crit}}=0.6$ in which case boundary effects are non-negligible and the phase transition becomes first order. The curves refer (bottom to top) to $T/T_{\mathrm{crit}}=0.00$, 0.76, 1.00, 1.13, 1.29.

Figure 2

Separation dependence of the effective potential at fixed temperature. The left-hand plot refers to the case of small temperature ($a_{\mathrm{crit}}\times T=0.003$) and the phase transition is first order. The curves (bottom to top) refer to the following values: $a/a_{\mathrm{crit}}=100$, 2.79, 1.39, 1.00, 0.83. In the right-hand plot we have set the temperature close to its critical value in the absence of boundaries and the curves refer (bottom to top) to $a\times T_{\mathrm{crit}}=247.00$, 7.41, 3.70, 2.47, 1.85.

Figure 3

The figure shows how the effective potential changes when both temperature and separation decrease linearly according to $T(\delta )=u-v\delta$ and $a(\delta )=q-p\delta$, where $u$, $v$, $q$, $p$ are constants and $\delta$ is varied. In the left- (right-)hand panel we have set $u=q=2$ and $v=p=1$ ($u=p=1$ and $1/v=q=30$). The curves 1–6 in the left- (right-)hand panel correspond, respectively, to the values of $\delta =-0.5$, 0.095, 1.15, 1.55, 1.638, 1.7 ($\delta =-60.0$, $-44.0$, 15.0, 29.4, 29.64, 29.72).

Figure 4

$aT$ phase diagram. The thick dashed (red) curve represents the critical line (the blue dots superposed are calculated numerically). The dots connected by the straight dotted brown (dot-dashed green) line refer to the values of $a$ and $T$ used in the left- (right-)panel in Fig. 3.

Figure 5

Left: Behavior of the condensate when temperature and separation change as in Fig. 3. The curves refer to the following choices of the parameters: (0) $u=p=1$, $q=1/v=30$; (1) $u=p=1$, $q=1/v=1$; (2) $u=p=1$, $q=1/v=0.7$; (3) $u=p=1$, $q=1/v=0.5$. Right: Casimir pressure ($P_c=F_c/S$) for three illustrative cases. Temperature and separation change as in Fig. 3 with (1) $u=p=1$, $q=1/v=30$; (2) $u=p=1$, $q=1/v=1$; (3) $u=p=1$, $q=1/v=0.7$. The orange continuous curve refers to the massless case.