Low temperature spin glass fluctuations: expanding around a spherical approximation

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.Comment: 16 pages, 2 figure

Mode-Coupling as a Landau Theory of the Glass Transition

We derive the Mode Coupling Theory (MCT) of the glass transition as a Landau theory, formulated as an expansion of the exact dynamical equations in the difference between the correlation function and its plateau value. This sheds light on the universality of MCT predictions. While our expansion generates higher order non-local corrections that modify the standard MCT equations, we find that the square root singularity of the order parameter, the scaling function in the \beta regime and the functional relation between the exponents defining the \alpha and \beta timescales are universal and left intact by these corrections.Comment: 6 pages, 1 figure, submitted to EPL; corrected typos in the abstract; corrected minor typo in reference

Predictive power of MCT: Numerics and Finite size scaling for a mean field spin glass

The aim of this paper is to test numerically the predictions of the Mode Coupling Theory (MCT) of the glass transition and study its finite size scaling properties in a model with an exact MCT transition, which we choose to be the fully connected Random Orthogonal Model. Surprisingly, some predictions are verified while others seem clearly violated, with inconsistent values of some MCT exponents. We show that this is due to strong pre-asymptotic effects that disappear only in a surprisingly narrow region around the critical point. Our study of Finite Size Scaling (FSS) show that standard theory valid for pure systems fails because of strong sample to sample fluctuations. We propose a modified form of FSS that accounts well for our results. {\it En passant,} we also give new theoretical insights about FSS in disordered systems above their upper critical dimension. Our conclusion is that the quantitative predictions of MCT are exceedingly difficult to test even for models for which MCT is exact. Our results highlight that some predictions are more robust than others. This could provide useful guidance when dealing with experimental data.Comment: 37 pages, 19 figure

Random transverse field Ising model in dimension d

For the quantum Ising model with ferromagnetic random couplings $J_{i,j}>0$ and random transverse fields $h_i>0$ at zero temperature in finite dimensions $d>1$, we consider the lowest-order contributions in perturbation theory in $(J_{i,j}/h_i)$ to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as : $\ln C(r) \sim - \frac{r}{\xi_{typ}} +r^{\omega} u$, where $\xi_{typ}$ is the typical correlation length, $u$ is a random variable, and $\omega$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the Directed Polymer with $D=(d-1)$ transverse directions. Our main conclusions are (i) whenever $\omega>0$, the quantum model is governed by an Infinite-Disorder fixed point : there are two distinct correlation length exponents related by $\nu_{typ}=(1-\omega)\nu_{av}$ ; the distribution of the local susceptibility $\chi_{loc}$ presents the power-law tail $P(\chi_{loc}) \sim 1/\chi_{loc}^{1+\mu}$ where $\mu$ vanishes as $\xi_{av}^{-\omega}$, so that the averaged local susceptibility diverges in a finite neighborhood $0<\mu<1$ before criticality (Griffiths phase) ; the dynamical exponent $z$ diverges near criticality as $z=d/\mu \sim \xi_{av}^{\omega}$ (ii) in dimensions $d \leq 3$, any infinitesimal disorder flows towards this Infinite-Disorder fixed point with $\omega(d)>0$ (for instance $\omega(d=2)=1/3$ and $\omega(d=3) \sim 0.24$) (iii) in finite dimensions $d > 3$, a finite disorder strength is necessary to flow towards the Infinite-Disorder fixed point with $\omega(d)>0$ (for instance $\omega(d=4) \simeq 0.19$), whereas a Finite-Disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension $d=\infty$ where $\omega=0$, we discuss the similarities and differences with the case of finite dimensions.Comment: 22 pages, v2=final versio