Eur. Phys. J. B 30, 71-76 (2002)
https://doi.org/10.1140/epjb/e2002-00360-1

Twist free energy in a spin glass

¹ Laboratoire de Physique Théorique, École Normale Supérieure (Unité Mixte de Recherche 8549 du Centre National de la Recherche Scientifique et de l'École Normale Supérieure) , 24 rue Lhomond 75231, Paris Cedex 05, France
² Service de Physique Théorique, CE Saclay, 91191 Gif-sur-Yvette, France

Corresponding author: ^a brezin@corto.lpt.ens.fr

Received: 12 April 2002
Revised: 30 July 2002
Published online: 19 November 2002

Abstract

The field theory of a short range spin glass with Gaussian random interactions, is considered near the upper critical dimension six. In the glassy phase, replica symmetry breaking is accompanied with massless Goldstone modes, generated by the breaking of reparametrization invariance of a Parisi type solution. Twisted boundary conditions are thus imposed at two opposite ends of the system in order to study the size dependence of the twist free energy. A loop-expansion is performed to first order around a twisted background. It is found, as expected but it is non trivial, that the theory does renormalize around such backgrounds, as well as for the bulk. However two main differences appear, in comparison with simple ferromagnetic transitions: (i) the loop expansion yields a (negative) anomaly in the size dependence of the free energy, thereby lifting the lower critical dimension to a value greater than two (ii) the free energy is lowered by twisting the boundary conditions. This situation is common in spin glasses, reflecting the non-positivity of mode multiplicity in replica symmetry breaking, but its physical meaning is still unclear.