Conformally Invariant Fractals and Potential Theory

The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a $Q$ -state Potts cluster, is solved in two dimensions. The dimension $\hat f(\theta)$ of the boundary set with local wedge angle $\theta$ is $\hat f(\theta)=\frac{\pi}{\theta} -\frac{25-c}{12} \frac{(\pi-\theta)^2}{\theta(2\pi-\theta)}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\rm EP} =sup_{\theta}\hat f(\theta)$ of the external perimeter and $D_{\rm H}$ of the hull of a Potts cluster obey the duality equation $(D_{\rm EP}-1)(D_{\rm H}-1)={1/4}$. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.Comment: 5 pages, 1 figur

"Screening" of universal van der Waals - Casimir terms by Coulomb gases in a fully-finite two-dimensional geometry

This paper is a continuation of a previous one [Jancovici and Samaj, 2004 J. Stat. Mech. P08006] dealing with classical Casimir phenomena in semi-infinite wall geometries. In that paper, using microscopic Coulomb systems, the long-ranged Casimir force due to thermal fluctuations in conducting walls was shown to be screened by the presence of an electrolyte between the walls into some residual short-ranged force. Here, we aim to extend the study of the screening (cancellation) phenomena to universal Casimir terms appearing in the large-size expansions of the grand potentials for microscopic Coulomb systems confined in fully-finite 2D geometries, in particular the disc geometry. Two cases are solved exactly: the high-temperature (Debye-H\"uckel) limit and the Thirring free-fermion point. Similarities and fundamental differences between fully-finite and semi-infinite geometries are pointed out.Comment: 21 pages, 1 figur

Surface Polymer Network Model and Effective Membrane Curvature Elasticity

A microscopic model of a surface polymer network - membrane system is introduced, with contact polymer surface interactions that can be either repulsive or attractive and sliplinks of functionality four randomly distributed over the supporting membrane surface anchoring the polymers to it. For the supporting surface perturbed from a planar configuration and a small relative number of surface sliplinks, we investigate an expansion of the free energy in terms of the local curvatures of the surface and the surface density of sliplinks, obtained through the application of the Balian - Bloch - Duplantier multiple surface scattering method. As a result, the dependence of the curvature elastic modulus, the Gaussian modulus as well as of the spontaneous curvature of the "dressed" membrane, ~{\sl i.e.} polymer network plus membrane matrix, is obtained on the mean polymer bulk end to end separation and the surface density of sliplinks.Comment: 15 pages with one included compressed uuencoded figure

Screening of classical Casimir forces by electrolytes in semi-infinite geometries

We study the electrostatic Casimir effect and related phenomena in equilibrium statistical mechanics of classical (non-quantum) charged fluids. The prototype model consists of two identical dielectric slabs in empty space (the pure Casimir effect) or in the presence of an electrolyte between the slabs. In the latter case, it is generally believed that the long-ranged Casimir force due to thermal fluctuations in the slabs is screened by the electrolyte into some residual short-ranged force. The screening mechanism is based on a "separation hypothesis": thermal fluctuations of the electrostatic field in the slabs can be treated separately from the pure image effects of the "inert" slabs on the electrolyte particles. In this paper, by using a phenomenological approach under certain conditions, the separation hypothesis is shown to be valid. The phenomenology is tested on a microscopic model in which the conducting slabs and the electrolyte are modelled by the symmetric Coulomb gases of point-like charges with different particle fugacities. The model is solved in the high-temperature Debye-H\"uckel limit (in two and three dimensions) and at the free fermion point of the Thirring representation of the two-dimensional Coulomb gas. The Debye-H\"uckel theory of a Coulomb gas between dielectric walls is also solved.Comment: 25 pages, 2 figure

Comment on the sign of the Casimir force

I show that reflection positivity implies that the force between any mirror pair of charge-conjugate probes of the quantum vacuum is attractive. This generalizes a recent theorem of Kenneth and Klich to interacting quantum fields, to arbitrary semiclassical bodies, and to quantized probes with non-overlapping wavefunctions. I also prove that the torques on charge-conjugate probes tend always to rotate them into a mirror-symmetric position.Comment: 13 pages, 1 figure, Latex file. Several points clarified and expanded, two references added

Spanning Forests on Random Planar Lattices

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q -> 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n -> -1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n = -2. Then, we show how to perform an expansion around the t = 0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.Comment: 43 pages, Dedicated to Edouard Brezin and Giorgio Parisi, on the occasion of their special birthda

Boundary Shape and Casimir Energy

Casimir energy changes are investigated for geometries obtained by small but arbitrary deformations of a given geometry for which the vacuum energy is already known for the massless scalar field. As a specific case, deformation of a spherical shell is studied. From the deformation of the sphere we show that the Casimir energy is a decreasing function of the surface to volume ratio. The decreasing rate is higher for less smooth deformations.Comment: 12 page

Casimir energy and geometry : beyond the Proximity Force Approximation

We review the relation between Casimir effect and geometry, emphasizing deviations from the commonly used Proximity Force Approximation (PFA). We use to this aim the scattering formalism which is nowadays the best tool available for accurate and reliable theory-experiment comparisons. We first recall the main lines of this formalism when the mirrors can be considered to obey specular reflection. We then discuss the more general case where non planar mirrors give rise to non-specular reflection with wavevectors and field polarisations mixed. The general formalism has already been fruitfully used for evaluating the effect of roughness on the Casimir force as well as the lateral Casimir force or Casimir torque appearing between corrugated surfaces. In this short review, we focus our attention on the case of the lateral force which should make possible in the future an experimental demonstration of the nontrivial (i.e. beyond PFA) interplay of geometry and Casimir effect.Comment: corrected typos, added references, QFEXT'07 special issue in J. Phys.