On the special properties of generalized harmonic systems within the n → 0 replica approach

Spin glass models are relevant until today because they provide the simplest models to investigate underlying concepts like disorder and frustration. They are endowed with a kind of phase transition and hence a kind of organization at low temperature. In order to study systems with quenched disorder, a proper treatment involves averaging the free energy, but this task, as well known, is very difficult since we have to deal with the logarithm of the partition function to calculate the free energy which is ``self-averaging" in the thermodynamic limit. However, the average cannot be carried out directly because the random variables occur inside the logarithm. Spin glasses are highly random systems, and in many of the standard models, the nonrandom part is zero. This technique is therefore not generally applicable, though even for spin glasses it is valid at high temperatures and can be used to generate high-temperature series expansions. At low temperatures and certainly below a spin glass transition temperature, one needs a non-perturbative way of averaging over disorder. And this is the idea behind the replica approach. In this work we will use the replica approach to analyze the spectral properties of generalized harmonic systems. A replica symmetry breaking analysis is accomplished too.