Casimir Sign Reversal from Finite-Range Bogoliubov Fluctuations

The gist: The zero-temperature Casimir effect for a three-dimensional Bose fluid with finite-range corrections to its Bogoliubov spectrum reveals that sign reversals are possible, occurring in the stable regime close to roton softening due to finite-momentum contributions

Finite-Range Corrections and Crossover

The study investigates the zero-temperature Casimir effect of a three-dimensional Bose fluid incorporating finite-range corrections to its Bogoliubov spectrum. Beyond the universal acoustic contribution scaling as L −3, the long-distance expansion contains a nonuniversal L −5 correction that can compete with the phononic term. The authors derive this crossover analytically within a minimal isotropic finite-range model and test this asymptotic result against the full dispersion. Finite-range and nonlocal interactions modify the dispersion away from the strictly linear regime, generating curvature corrections, maxon-roton features, and, in more complete anisotropic models, supersolid behavior with vortices. The authors interpret Rc as an effective microscopic range relevant to strongly dipolar atoms rather than a literal Rydberg blockade radius.

Analytical Results and Sign Reversal

The formal zero-point energy per unit area is calculated by considering the discrete sum of modes in a confined geometry and subtracting the continuous integral over the bulk space using an exponential regulator. The Euler-Maclaurin formula yields an asymptotic result for the Casimir energy E(l) at large separations compared to the effective range. In the case of periodic boundary conditions, it is shown that EP (l) = 2E(l/2) so that EP (l) = −E0/4R 2c16π/2√Λ + 3B 360√3l cubed + 64π/4(Λ − 30A) 25200√3l 5√Λ + 3B. The full Poisson-resummed calculation confirms that the apparent low-coupling zero inferred by balancing only the L −3 and L −5 terms is not quantitatively controlled in the low-coupling scaling regime. Robust sign changes instead occur in the stable regime close to roton softening, where the Casimir energy acquires an oscillatory finite-momentum contribution.

Near-Roton Physics and Oscillations

The full finite-range spectrum contains a distinct finite-momentum scale, and for A = 1 and B = 0, it develops a zero-energy roton only at Λc ≃ 1091.45. Numerical evaluation of the full Casimir sum–integral difference reveals that robust sign changes occur in the stable near-roton regime, before the excitation gap closes. The Poisson representation directly exposes the finite-momentum origin of this oscillatory behavior, showing that if the non-acoustic part of kR(k) is concentrated around a roton minimum kr, the leading Poisson image contains a contribution of the form δEr(l) ∼ A(l) sin(2lkr + ϕ), where A(l) is a slowly varying envelope. The expected oscillation period is therefore ∆l ≃ π/kr, with ∆lzero ≃ π/2kr.

Nanomechanical Probing

The analysis leads to a Duffing equation describing the mechanical response of a movable plate displaced by x from the static equilibrium gap d. The simple L −4 + L −6 force law follows from the asymptotic energy only, but in the near-roton regime, one should replace it by the force derived from the full Poisson-resummed energy F full Cas(L) = −S∂E full Cas(L)/∂L. This form is inspired by recent proposals and measurements that infer Casimir forces from nonlinear mechanical response in superconducting drum resonators. The stationary primary-resonance response relation gives the value(s) of the physical detuning compatible with a chosen amplitude a.

Future Directions

The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction, including the geometry-dependent roton minimum and Lee-Huang-Yang stabilization. A distinct nanostructure-compatible direction is provided by spatially indirect excitons in semiconductor double layers where relevant geometric length scales are naturally nanometric. The interaction in these systems is not of soft-core top-hat form, but rather dipolar and geometry-dependent. This extension would therefore require replacing the finite-range Bogoliubov spectrum by the appropriate excitonic dipolar dispersion. The present calculation should be viewed as a controlled analytic benchmark rather than a microscopic theory of a dipolar supersolid.

Conclusions

The paper concludes that while the analytical zero is useful as an indicator of competition between the acoustic and finite-range terms, it is not an exact location of the full sign reversal. The full dispersion does support genuine sign reversals in the stable region close to roton softening. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The microscopic origin and geometry are, however, different in this problem compared to those discussed in other contexts. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction. The natural next step is to replace the isotropic effective potential by the full anisotropic dipole-dipole interaction.

1. The AI system can perform quantitative analysis of Casimir energy sign reversals by evaluating the full Poisson-resummed evaluation instead of relying on truncations, as the apparent low-coupling zero obtained by balancing only these two terms is not quantitatively controlled. This allows the AI to distinguish between a crossover scale of the truncated asymptotic expansion rather than as an exact physical threshold and a true physical zero.

2. The system can predict the dynamical response of nanomechanical resonators near instability by utilizing the full Poisson-resummed energy to derive the force, as suggested by replacing the asymptotic L−4 + L−6 force law by the force derived from the full Poisson-resummed energy. This enables simulation of a strong nonmonotonic Casimir interaction where different plate separations can have opposite Casimir-energy signs at the same value of Λ.

3. The AI can identify physical regimes for experimental probing by correlating theoretical predictions with nonlinear mechanical signatures, specifically by analyzing the stationary response relation: σ(a) = 3αeff/8ω0 a squared ± f 2/4ω 20 a squared − α 21/2. This allows the system to determine if a detected frequency shift corresponds to a change in the finite-range coefficient or simply an artifact of the perturbative expansion.

4. The AI can map physical parameters to interaction regimes by calculating key characteristic scales, such as determining if the Casimir sign reversal occurs while the homogeneous Bogoliubov spectrum is still stable, which is confirmed by comparing calculated zero locations with Λc ≃ 1091.45. This provides a quantitative criterion for interpreting experimental results in the near-roton regime.

We study the zero-temperature Casimir effect of a three-dimensional Bose fluid with finite-range corrections to its Bogoliubov spectrum. Beyond the universal acoustic contribution scaling as L-3, the long-distance expansion contains a nonuniversal L-5 correction that can compete with the phononic term. We derive this crossover analytically within a minimal isotropic finite-range model and test the asymptotic result against the full dispersion. A regulator-independent Poisson-resummed calculation confirms that sign reversals of the Casimir energy are possible, while showing that the apparent low-coupling zero inferred by balancing only the L-3 and L-5 terms is not quantitatively controlled. Robust sign changes instead occur in the stable regime close to roton softening, where the Casimir energy acquires an oscillatory finite-momentum contribution. Strongly dipolar atomic gases provide a natural motivation for microscopic interaction lengths in the nanometric regime. We further discuss how the resulting nonlinear force law could be probed through the frequency response of a driven nanomechanical resonator.