[Paper Review] Thermodynamic Casimir Effect in the large-n limit

Denis Comtesse, Alfred Hucht+1 more|arXiv (Cornell University)|Apr 23, 2009

Quantum Electrodynamics and Casimir Effect3 references3 citations

TL;DR

This paper develops a numerical method to compute thermodynamic Casimir amplitudes in O(n)-symmetric systems with slab geometry in the large-n limit, focusing on non-periodic boundary conditions such as Dirichlet and open. It confirms asymptotic equivalence of Casimir amplitudes for Dirichlet and open conditions and validates results against known analytical solutions for periodic and antiperiodic boundaries.

ABSTRACT

We consider systems with slab geometry of finite thickness L that undergo second order phase transitions in the bulk limit and belong to the universality class of O(n)-symmetric systems with short-range interactions. In these systems the critical fluctuations at the bulk critical temperature T_c induce a long-range effective force called the "thermodynamic Casimir force". We describe the systems in the framework of the O(n)-symmetric phi^4-model, restricting us to the large-n limit n->infty. In this limit the physically relevant case of three space dimensions d=3 can be treated analytically in systems with translational symmetry as, e.g., in the bulk or slabs with periodic or antiperiodic boundary conditions. We consider Dirichlet and open boundary conditions at the surfaces that break the translational invariance along the axis perpendicular to the slab. From the broken translational invariance we conclude the necessity to solve the systems numerically. We evaluate the Casimir amplitudes for Dirichlet and open boundary conditions on both surfaces and for Dirichlet on one and open on the other surface. Belonging to the same surface universality class we find the expected asymptotic equivalence of Dirichlet and open boundary conditions. To test the quality of our method we confirm the analytical results for periodic and antiperiodic boundary conditions.

Motivation & Objective

• To compute thermodynamic Casimir amplitudes for systems with non-translationally invariant boundary conditions in the large-n limit.
• To investigate the asymptotic equivalence of Casimir amplitudes for Dirichlet and open boundary conditions within the same surface universality class.
• To develop and validate a numerical method for evaluating excess free energy and extracting Casimir amplitudes in finite-size systems.
• To test the accuracy of the method by comparing numerical results with known analytical solutions for periodic and antiperiodic boundary conditions.
• To assess the magnitude of non-universal corrections to scaling in different boundary condition configurations.

Proposed method

• Formulates the O(n)-symmetric φ⁴ model in the large-n limit to enable analytical treatment of d=3 systems with translational symmetry.
• Derives the excess free energy per unit area from the effective action, leading to a self-consistent eigenvalue problem for the correlation function.
• Solves the eigenvalue problem numerically for finite slab thickness L, using a discretized lattice representation of the system.
• Fits the numerically computed excess free energy to a power-law ansatz f_ex = a₀ + a₂L⁻² + a₃L⁻³ + a₄L⁻⁴ to extract the Casimir amplitude Δ_C as a₂.
• Validates the method by comparing numerical Casimir amplitudes for periodic and antiperiodic boundary conditions with known analytical results.
• Analyzes non-universal corrections (O(L⁻³), O(L⁻⁴)) to assess the convergence and accuracy of the fitting procedure.

Experimental results

Research questions

• RQ1 Do Dirichlet and open boundary conditions in the same surface universality class yield asymptotically equivalent Casimir amplitudes in the large-n limit?
• RQ2 How do non-universal corrections to the leading scaling behavior differ between open-open, Dirichlet-open, and Dirichlet-Dirichlet boundary conditions?
• RQ3 Can the numerical method accurately reproduce known analytical Casimir amplitudes for periodic and antiperiodic boundary conditions?
• RQ4 What is the quantitative value of the Casimir amplitude for open-open, Dirichlet-open, and Dirichlet-Dirichlet boundary conditions in the large-n limit?
• RQ5 How does the numerical method handle the breakdown of translational invariance due to non-periodic boundary conditions?

Key findings

• The Casimir amplitude for open-open boundary conditions is Δ_C,OO = -0.012(1), with error bar indicating high precision.
• The Casimir amplitude for Dirichlet-open boundary conditions is Δ_C,DO = -0.012(3), showing consistency with the open-open case within error margins.
• The Casimir amplitude for Dirichlet-Dirichlet boundary conditions is Δ_C,DD = -0.012(3), confirming asymptotic equivalence with the other two cases.
• The numerical results confirm the expected asymptotic equivalence of Casimir amplitudes for Dirichlet and open boundary conditions, as both belong to the same surface universality class.
• The non-universal corrections (O(L⁻³), O(L⁻⁴)) are significantly smaller in the open-open case compared to Dirichlet-Dirichlet and Dirichlet-open configurations, indicating simpler convergence.
• The method successfully reproduces the analytical Casimir amplitude for periodic boundary conditions, Δ_C,PBC = -2ζ(3)/(5π) ≈ -0.153050, confirming numerical accuracy.