Abstract
We characterize the Casimir forces for the Dirac vacuum on free-fermionic chains with smoothly varying hopping amplitudes, which correspond to (1+1)D curved spacetimes with a static metric in the continuum limit. The first-order energy potential for an obstacle on that lattice corresponds to the Newtonian potential associated to the metric, while the finite-size corrections are described by a curved extension of the conformal field theory predictions, including a suitable boundary term. We show that, for weak deformations of the Minkowski metric, Casimir forces measured by a local observer at the boundary are metric-independent. We provide numerical evidence for our results on a variety of (1+1)D deformations: Minkowski, Rindler, anti-de Sitter (the so-called rainbow system) and sinusoidal metrics.
Highlights
The quantum vacuum on a static spacetime is nothing but the ground state (GS) of a certain Hamiltonian
We have considered the sinusoidal metric, Eq (6), where the boundary term dominates the force for large N, while the conformal field theory (CFT) term dominates for low N, as we can see in JN = 1 + A sin(kN2), (31)
We have derived an expression for the ground-state energy of the discretized version of the Dirac equation in a deformed (1 + 1)D medium, which corresponds to the vacuum state in static curved metrics
Summary
The quantum vacuum on a static spacetime is nothing but the ground state (GS) of a certain Hamiltonian. Even if our technological abilities do not allow us to access direct measurements of the Casimir effect in curved spacetimes, we are aware of possible strategies to develop quantum simulators using current technologies, such as ultracold atoms in optical lattices [19]. It has been shown that the Dirac vacuum on certain static spacetimes can be characterized in such a quantum simulator [20], and an application has been devised to measure the Unruh radiation, including its nontrivial dimensional dependence [21–23]. The key insight is the use of curved optical lattices, in which fermionic atoms are distributed on a flat optical lattice with inhomogeneous hopping amplitudes, simulating a position-dependent index of refraction or, in other terms, an optical metric. The aim of this paper is to extend the aforementioned (1 + 1)D CFT predictions on curved backgrounds to characterize the Casimir force for the fermionic vacuum on curved optical lattices. The paper closes with a series of conclusions and proposals for further work
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