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Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-Dimensional Topological Insulators
No full textTwo-dimensional topological insulators are characterized by an insulating bulk and conductive edge states protected by the nontrivial topology of the bulk electronic structure. They remain robust against moderate disorder until Anderson localization occurs and destroys the topological phase. Interestingly, disorder can also induce a topological phase - known as a topological Anderson insulator - starting from an otherwise pristine trivial phase. While topological invariants are generally regarded as global quantities, we argue that space-resolved topological markers can act as local order parameters, revealing the role of fluctuations and correlations in the local topology under Anderson disorder and vacancies. With this perspective, we perform numerical simulations of disorder-driven topological phase transitions in the Haldane and Kane-Mele models, using supercells with both open and periodic boundary conditions. We find that short-scale fluctuations of topological markers vanish upon coarse graining, except at the topological phase transition, where their correlation length peaks and large-scale fluctuations remain. Notably, such a topological correlation function is characterized by critical exponents that appear universal across disorder types, yet they can resolve different topological phase transitions
Axion-photon conversion down to the nonrelativistic regime
No full textIn the presence of a magnetic field, axions can convert into photons and vice versa. The phenomenology of the conversion is captured by a system of two coupled Klein-Gordon equations, which, assuming that the axion is relativistic, is usually recast into a pair of first-order Schrödinger-like equations. In such a limit, focusing on a constant magnetic field and plasma frequency, the equations admit an exact analytic solution. The relativistic limit significantly simplifies the calculations and, therefore, it is widely used in phenomenological applications. In this work, we discuss how to evaluate the axion-photon system evolution without relying on such relativistic approximation. In particular, we give an exact analytical solution, valid for any axion energy, in the case that both the magnetic field and plasma frequency are constant. Moreover, we devise an analytic perturbative expansion that allows for tracking the conversion probability in a slightly inhomogeneous magnetic field or plasma frequency, whose characteristic scale of variation is much larger than the typical axion-photon oscillation length. Finally, we discuss a case of resonant axion-photon conversion giving useful simplified formulae that might be directly applied to dark matter axions converting in neutron star magnetospheres
Quenching from superfluid to free bosons in two dimensions: Entanglement, symmetries, and the quantum Mpemba effect
No full textWe study the nonequilibrium dynamics of bosons in a two-dimensional optical lattice after a sudden quench from the superfluid phase to the free-boson regime. The initial superfluid state is described approximately using both the Bogoliubov theory and the Gaussian variational principle. The subsequent time evolution remains Gaussian, and we compare the results from each approximation of the initial state by examining different aspects of the dynamics. First, we analyze the entanglement entropy and observe that, in both cases, it increases linearly with time before reaching a saturation point. This behavior is attributed to the propagation of entangled pairs of quantum depletions in the superfluid state. Next, we explore the fate of particle-number symmetry, which is spontaneously broken in the superfluid phase. To do so, we use the entanglement asymmetry, a recently introduced observable that enables us to track symmetry breaking within a subsystem. We observe that its evolution varies qualitatively depending on the theory used to describe the initial state. However, in both cases, the symmetry remains broken and is never restored in the stationary state. Finally, we assess the time it takes to reach the stationary state by evaluating the quantum fidelity between the stationary reduced density matrix and the time-evolved one. Interestingly, within the Gaussian variational principle, we find that an initial state further from the stationary state can relax more quickly than one closer to it, indicating the presence of the recently discovered quantum Mpemba effect. We derive the microscopic conditions necessary for this effect to occur and demonstrate that these conditions are never met in the Bogoliubov theory
Dissipative and Measurement-Induced Phases in Many-Body Quantum Systems
No full textOpen quantum many-body systems are an important setting that allows us to study the interplay between dissipation and interactions, contributing to a more accurate description of real-life phenomena and experiments, and providing new features to be exploited both on theoretical and technological grounds. This thesis deals with the study of such systems across a wide range of settings. Beyond providing the theoretical tools to analyze and understand this framework, the thesis also addresses some important and modern concepts. A key goal is to show how dissipation enables the realization of new phases of matter, such as quantum synchronization and time-crystal phases. To achieve this, various theoretical tools will be employed, ranging from path integral formalism to a Lindblad master equation approach. These same tools will also allow us to pursue another important goal: studying the dynamics of many-body systems subject to continuous measurements. This will lead to observing the emergence of a recently discovered phenomenon, known as measurement-induced phase transitions, and will provide a natural framework for addressing the complexity arising from the stochastic nature of quantum measurements. This thesis constitutes a modern and comprehensive view of the field of open quantum many-body systems, with a special focus on novel phases of matter emerging from interactions with an external environment or with a measuring device
VC dimension of Graph Neural Networks with Pfaffian activation functions
No full textGraph Neural Networks (GNNs) have emerged in recent years as a powerful tool to learn tasks across a wide range of graph domains in a data-driven fashion. Based on a message passing mechanism, GNNs have gained increasing popularity due to their intuitive formulation, closely linked to the Weisfeiler-Lehman (WL) test for graph isomorphism, to which they were demonstrated to be equivalent (Morris et al., 2019 and Xu et al., 2019). From a theoretical point of view, GNNs have been shown to be universal approximators, and their generalization capability - related to the Vapnik Chervonekis (VC) dimension (Scarselli et al., 2018) - has recently been investigated for GNNs with piecewise polynomial activation functions (Morris et al., 2023). The aim of our work is to extend this analysis on the VC dimension of GNNs to other commonly used activation functions, such as the sigmoid and hyperbolic tangent, using the framework of Pfaffian function theory. Bounds are provided with respect to the architecture parameters (depth, number of neurons, input size) as well as with respect to the number of colors resulting from the 1-WL test applied on the graph domain. The theoretical analysis is supported by a preliminary experimental study
Resonant Large Deviations Principle for the Beating NLS Equation
No full textWe prove a large deviations principle for the solution to the beating nonlinear Schrödinger equation on the torus with random initial data supported on two Fourier modes. When these modes have different initial variance, we prove that the resonant energy exchange between them increases the likelihood of extreme wave formation. Our results show that nonlinear focusing mechanisms can lead to tail fattening of the probability measure of the sup-norm of the solution to a nonlinear dispersive equation
Going deeper into the dark with COSMOS-Web JWST unveils the total contribution of radio-selected NIR-faint galaxies to the cosmic star formation rate density
No full textWe present the first follow-up with JWST of radio-selected near-infrared (NIR)-faint galaxies as part of the COSMOS-Web survey. By selecting galaxies detected at radio frequencies (S3 GHz > 11.5 μJy; i.e., S/N > 5) and with faint counterparts at NIR wavelengths (F150W > 26.1 mag), we collected a sample of 127 likely dusty star-forming galaxies (DSFGs). We estimated their physical properties through SED fitting, computed the first radio luminosity function for these types of sources and their contribution to the total cosmic star formation rate density. Our analysis confirms that these sources represent a population of highly dust-obscured (hAvi ∼ 3.5 mag) massive (hM?i ∼ 1010.8 M) and star-forming galaxies (hSFRi ∼ 300 M yr−1) located at hzi ∼ 3.6, representing the high-redshift tail of the full distribution of radio sources. Our results also indicate that these galaxies could dominate the bright end of the radio luminosity function and reach a total contribution to the cosmic star formation rate density equal to that estimated only considering NIR-bright sources at z ∼ 4.5. Finally, our analysis further confirms that the radio selection can be employed to collect statistically significant samples of DSFGs, representing a complementary alternative to the other selections based on JWST colors or detection at FIR/(sub)millimeter wavelengths
Variational problems in a nonsmooth geometric setting
No full textIn its long history, the Calculus of Variations has developed a rich bag of words, ideas and tools to deal with nonsmoothness. This thesis presents the results of three projects where such nonsmoothness appears with a distinct geometric flavour, be it in the very objects that are described or in the domains where they are defined.
The first project deals with metric measure geometry in high dimension. The first
part is devoted to the study of quantities which are stable with respect to convergence in concentration, a notion well suited for sequences of spaces with unbounded dimension. The main result that we obtain is the convergence of the heat flow in presence of a uniform Ricci lower bound. In order to prove that, we show – in the same assumptions – the Gamma convergence of the slope of the entropy and the Mosco convergence of the Cheeger energy.
As a byproduct, we are able to define a meaningful convergence of vector fields and of Sobolev functions and to prove the stability of the solution to continuity equations and of the Laplacian eigenvalues. The second part is devoted to the study of Gromov’s pyramids and of extended metric measure spaces, two possible notions of infinite dimensional metric measure spaces. We describe a way to associate – in a possibly non-canonical fashion – to each pyramid an extended metric measure space and to each extended metric measure space a pyramid. We also discuss a natural condition under which the two operations are consistent with each other, encompassing interesting examples such as the Wiener space.
The second project deals with nonsmooth Lorentzian geometry. The first part is devoted to the definition and study of a relaxed p-Cheeger energy – obtained by relaxation – and of its gradient flow – the p-heat flow. We show that this energy produces a notion of maximal subslope and prove natural calculus rules and an interesting Kuwada-type lemma, linking the energy to an entropy functional on the Wasserstein space. Finally, we prove that under some assumptions – that we believe to be technical – our maximal subslope coincides with the one defined via test plans. The second part is devoted to the definition and study of causally convex functions and of their gradient flows. We define basic convex-analytic objects and quantities – such as the slope and the subdifferential – and prove basic regularity properties for causally convex functions. Finally, we define a notion of (p, q)-EDE gradient flow via an energy-dissipation-like property and an EVI-gradient flow mimicking a differential inclusion and prove local existence, a partial uniqueness result and that – up to a suitable reparametrization – the flows coincide.
The last project deals with the study of the asymptotics of a hyperelastic-type energy
functional defined on vector functions on a perforated domain, in presence of nonlinear pointwise constraints in the perforation sites. We prove the Gamma convergence to a limit functional under quite general assumptions on the energy and the constraints and conclude with some numerical simulations
Simulation-based population inference of LISA’s Galactic binaries: Bypassing the global fit
No full textThe Laser Interferometer Space Antenna (LISA) is expected to detect thousands of individually resolved gravitational wave sources, overlapping in time and frequency, on top of unresolved astrophysical and/or primordial backgrounds. Disentangling resolved sources from backgrounds and extracting their parameters in a computationally intensive "global fit" is normally regarded as a necessary step toward reconstructing the properties of the underlying astrophysical populations. Here, we show that it is in principle feasible to infer the population properties of the most numerous of LISA sources-Galactic double white dwarfs- directly from the frequency (or, equivalently, time) strain series by adopting a simulation-based approach, without extracting and estimating the parameters of each single source. By training a normalizing flow on a custom-designed compression of simulated LISA frequency series from the Galactic double white dwarf population, we demonstrate how to infer the posterior distribution of population parameters (e.g., mass function, frequency, and spatial distributions). This allows for extracting information on the population parameters from both resolved and unresolved sources simultaneously and in a computationally efficient manner. This approach can be extended to other source classes (e.g., massive and stellar-mass black holes, extreme mass ratio inspirals) and to scenarios involving non-Gaussian or nonstationary noise (e.g., data gaps), provided that fast and accurate simulations are available
Fourier Optimization, de Branges Spaces, and Zeros of L-functions
No full textThis thesis deals with a few topics at the intersection of Fourier analysis, number theory, and complex analysis. Using the framework of Fourier optimization we obtain new bounds related to the following questions in number theory: the least quadratic non-residue, the least prime in an arithmetic progression, and Montgomery's pair correlation conjecture. We also make contributions related to Hilbert spaces of entire functions, namely, studying norms of embeddings between weighted Paley--Wiener spaces, finding the sharp constant for an operator of multiplication in certain de Branges spaces, and introducing new sign uncertainty principles for functions of exponential type