Level Of Replica Symmetry Breaking Research Articles

Parisi Formula, Disorder Chaos and Fluctuation for the Ground State Energy in the Spherical Mixed p-Spin Models

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution

.In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai and Wolynes is realized in a strong sense in the mean-field limit. We also suggest how the computation could be generalized in an approximate way to finite-dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results.

Free energy of mean-field spin-glass models: Evolution operator and perturbation expansion

The full mean-field solution of spin glass models with a continuous order-parameter function is not directly available and approximate schemes must be used to assess its properties. The averaged physical quantities are to be represented via the replica trick and the limit to zero number of replicas is to be performed for each of them. To avoid this we introduce a perturbation expansion for a mean-field free-energy functional with a continuous order-parameter function without the need to refer to the replica trick. The expansion can be used to calculate all physical quantities in all mean-field spin-glass models and at all temperatures, including zero temperature. The small expansion parameter is a difference between the continuous order-parameter function and the corresponding order parameter from the solution with one level of replica-symmetry breaking. The first correction beyond the approximation with one level of replica-symmetry breaking is explicitly evaluated in the glassy phase of the Sherrington-Kirkpatrick model.

Mixtures of ultracold atoms in one-dimensional disordered potentials

We study interacting 1D two-component mixtures of cold atoms in a random potential, and extend the results reported earlier [{\it Phys. Rev. Lett.} {\bf 105}, 115301 (2010)]. We construct the phase diagram of a disordered Bose-Fermi mixture as a function of the strength of the Bose-Bose and Bose-Fermi interactions, and the ratio of the bosonic sound velocity and the Fermi velocity. Performing renormalization group and variational calculations, three phases are identified: (i) a fully delocalized two-component Luttinger liquid with superfluid bosons and fermions (ii) a fully localized phase with both components pinned by disorder, and (iii) an intermediate phase where fermions are localized but bosons are superfluid. Within the variational approach, each phase corresponds to a different level of replica symmetry breaking. In the fully localized phase we find that the bosonic and fermionic localization lengths can largely differ. We also compute the momentum distribution as well as the structure factor of the atoms (both experimentally accessible), and discuss how the three phases can be experimentally distinguished.

Survey propagation for the cascading Sourlas code

We investigate how insights from statistical physics, namely survey propagation, can improve decoding of a particular class of sparse error correcting codes. We show that a recently proposed algorithm, time averaged belief propagation, is in fact intimately linked to a specific survey propagation for which Parisi's replica symmetry breaking parameter is set to zero, and that the latter is always superior to belief propagation in the high connectivity limit. We briefly look at further improvements available by going to the second level of replica symmetry breaking.

On the formal equivalence of the TAP and thermodynamic methods in the SK model

We revisit two classic Thouless-Anderson-Palmer (TAP) studies of the Sherrington-Kirkpatrick model [Bray A J and Moore M A 1980 J. Phys. C 13, L469; De Dominicis C and Young A P, 1983 J. Phys. A 16, 2063]. By using the Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry, we prove the general equivalence of TAP and replica partition functions, and show that the annealed calculation of the TAP complexity is formally identical to the quenched thermodynamic calculation of the free energy at one step level of replica symmetry breaking. The complexity we obtain by means of the BRST symmetry turns out to be considerably smaller than the previous non-symmetric value.

Mean-field theory and fluctuations in Potts spin glasses. I

A short-range Potts spin glass is studied. A stable, non-marginal, mean-field theory is found with one level of replica symmetry breaking and a discontinuous transition for p>4. A complete stability analysis is provided, and two different correlation lengths are found above eight dimensions. The fluctuations in the ordered phase around this solution are incorporated in a renormalisation group approach and it is found that for a small range of the parameters they restore scaling close to the upper critical dimension. For the three-state case it appears that fluctuations destroy the stability of the solution for d<or=8 and cause the system to undergo a first-order phase transition. Non-universal corrections to the equation of state above the upper critical dimension are discussed.