The Replica Fourier Transform is the generalization of the discrete Fourier Transform to quantities defined on an ultrametric tree. It finds use in con- junction of the replica method used to study thermodynamics properties of disordered systems such
as spin glasses. Its definition is presented in a system- atic and simple form and its use illustrated with some representative examples. In particular we give a detailed discussion of the diagonalization in the Replica Fourier Space of the Hessian matrix of the Gaussian fluctuations about the mean field saddle point of spin glass theory. The general results are finally discussed for a generic spherical spin glass model, where the Hessian can be computed analytically.
Some recent results concerning the Sherrington-Kirkpatrick model are reported. For $T$ near the critical temperature $T_c$, the replica free energy of the Sherrington-Kirkpatrick model is taken as the starting point of an expansion in powers of $delt
a Q_{ab} = (Q_{ab} - Q_{ab}^{rm RS})$ about the Replica Symmetric solution $Q_{ab}^{rm RS}$. The expansion is kept up to 4-th order in $delta{bm Q}$ where a Parisi solution $Q_{ab} = Q(x)$ emerges, but only if one remains close enough to $T_c$. For $T$ near zero we show how to separate contributions from $xll Tll 1$ where the Hessian maintains the standard structure of Parisi Replica Symmetry Breaking with bands of eigenvalues bounded below by zero modes. For $Tll x leq 1$ the bands collapse and only two eigenvalues, a null one and a positive one, are found. In this region the solution stands in what can be called a {sl droplet-like} regime.
To test the stability of the Parisi solution near T=0, we study the spectrum of the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In th
e limit $Tll 1$ two regions can be identified. In the first region, for $x$ close to 0, where $x$ is the Parisi replica symmetry breaking scheme parameter, the spectrum of the Hessian is not trivial and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region $Tll x leq 1$ as $Tto 0$, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full replica symmetry breaking state collapse. In this region only two eigenvalues are found: a null one and a positive one, ensuring stability for $Tll 1$. In the limit $Tto 0$ the width of the first region shrinks to zero and only the positive and null eigenvalues survive. As byproduct we enlighten the close analogy between the static Parisi replica symmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it equals the static limit of the dynamic susceptibility computed via the modified fluctuation dissipation theorem.
We study the spectrum of the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit $Tll 1$ two regions can be identified. The firs
t for $x$ close to 0, where $x$ is the Parisi replica symmetry breaking scheme parameter. In this region the spectrum of the Hessian is not trivial, and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region $Tll x leq 1$ as $Tto 0$, the bands typical of the full replica symmetry breaking state collapse and only two eigenvalues are found: a null one and a positive one. We argue that this region has a droplet-like behavior. In the limit $Tto 0$ the width of the full replica symmetry breaking region shrinks to zero and only the droplet-like scenario survives.
An expansion for the free energy functional of the Sherrington-Kirkpatrick (SK) model, around the Replica Symmetric SK solution $Q^{({rm RS})}_{ab} = delta_{ab} + q(1-delta_{ab})$ is investigated. In particular, when the expansion is truncated to fou
rth order in. $Q_{ab} - Q^{({rm RS})}_{ab}$. The Full Replica Symmetry Broken (FRSB) solution is explicitly found but it turns out to exist only in the range of temperature $0.549...leq Tleq T_c=1$, not including T=0. On the other hand an expansion around the paramagnetic solution $Q^{({rm PM})}_{ab} = delta_{ab}$ up to fourth order yields a FRSB solution that exists in a limited temperature range $0.915...leq T leq T_c=1$.
The spin glass behavior near zero temperature is a complicated matter. To get an easier access to the spin glass order parameter $Q(x)$ and, at the same time, keep track of $Q_{ab}$, its matrix aspect, and hence of the Hessian controlling stability,
we investigate an expansion of the replicated free energy functional around its ``spherical approximation. This expansion is obtained by introducing a constraint-field and a (double) Legendre Transform expressed in terms of spin correlators and constraint-field correlators. The spherical approximation has the spin fluctuations treated with a global constraint and the expansion of the Legendre Transformed functional brings them closer and closer to the Ising local constraint. In this paper we examine the first contribution of the systematic corrections to the spherical starting point.
