A note on the Guerra and Talagrand theorems for Mean Field Spin Glasses: the simple case of spherical models

The aim of this paper is to discuss the main ideas of the Talagrand proof of the Parisi Ansatz for the free-energy of Mean Field Spin Glasses with a physicist's approach. We consider the case of the spherical $p$-spin model, which has the following advantages: 1) the Parisi Ansatz takes the simple ``one step replica symmetry breaking form'', 2) the replica free-energy as a function of the order parameters is simple enough to allow for numerical maximization with arbitrary precision. We present the essential ideas of the proof, we stress its connections with the theory of effective potentials for glassy systems, and we reduce the technically more difficult part of the Talagrand's analysis to an explicit evaluation of the solution of a variational problem.Comment: 20 pages, 5 figures. Added references and minor language correction

Metastable States, Relaxation Times and Free-energy Barriers in Finite Dimensional Glassy Systems

In this note we discuss metastability in a long-but-finite range disordered model for the glass transition. We show that relaxation is dominated by configuration belonging to metastable states and associate an in principle computable free-energy barrier to the equilibrium relaxation time. Adam-Gibbs like relaxation times appear naturally in this approach.Comment: 4 pages, 2 figures. Typos correcte

Local Spin Glass Order in 1D

We study the behavior of one dimensional Kac spin glasses as function of the interaction range. We verify by Montecarlo numerical simulations the crossover from local mean field behavior to global paramagnetism. We investigate the behavior of correlations and find that in the low temperature phase correlations grow at a faster rate then the interaction range. We completely characterize the growth of correlations in the vicinity of the mean-field critical region

Non-neutral theory of biodiversity

We present a non-neutral stochastic model for the dynamics taking place in a meta-community ecosystems in presence of migration. The model provides a framework for describing the emergence of multiple ecological scenarios and behaves in two extreme limits either as the unified neutral theory of biodiversity or as the Bak-Sneppen model. Interestingly, the model shows a condensation phase transition where one species becomes the dominant one, the diversity in the ecosystems is strongly reduced and the ecosystem is non-stationary. This phase transition extend the principle of competitive exclusion to open ecosystems and might be relevant for the study of the impact of invasive species in native ecologies.Comment: 4 pages, 3 figur

Replica bounds for diluted non-Poissonian spin systems

In this paper we extend replica bounds and free energy subadditivity arguments to diluted spin-glass models on graphs with arbitrary, non-Poissonian degree distribution. The new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method. As a byproduct we obtain self-averaging identities that generalize the Ghirlanda-Guerra ones to the multi-overlap case.Comment: Latex file, 15 pages, 2 eps figures; Weak point revised and corrected; Misprints correcte

Quasiparticle interference patterns as a test for the nature of the pseudogap phase in the cuprate superconductors

Electrons, when scattered by static random disorder, form standing waves that can be imaged using scanning tunneling microscopy. Such interference patterns, observable by the recently developed technique of Fourier transform scanning tunneling spectroscopy (FT-STS), are shown to carry unique fingerprints characteristic of the electronic order present in a material. We exploit this feature of the FT-STS technique to propose a test for the nature of the enigmatic pseudogap phase in the high-$T_c$ cuprate superconductors. Through their sensitivity to the quasiparticle spectra and coherence factors, the FT-STS patterns in principle carry enough information to unambiguously determine the nature of the condensate responsible for the pseudogap phenomenon. We argue that the next generation of FT-STS experiments, currently underway, should be able to distinguish between the pseudogap dominated by the remnants of superconducting order from the pseudogap dominated by some competing order in the particle-hole channel. Using general arguments and detailed numerical calculations, we point to certain fundamental differences between the two scenarios and discuss the prospects for future experiments.Comment: 15 pages REVTeX + 9 ps figures. For related work and info visit http://www.physics.ubc.ca/~franz; version 2 to appear in IJMP

How glasses explore configuration space

We review a statistical picture of the glassy state derived from the analysis of the off-equilibrium fluctuation-dissipation relations. We define an ultra-long time limit where ``one time quantities'' are close to equilibrium while response and correlation can still display aging. In this limit it is possible to relate the fluctuation-response relation to static breaking of ergodicity. The resulting picture suggests that even far from that limit, the fluctuation-dissipation ratio relates to the rate of growth of the configurational entropy with free-energy density.Comment: To appear in the proceedings of the "3rd workshop on non-equilibrium phenomena in supercooled fluids, glasses and amorphous materials" Pisa 22-27 September 200

Stability of Majorana Fermions in Proximity-Coupled Topological Insulator Nanowires

It has been shown previously that a finite-length topological insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, realizes a one-dimensional topological superconductor with an unpaired Majorana fermion (MF) localized at each end of the nanowire. Here, we study the stability of these MFs with respect to various perturbations that are likely to occur in a physical realization of the proposed device. We show that the unpaired Majorana fermions persist in this system for any value of the chemical potential inside the bulk band gap of order 300 meV in Bi$_2$Se$_3$ by computing the Majorana number. From this calculation, we also show that the unpaired Majorana fermions persist when the magnetic flux through the nanowire cross-section deviates significantly from half flux quantum. Lastly, we demonstrate that the unpaired Majorana fermions persist in strongly disordered wires with fluctuations in the on-site potential ranging in magnitude up to several times the size of the bulk band gap. These results suggest this solid-state system should exhibit unpaired Majorana fermions under accessible conditions likely important for experimental study or future applications.Comment: 17 pages, 13 figure

Nonequilibrium dynamics of a simple stochastic model

We investigate the low-temperature dynamics of a simple stochastic model, introduced recently in the context of the physics of glasses. The slowest characteristic time at equilibrium diverges exponentially at low temperature. On smaller time scales, the nonequilibrium dynamics of the system exhibits an aging regime. We present an analytical study of the scaling behaviour of the mean energy, of its local correlation and response functions, and of the associated fluctuation-dissipation ratio throughout the regime of low temperature and long times. This analysis includes the aging regime, the convergence to equilibrium, and the crossover behaviour between them.Comment: 36 pages, plain tex, 7 figures, to be published by Journal of Physics