Spin-glass theory is one of the leading paradigms of complex physics and describes condensed matter, neural networks and biological systems, ultracold atoms, random photonics and many other research fields. According to this theory, identical systems under identical conditions may reach different states. This effect is known as replica symmetry breaking and is revealed by the shape of the probability distribution function of an order parameter named the Parisi overlap. However, a direct experimental evidence in any field of research is still missing. Here we investigate pulse-to-pulse fluctuations in random lasers, we introduce and measure the analogue of the Parisi overlap in independent experimental realizations of the same disordered sample, and we find that the distribution function yields evidence of a transition to a glassy light phase compatible with a replica symmetry breaking.

We introduce a diluted version of the one dimensional spin-glass model with interactions decaying in probability as an inverse power of the distance. In this model varying the power corresponds to change the dimension in short-range models. The spin-glass phase is studied in and out of the range of validity of the mean-field approximation in order to discriminate between different theories. Since each variable interacts only with a finite number of others the cost for simulating the model is drastically reduced with respect to the fully connected version and larger sizes can be studied. We find both static and dynamic evidence in favor of the so-called replica symmetry breaking theory.PACS numbers: 75.10. Nr,71.55.Jv,05.70.Fh Mean-field spin glass models are known to have a rather complex low-temperature phase [1], which has not been clearly observed so far in numerical simulations of finite-dimensional models with short range interactions. Theories alternative to the mean-field one have been proposed [2], but short-range systems with quenched disorder are very tough to study analytically [3]. Numerical simulations have been, thus, extensively employed, developing more and more refined algorithms over the years, though with no conclusive indication on the nature of the spin-glass phase in finite dimension.Long-range models are such that their lower critical dimension is lower than the one of the corresponding short range model. In particular, one can have a phase transition even in one dimensional systems, provided the range of interaction is large enough. One dimensional spin glass models with power-law decaying interactions actually allow to explore both long-and short-range regimes by changing the power [4,5,6,7]. These models would be perfect candidates for comparing the spin glass phase in and out of the range of validity of the mean field approximation. Unfortunately, since each variable interacts with all the others, numerical simulations are very computer demanding and it is hard to get a clear numerical evidence supporting a specific spin glass theory [6,7].We, therefore, introduce a diluted version of the model, where the mean coordination number is fixed (see also Ref. 8). In diluting, the run time grows as the size N of the system, rather than proportionally to N 2 . This is a fundamental issue because finite volume effects are strong in these models: previous studies were restricted to N ≤ 512, while we can now thermalize systems up to N = 16384, thus keeping these effects under control.We are interested in analyzing the difference among the predictions on the spin glass phase of the droplet theory [2], the trivial-non-trivial (TNT) scenario [9] and the replica symmetry breaking (RSB) theory [1]. Studying the thermodynamics, we focus on site and link overlaps, providing strong evidence that both fluctuate in the infinite volume limit. From the dynamic behavior we learn that the four-point correlation function goes to zero at large distances when extrapolated at infinite times. In this framework w...

We present a unified approach to the theory of multimodal laser cavities including a variable amount of structural disorder. A general mean-field theory is studied for waves in media with variable nonlinearity and randomness. Phase diagrams are reported in terms of optical power, degree of disorder, and degree of nonlinearity, tuning between closed and open cavity scenarios. In the thermodynamic limit of infinitely many modes, the theory predicts four distinct regimes: a continuous wave behavior for low power, a standard mode-locking laser regime for high power and weak disorder, a random laser for high pumped power and large disorder, and a novel intermediate regime of phase locking occurring in the presence of disorder but below the lasing threshold.

A careful critical analysis of the complexity, at the annealed level, of the Sherrington-Kirkpatrick model has been performed. The complexity functional is proved to be always invariant under the Becchi-Rouet-StoraTyutin supersymmetry, disregarding the formulation used to define it. We consider two different saddle points of such functional, one satisfying the supersymmetry ͓A. Cavagna et al., J. Phys. A 36, 1175 ͑2003͔͒ and the other one breaking it ͓A. J. Bray and M. A. Moore, J. Phys. C 13, L469 ͑1980͔͒. We review the previews studies on the subject, linking different perspectives and pointing out some inadequacies and even inconsistencies in both solutions.

Abstract.A picture for thermodynamics of the glassy state is introduced. It assumes that one extra parameter, the effective temperature, is needed to describe the glassy state. This explains the classical paradoxes concerning the Ehrenfest relations and the PrigogineDefay ratio.As a second part, the approach connects the response of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized non-equilibrium way. I INTRODUCTIONNon-equilibrium thermodynamics for systems far from equilibrium has long been a field of confusion. A typical application is window glass. Such a system is far from equilibrium: a cubic micron of glass is neither a crystal nor an ordinary undercooled liquid. It is an under-cooled liquid that, in the glass formation process, has fallen out of its meta-stable equilibrium.Until our recent works on this field, the general consensus reached after more than half a century of research was: Thermodynamics does not work for glasses, because there is no equilibrium [1]. This conclusion was mainly based on the failure to understand the Ehrenfest relations and the related Prigogine-Defay ratio. It should be kept in mind that, so far, the approaches leaned very much on equilibrium ideas. Well known examples are the 1951 Davies-Jones paper [2], the 1958 GibbsDiMarzio [3] and the 1965 Adam-Gibbs [4] papers, while a 1981 paper by DiMarzio has title "Equilibrium theory of glasses" and a subtitle "An equilibrium theory of glasses is absolutely necessary" [5]. We shall stress that such approaches are not applicable, due to the inherent non-equilibrium character of the glassy state.Thermodynamics is the most robust field of physics. Its failure to describe the glassy state is quite unsatisfactory, since up to 25 decades in time can be involved. Naively we expect that each decade has its own dynamics, basically independent of the other ones. We have found support for this point in models that can be solved exactly. Thermodynamics then means a description of system properties under smooth enough non-equilibrium conditions.

The statistical properties of the phases of several modes nonlinearly coupled in a random system are investigated by means of a Hamiltonian model with disordered couplings. The regime in which the modes have a stationary distribution of their energies and in which the phases are coupled is studied for arbitrary degrees of randomness and energy. The complexity versus temperature and strength of nonlinearity is calculated. A phase diagram is derived in terms of the stored energy and amount of disorder. Implications in random lasing, nonlinear wave propagation, and finite-temperature Bose-Einstein condensation are discussed

We study the quenched complexity in spin-glass mean-field models satisfying the Becchi-Rouet-Stora-Tyutin supersymmetry. The outcome of such study, consistent with recent numerical results, allows, in principle, to conjecture the absence of any supersymmetric contribution to the complexity in the Sherrington-Kirkpatrick model. The same analysis can be applied to any model with a full replica symmetry breaking phase, e.g., the Ising p-spin model below the Gardner temperature. The existence of different solutions, breaking the supersymmetry, is also discussed.

We investigate mode-locking processes in lasers displaying a variable degree of structural randomness. By a spin-glass theoretic approach, we analyze the mean-field Hamiltonian and derive a phase diagram in terms of pumping rate and degree of disorder. Paramagnetic (noisy continuous wave emission), ferromagnetic (standard passive mode locking), and spin-glass phases with an exponentially large number of configurations are identified. The results are also relevant for other physical systems displaying a random Hamiltonian, such as Bose-condensed gases and nonlinear optics.