J. Phys. I France
Volume 7, Numéro 12, December 1997
Page(s) 1513 - 1521
DOI: 10.1051/jp1:1997152
J. Phys. I France 7 (1997) 1513-1521

Variational Approach to Interfaces in Random Media: Negative Variances and Replica Symmetry Breaking

D.B. Saakian1 and Th.M. Nieuwenhuizen2

1  Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia
2  Van der Waals-Zeeman Institute, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

(Received 1 April 1997, received in final form 26 August 1997, accepted 2 September 1997)

A Gaussian variational approximation is often used to study interfaces in random media. By considering the 1+1 dimensional directed polymer in a random medium, it is shown here that the variational Ansatz typically leads to a negative variance of the free energy. The situation improves by taking into account more and more steps of replica symmetry breaking. For infinite order breaking the variance is zero (i.e. subextensive). This situation is reminiscent of the negative entropies in mean field spin glass models, which were also eliminated by considering infinite order replica symmetry breaking.

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