On infrared divergences in spin glasses

By studying the structure of infrared divergences in a toy propagator in the replica approach to the Ising spin glass below $T_c$, we suggest a possible cancellation mechanism which could decrease the degree of singularity in the loop expansion.Comment: 13 pages, Latex , revised versio

Dynamics versus replicas in the random field Ising model

In a previous article we have shown, within the replica formalism, that the conventional picture of the random field Ising model breaks down, by the effect of singularities in the interactions between fields involving several replicas, below dimension eight. In the zero-replica limit several coupling constants have thus to be considered, instead of just one. As a result we had found that there is no stable fixed point in the vicinity of dimension six. It is natural to reconsider the problem in a dynamical framework, which does not require replicas, although the equilibrium properties should be recovered in the large time limit. Singularities in the zero-replica limit are a priori not visible in a dynamical picture. In this note we show that in fact new interactions are also generated in the stochastic approach. Similarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin $t_0$ at which initial data are given. The dynamical properties are thus dependent upon the waiting time. It is shown here that one can indeed find a complete correspondence between the equilibrium singularities in the $n=0$ limit, and the singularities in the dynamics when the initial time $t_0$ goes to minus infinity, with $n$ replaced by $-\frac{1}{t_0}$. There is thus complete coherence between the two approaches.Comment: 8 pages, latex, no figur

The Sherrington-Kirkpatrick model near T_c and near T=0

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 $\delta Q_{ab} = (Q_{ab} - Q_{ab}^{\rm RS})$ 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 $x\ll T\ll 1$ where the Hessian maintains the standard structure of Parisi Replica Symmetry Breaking with bands of eigenvalues bounded below by zero modes. For $T\ll x \leq 1$ 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.Comment: 11 pages, 3 figures, Published versio

Twist Free Energy in a Spin Glass

The field theory of a short range spin glass with Gaussian random interactions, is considered near the upper critical dimension six. In the glassy phase, replica symmetry breaking is accompanied with massless Goldstone modes, generated by the breaking of reparametrization invariance of a Parisi type solution. Twisted boundary conditions are thus imposed at two opposite ends of the system in order to study the size dependence of the twist free energy. A loop-expansion is performed to first order around a twisted background. It is found, as expected but it is non trivial, that the theory does renormalize around such backgrounds, as well as for the bulk. However two main differences appear, in comparison with simple ferromagnetic transitions : (i) the loop expansion yields a (negative) anomaly in the size dependence of the free energy, thereby lifting the lower critical dimension to a value greater than two given by $d_c = 2-\eta(d_c)$ (ii) the free energy is lowered by twisting the boundary conditions. This sign may reflect a spontaneous spatial non-uniformity of the order parameter.Comment: 15 pages, latex, no figur

Twist Free Energy

One may impose to a system with spontaneous broken symmetry, boundary conditions which correspond to different pure states at two ends of a sample. For a discrete Ising-like broken symmetry, boundary conditions with opposite spins in two parallel limiting planes, generate an interface and a cost in free energy per unit area of the interface. For continuum symmetries the order parameter interpolates smoothly between the end planes carrying two different directions of the order parameter. The cost in free energy is then proportional to $L^{d-2}$ for a system of characteristic size L. The power of $L$ is related to the lower critical dimension, and the coefficient of this additional free energy vanishes at the critical temperature. In this note it is shown within a loop expansion that one does find the expected behavior of this twist free energy. This is a preamble to the study of situations where the broken continuum symmetry is believed to be more complex, as in Parisi's ansatz for the Edwards-Anderson spin glass.Comment: 15 pages, latex, no figur

Spin Glass Field Theory with Replica Fourier Transforms

We develop a field theory for spin glasses using Replica Fourier Transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas $n$ and the number of replica symmetry breaking steps $R$ generic integers. We show how the RFT applied to the two-replica fields allows to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows to express any $i$-replica vertex in the new RFT basis and hence enables to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass.Comment: 39 pages, 3 figure

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w^2. Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the \epsilon-expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w^2 and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed.Comment: 14 page

On the structure of correlations in the three dimensional spin glasses

We investigate the low temperature phase of three-dimensional Edwards-Anderson model with Bernoulli random couplings. We show that at a fixed value $Q$ of the overlap the model fulfills the clustering property: the connected correlation functions between two local overlaps decay as a power whose exponent is independent of $Q$ for all $0\le |Q| < q_{EA}$. Our findings are in agreement with the RSB theory and show that the overlap is a good order parameter.Comment: 5 pages, 5 figure