The Distributional Zeta-Function in Disordered Field Theory

TL;DR

This paper introduces a rigorous mathematical method using a distributional zeta-function to compute the average free energy in disordered systems, simplifying the replica approach and avoiding complex permutation group analysis.

Contribution

It presents a new formalism based on a distributional zeta-function for calculating average free energy, bypassing the need for permutation group understanding in replica methods.

Findings

The method accurately computes the average free energy of disordered systems.

It demonstrates the approach on the disordered λφ^4 model in arbitrary dimensions.

The second contribution to the free energy can be made arbitrarily small.

Abstract

In this paper we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which can not be written as a series of the integer moments, can be made as small as desired. This result supports the use of integer moments of the partition function, computed via replicas, for expressing the average free energy of the system. One advantage of the proposed formalism is that it does not require the understanding of the properties of the permutation group when the number of replicas goes to zero. Moreover, the symmetry is broken using the saddle-point equations of the model. As an application for the distributional zeta-function technique, we obtain the average free energy of the disordered $\lambda\varphi^{4}$ model defined in a $d$-dimensional Euclidean space.