Generic replica symmetric field-theory for short range Ising spin glasses

Abstract

Symmetry considerations and a direct, Hubbard-Stratonovich type, derivation are used to construct a replica field-theory relevant to the study of the spin glass transition of short range models in a magnetic field. A mean-field treatment reveals that two different types of transitions exist, whenever the replica number n is kept larger than zero. The Sherrington-Kirkpatrick critical point in zero magnetic field between the paramagnet and replica magnet (a replica symmetric phase with a nonzero spin glass order parameter) separates from the de Almeida-Thouless line, along which replica symmetry breaking occurs. We argue that for studying the de Almeida-Thouless transition around the upper critical dimension d = 6, it is necessary to use the generic cubic model with all the three bare masses and eight cubic couplings. The critical role n may play is also emphasized. To make perturbative calculations feasible, a new representation of the cubic interaction is introduced. To illustrate the method, we compute the masses in one-loop order. Some technical details and a list of vertex rules are presented to help future renormalisation-group calculations.

Critical dynamics in glassy systems

“…Note that this definition is different from those we employed in the previous case where δq ab was the difference between the order parameter and solution at the given value of r and therefore there was no constant term in the equation. At n = 1 not only the replicon eigenvalue vanishes but also the longitudinal one [29,30] this is connected with the fact that although m 2 is finite at the transition it gives a vanishing contribution at n = 1 i.e. for δq ab = δq we have:…”

“…In this context it is convenient to consider the fluctuations of the overlaps between different replicas of the same system (additionally they must have the same initial condition in the n = 1 case). As explained in appendix B one has to consider at least six different replicas in order to evaluate the following eight cubic overlaps: then the six-point cumulants ω 1 and ω 2 can be obtained using the following formulas [30]:…”

“…It follows from the fact that summing over a replica index we are removing the singular replicon component from δQ ab and we are left with only the regular anomalous and longitudinal component. Another possible way of obtaining the above result is by considering the expression of the longitudinal and anomalous eigenvalues in terms of G 1 , G 2 and G 3 (see [30]) and impose that, contrary to the replicon, they remains finite, i.e. the singular part of their inverse is zero.…”

Gardner transition in finite dimensions

“…As a consequence, one has to consider all quadratic and cubic terms allowed by replica symmetry. By analyzing the quadratic terms of the expansion of the action one recognizes that the replica field theory has a mass matrix that can be easily diagonalized [40]. Three distinct eigenvalues are found: the replicon, the longitudinal and the anomalous one.…”

Replica symmetry breaking in and around six dimensions

“…This is the generic replica symmetric field theory elaborated in Refs. [9,15]. The vicinity of the (hypothetical) zero temperature fixed point can be studied in this field theory by assuming a hard (practically infinite) longitudinal mass, thus projecting the theory into the replicon sector.…”