Dynamic Fluctuations in a Short-range Spin Glass Model

Abstract

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for T→Tc+. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order O(1/N) and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian L for the fields δQαβi(t1, t2) up to the cubic order in the series expansion around the dynamic Mean-Field value $\overline{Q^{\alpha \beta}}(t_{1}, t_{2})$. We find the more general expression for the time depending non-local fluctuations, the propagators [〈δQαβi(t1, t2)δQαβj(t3, t4)〉ξ]J, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices)

Mean field dynamical exponents in finite-dimensional Ising spin glass

“…In the thermodynamic limit, as shown in [16], the Hartree-Fock approximation is exact and we can linearize the Langevin equation:…”

Dynamical fluctuations in an exactly solvable model of spin glasses

“…At the dynamical critical temperature, where dynamical scaling is supposed to hold and the off-equilibrium features are not relevant, these equations have been solved for some models [7].…”

One-loop contribution to the dynamical exponents in spin glasses

“…In a previous work, [6], we evaluated the Gaussian dynamical fluctuations of the order parameter around the MF limit. The aim of this letter is to pursue this analysis, by considering the 1-loop correction to the mean field (MF) theory in a renormalization group calculation.…”

“…(20) Let us consider the 1-loop correction to the "free" theory. We intend to use the propagators derived in [6] to evaluate the contribution of the 1-loop Feynman diagrams to the mean value Q αβ , to the bare propagators G αβγδ , and to the bare cubic vertices. We consider the 1-loop Feynman diagrams as a g-series expansion, by using the correspondent propagators.…”

“…We succeed in evaluating the singular behaviour of the diagrams A 1 and A 2 , while we have to use a trick to take into account the contribution from the others. By using the series expansion in g of the propagators in the loop, see [6], we obtain a correspondent g series of the two 1P.I. diagrams, but the zero term is missing.…”