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arXiv:2603.08601

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

Aitor Fernández, César D. Fosco

hep-thquant-phSubmitted Mar 9, 2026• Announced Mar 10, 2026• Revised May 16, 2026 (v2)
v1 · Mar 9, 2026v2 · May 16, 2026
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Abstract

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.

Comments
Reference added, corrected numerical coefficient and figures. Added appendix on generalization to the case of more general coupling constants

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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.
Abstract We analyze the thermodynamic Casimir effect occurring in a gas of non-interacting bosons confined by two parallel walls with a strongly anisotropic dispersion inherited from an underlying lattice. In the direction perpendicular to the confining walls, the standard quadratic dispersion is replaced by the term | p | α with α ⩾ 2 treated as a parameter. We derive a closed, analytical expression for the Casimir force depending on the dimensionality d and the exponent α , and analyze it for thermodynamic states in which the Bose–Einstein condensate is present. For α ∈ {4, 6, 8, …} the exponent governing the decay of the Casimir force with increasing distance between the walls becomes modified and the Casimir amplitude Δ α ( d ) exhibits oscillations of sign as a function of d . Otherwise, we find that Δ α ( d ) features singularities when viewed as a function of d and α . Recovering the known previous results for the isotropic limit α = 2 turns out to occur via a cancellation of singular terms.
We investigate the Casimir effect for a massive real scalar field confined between two perfectly reflecting parallel plates in the presence of a position-dependent effective mass, a mechanism for coupling a scalar background to a scalar field. Exact normal modes are obtained by solving the corresponding Klein-Gordon equation, leading to a transverse energy spectrum that exhibits a characteristic Landau-like structure despite the absence of an external magnetic field. Upon quantization of the field, the vacuum energy is evaluated by means of generalized zeta-function regularization together with an appropriate renormalization procedure. The renormalized vacuum energy naturally separates into a Landau-like contribution and an additional term induced by the spatial dependence of the effective mass. We show analytically and numerically that both contributions are exponentially suppressed in the strong-coupling regime. In the opposite limit, the Landau-like contribution smoothly reproduces the standard vacuum energy for a massive scalar field confined between parallel plates, whereas the additional contribution becomes singular owing to the restricted domain of validity of the exact spectrum. Except in the vicinity of this singular limit, the vacuum energy is shown to be dominated by the Landau-like sector. Our results establish a direct connection between position-dependent effective masses and boundary-induced quantum vacuum phenomena, providing a new exactly solvable framework for the investigation of Casimir effects in spatially inhomogeneous relativistic systems.
We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number C=1C=1C=1 is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity σxy=12e2hσ_{xy}=\frac{1}{2}\frac{e^2}{h}σxy​=21​he2​. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.
We investigate the fluctuation-induced Casimir interactions between two parallel graphene sheets carrying steady-state drift currents. The graphene properties are modeled based on the shifted Fermi disk model to capture the non-equilibrium optical response of the system. We find that the drift current introduces a repulsive correction to the perpendicular to the layers Casimir interaction, thereby reducing the overall attractive force. Although the correction is repulsive, it does not overcome the underlying attraction between the layers. It also generates a lateral force that opposes the carrier flow direction. Both contributions are studied in terms of distance and drift velocity functionalities showing pathways for Casimir force control.
For NNN spin-polarized fermions with harmonic pair interactions in a one-dimensional trap an odd-even effect is found. The spectrum of the one-particle reduced density matrix of the system's ground state differs qualitatively for NNN odd and NNN even. This effect only occurs for strong attractive and repulsive interactions. Since it does not exist for bosons, it must originate from the repulsive nature implied by the fermionic exchange statistics. In contrast to the spectrum, the one-particle density and correlation function for strong attractive interactions do not show any sensitivity on the number parity. This also suggests that reduced-density-matrix-functional theory has a more subtle NNN dependency than density-functional theory.
Casimir torque is conventionally associated with explicit breaking of rotational symmetry, arising from material dielectric anisotropy, geometric asymmetry, or externally applied fields that themselves break rotational invariance. Here we demonstrate a fundamentally different mechanism: an axially symmetric magnetic field can generate a Casimir torque by inducing an axially asymmetric Casimir energy - and can even reverse the torque's sign. Focusing on two-dimensional altermagnets, we show that a magnetic field applied perpendicular to the plane - while preserving in-plane rotational symmetry - activates an orientation-dependent vacuum interaction through the combined crystalline symmetry CnT\mathrm{C_n T}Cn​T inherent to altermagnetic order. The resulting torque emerges continuously and scales quadratically with the magnetic field strength. We further analyze its temperature and distance dependence, revealing scaling behaviors that are qualitatively different from those found in uniaxial bulk materials. Our results identify time-reversal symmetry breaking as a powerful route for engineering both the sign and strength of Casimir torque and establish altermagnets as an exciting platform for exploring phenomena driven by vacuum quantum fluctuations.