Day 3 - Talks

Entanglement is a fundamental feature of quantum physics. It is often characterized and quantified in terms of the purity of the subsystems of a quantum object. We discuss the phase transitions undergone by the entanglement of a large bipartite quantum system. We frame the problem in terms of random matrices and characterize the spectra of the reduced density matrices. For a given subsystem, purity can always be minimized by taking a suitable (pure) state. When many subsystems are considered, the requirement that purity be minimal for all of them can engender conflicts and frustration arises. This unearths an interesting link between frustration and multipartite entanglement.

One of the fundamental contributions of Giorgio Parisi to our understanding of complex systems is the definition and mathematical description of a prominent mechanism for the breakdown of ergodicity, now widely known as replica symmetry breaking. Recently, a major breakthrough in the field of disordered quantum systems has identified another universal mechanism for breakdown of ergodicity, which bears the name of many-body localization. I will discuss the latter in connection with the former and their interplay in some toy models.

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linked to the O(N) scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, N → ∞, we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the N → ∞ limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N−1 integrals of motion, notably, their scaling with N, and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent GibbsBoltzmann behaviour of the global observables. We elaborate on the role played by these constants of motion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a generalised Gibbs ensemble.

Spontaneous synchronization is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronization of phase oscillators. In this talk, I will summarize recent results on the study of a generalized Kuramoto model that includes inertial effects and stochastic noise. I will describe the dynamics from a different perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. Using tools of statistical physics, I will highlight the equilibrium and nonequilibrium aspects of the dynamics and uncover the rich and complex phase diagram that the model exhibits.

Nonequilibrium pervades in nature. From living cells to the expanding universe virtually all energy processes in nature occur in nonequilibrium conditions. The possibility of using electromagnetic fields to exert forces in the range of the piconewton has spurred the development of new experimental techniques such as optical tweezers that are capable of manipulating single molecules with unprecedented accuracy. In this talk I will present recent investigations carried out in my lab that apply the finest tools from statistical physics to characterize equilibrium breakdown in single molecules and cells. I will show how it is possible to measure violations of the fluctuation dissipation with single molecules and cells manipulated by optical traps. These experiments lay the ground to explore new physical concepts and tools essential for our understanding of nonequilibrium phenomena in physics and beyond.

We consider the Sherrington-Kirkpatrick (SK) model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as “Z2 synchronization”. Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the Thouless-Anderson-Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, provided the ferromagnetic bias is larger than a constant (i.e. the noise level is small enough in synchronization). Namely, we prove that the scaled distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. As a crucial technical step (of independent interest) we compute a tight upper bound on the annealed complexity of the SK model using the Kac-Rice formula.

Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? In this talk we consider a simple case, namely two-layers neural networks, and prove that in a suitable scaling limit SGD dynamics is captured by a certain non-linear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples, and show how DD can be used to prove convergence of SGD to networks with nearly-ideal generalization error. This description allows to `average-out' some of the complexities of the landscape of neural networks, and can be used to prove a general convergence result for noisy SGD.

In recent years, ideas from statistical physics of disordered systems, notably the cavity method, have helped to develop new algorithms for important inference problems, ranging from community detection to compressed sensing, machine learning (neural networks) and generalized linear regression. The talk will review these developments and explain how they can be used, together with the replica method, to identify phase transitions in benchmark ensembles of inference problems.