Temperature dependence of the Casimir force between plane-layered structures: an approach based on the principle of argument

Technical Summary: Temperature dependence of the Casimir force between plane-layered structures: an approach based on the principle of argument

Problem Statement
The paper addresses the theoretical derivation of the temperature-dependent Casimir-Lifshitz dispersion force between two arbitrary plane-layered structures separated by a vacuum gap or a layer of dissipative dielectric. While the zero-temperature Casimir force and the finite-temperature Lifshitz formula are well-established, the author identifies a specific limitation in the standard Lifshitz approach when applied to small distances ($d$) at finite temperatures ($T$). Specifically, the standard Lifshitz formula (expressed as a sum over Matsubara frequencies) fails to correctly recover the zero-temperature Casimir result in the limit where $d \to 0$ at finite $T$. The paper seeks to derive a unified formula that remains valid across all distance and temperature regimes, particularly for ideally conducting structures, by rigorously applying complex analysis theorems.

Methodology
The author employs the "principle of argument" from complex analysis to derive the free energy and the resulting force. The methodology involves the following steps:

1. Free Energy Formulation: The free energy of the resonator is defined using the energy of a quantum oscillator in a thermal field, incorporating the hyperbolic cotangent function $\coth(\hbar\omega/2k_BT)$.
2. Complex Plane Analysis: The summation over discrete frequencies is transformed into a contour integral in the complex frequency plane ($\omega$). The author utilizes the symmetry properties of the reflection coefficients and the dielectric permittivity (DP) to analyze the distribution of zeros and poles.
3. Wick Rotation: The integration path is rotated from the real frequency axis to the imaginary frequency axis ($\omega = i\xi$). This transformation introduces the Matsubara frequencies ($\xi_n = 2\pi n k_B T / \hbar$) as poles of the hyperbolic cotangent function.
4. Contour Decomposition: By applying the principle of argument, the total force is decomposed into two distinct components:
   • A discrete sum over the Matsubara frequencies (the standard Lifshitz term).
   • A continuous contour integral along the imaginary frequency axis, which arises from the "principal value" integration around the poles of the cotangent function.

Key Contributions and Results
The primary contribution is the derivation of a generalized temperature-dependent formula for the Casimir force that consists of the standard Lifshitz term plus a temperature-dependent contour integral.

• Generalized Formula: The derived force $F(d, T)$ is expressed as the sum of the Lifshitz formula (a sum over Matsubara frequencies) and a contour integral term. The contour integral accounts for the continuous spectrum of frequencies that the discrete sum misses, particularly in the limit of small distances.
• Recovery of Known Limits:
  • Zero Temperature ($T \to 0$): The contour integral vanishes, and the formula reduces to the Schram/van Kampen formula (which coincides with the Lifshitz formula for vacuum gaps), recovering the standard zero-temperature Casimir result.
  • Ideal Conductors at Large Distances ($d \to \infty$): The results align with the well-known high-temperature limit $F \propto T/d^3$.
  • Ideal Conductors at Small Distances ($d \to 0$): The paper demonstrates that the standard Lifshitz formula alone significantly underestimates the force at small $d$ and finite $T$. The inclusion of the contour integral term corrects this discrepancy, allowing the total force to converge to the exact Casimir result ($F \propto 1/d^4$) as $d \to 0$.
• Quantitative Analysis: Numerical and analytical estimates for ideally conducting plates at room temperature ($300$ K) show that the contour integral contribution becomes significant for distances $d < 100$ nm. At $d = 10$ nm, the standard Lifshitz formula yields a value of approximately $0.0199 \times 10^6$ N/m², which is roughly 85% lower than the Casimir result of $0.1295 \times 10^6$ N/m². The sum of the Lifshitz term and the contour integral restores the correct magnitude.
• Physical Interpretation: The author interprets the contour integral as a correction arising from the specific way thermal fluctuations interact with the boundaries at short scales, distinct from the discrete Matsubara modes. The paper clarifies that the imaginary parts of complex frequencies are compensated during the summation process, ensuring the final force is real.

Significance and Claims
The paper claims to provide a mathematically rigorous generalization of previous results by van Kampen, Schram, and Lifshitz. Its significance lies in resolving the inconsistency where the standard Lifshitz formula fails to transition smoothly to the zero-temperature Casimir result in the limit of small separation at finite temperature.

The author asserts that for distances greater than $100$ nm at room temperature, the standard Lifshitz formula is sufficiently accurate. However, for sub-100 nm separations, the additional contour integral term is necessary to obtain the correct physical force. The work does not propose new experimental setups but rather offers a theoretical correction to existing dispersion force calculations, ensuring consistency with causality principles and the principle of argument in complex analysis. The results confirm that the total interaction is the sum of the discrete thermal modes (Lifshitz) and a continuous contribution (contour integral), which together satisfy the known asymptotic behaviors of the Casimir force.