Finite temperature fermionic condensate in a conical space with a circular boundary and magnetic flux

We investigate the edge effects on the finite temperature fermionic condensate (FC) for a massive fermionic field in a ($2+1$)-dimensional conical spacetime with a magnetic flux located at the cone apex. The field obeys the bag boundary condition on a circle concentric with the apex. The analysis is presented for both the fields realizing two irreducible representations of the Clifford algebra and for the general case of the chemical potential. In both the regions outside and inside the circular boundary, the FC is decomposed into the boundary-free and boundary-induced contributions. They are even functions under the simultaneous change of the signs for the magnetic flux and the chemical potential. The dependence of the FC on the magnetic flux becomes weaker with decreasing planar angle deficit. For points near the boundary, the effects of finite temperature, of planar angle deficit, and of magnetic flux are weak. For a fixed distance from the boundary and at high temperatures the FC is dominated by the Minkowskian part. The FC in parity and time-reversal symmetric ($2+1$)-dimensional fermionic models is discussed and applications are given to graphitic cones.

Figure 1

FC in the exterior region for a massless field with a zero chemical potential versus the parameter ${\alpha }_0$ (left) and the temperature (right). The graphs are plotted for $r/a=1.5$ and the numbers near the curves correspond to the values of $q$. For the left panel we have taken $Ta=0.5$. The full/dashed curves in the right panel correspond to ${\alpha }_0=0/{\alpha }_0=0.4$.

Figure 2

FC in the exterior region for a massless field with a zero chemical potential as a function of the radial coordinate (left) and of the parameter $q$ (right). For the left panel $Ta=0.5$ and the full/dashed curves correspond to ${\alpha }_0=0/{\alpha }_0=0.4$. The numbers near the curves are the values of $q$. The right panel is plotted for $r/a=1.5$, $Ta=0.5$ (full curves), $Ta=0.25$ (dashed curves) and the numbers near the curves are the corresponding values of ${\alpha }_0$.

Figure 3

The same as in Fig. 1 for the interior region with fixed $r/a=0.5$.

Figure 4

The same as in Fig. 2 for the interior region. For the right panel we have taken $r/a=0.5$.

Figure 5

Boundary-induced FC as a function of the field mass in the case of a zero chemical potential in the exterior (left panel, $r/a=1.5$) and interior (right panel, $r/a=0.5$) regions. The full and dashed curves correspond to $s=1$ and $s=-1$, respectively. The numbers near the curves are the values of $Ta$ and for remaining parameters we have taken $q=2.5$, ${\alpha }_0=0.4$.

Figure 6

Boundary-free part in the FC (full curves) versus the field mass in the case of a zero chemical potential and for the field with $s=1$. The left and right panels are plotted for $r/a=1.5$ and $r/a=0.5$, respectively. The dashed curves present the FC in ($2+1$)-dimensional Minkowski spacetime when the magnetic flux is absent. The values of the remaining parameters are the same as those for Fig. 5.