Surface states and the Casimir interaction in the Ising model

¹ Theoretical Physics, Department of Physics, University of Oxford - 1 Keble Road, Oxford OX1 3NP, UK, EU
² Centre for Non-linear Studies, Los Alamos National Laboratory - Los Alamos, NM 87545, USA
³ Max-Planck-Institut für Intelligente Systeme - Heisenbergstraße 3, D-70569 Stuttgart, Germany, EU
⁴ IV Institut für Theoretische Physik, Universität Stuttgart - Pfaffenwaldring 57, D-70569 Stuttgart, Germany, EU
⁵ Institute of Physical Chemistry, Polish Academy of Sciences, Department III - Kasprzaka 44/52, PL-01-224 Warsaw, Poland, EU

Received: 30 October 2012
Accepted: 9 January 2013

Abstract

Using exact calculations, we elucidate the significance of surface states for the Casimir interactions in an Ising strip with a finite width. The surface states are responsible for the strong asymmetry between the super- and sub-critical regimes of the Casimir forces. We introduce an enhanced version of Fisher-Privman theory and justify it using another exact calculation. This gives a rather accurate account of the Casimir scaling function in the sub-critical regime and its exact asymptotic behaviour. We apply analogous ideas in three dimensions and obtain the Casimir scaling function which is in striking agreement with Monte Carlo simulations results for large negative scaling variable.