Parity-dependent Casimir forces and Hall currents for a confined Dirac field

We study a massless Dirac field subjected to two alternative boundary conditions on two parallel thin walls, in d + 1 dimensions. The two configurations correspond to the system being even or odd under reflection about the midplane between the two walls, and lead to qualitatively different behaviors. The even (symmetric) configuration produces an attractive Casimir force, whereas the odd (antisymmetric) one yields repulsion, in agreement with a general theorem linking parity to the sign of the fermionic Casimir effect. We complement this result by studying two phenomena associated with the vacuum fluctuations responsible for the Casimir interaction, both of which are also sensitive to parity: the correlation between currents concentrated on the walls, and the induced bulk current under the influence of an external electric field. For the latter we show that, in 2 + 1 dimensions, an induced transverse (Hall-like) current arises, whose spatial profile inherits the symmetry of the confining potential.

💡 Research Summary

The paper investigates a massless Dirac field confined between two parallel, infinitesimally thin walls in d + 1 dimensions. The walls are modeled by singular delta‑function potentials V(x_d)=g_L δ(x_d−a_L)+g_R δ(x_d−a_R) with coupling constants g_{L,R}=±2, which enforce MIT bag boundary conditions on both sides of each wall. The sign choices η_{L,R}=±1 determine the parity of the configuration: η_L=η_R yields a parity‑even (symmetric) setup, while η_L=−η_R gives a parity‑odd (antisymmetric) one.

Using functional determinant techniques, the renormalized Casimir energy per unit area is obtained as
E(a)=−2⌊n/2⌋∫(d^dp_q/(2π)^d) ln