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We study the memorization and generalization capabilities of a Diffusion Model (DM) in the case of structured data defined on a latent manifold. We specifically consider a set of $P$ mono-modal data points in $N$ dimensions lying on a latent subspace of dimension $D = \alpha_D N$, according to the Hidden Manifold Model (HMM). Our analysis leverages the recently introduced formalism based on the statistical physics of the Random Energy Model (REM). We provide evidence for the existence of an onset time $t_{o} > t_c$ when traps appear in the potential without affecting the typical diffusive trajectory. The size of the basins of attraction of such traps is computed as a function of time. Moreover, we derive the collapse time $t_{c}$ at which trajectories fall in the basin of one of the training points, implying memorization. An explicit formula for $t_c$ is given as a function of $P$ and the ratio $\alpha_D$, proving that the curse of dimensionality issue does not hold for highly structured data, i.e. $\alpha_D\ll 1$, regardless of the non-linearity of the manifold surface. We also prove that collapse coincides with the condensation transition in the REM. Eventually, the degree of generalization of DMs is formulated in terms of the Kullback-Leibler divergence between the exact and the empirical distribution of the sampled configurations: we show the existence of an additional time $t_{g}

Generative diffusion processes are state-of-the-art machine learning models deeply connected with fundamental concepts in statistical physics. Depending on the dataset size and the capacity of the network, their behavior is known to transition from an associative memory regime to a generalization phase in a phenomenon that has been described as a glassy phase transition. Here, using statistical physics techniques, we extend the theory of memorization in generative diffusion to manifold-supported data. Our theoretical and experimental findings indicate that different tangent subspaces are lost due to memorization effects at different critical times and dataset sizes, which depend on the local variance of the data along their directions. Perhaps counterintuitively, we find that, under some conditions, subspaces of higher variance are lost first due to memorization effects. This leads to a selective loss of dimensionality where some prominent features of the data are memorized without a full collapse on any individual training point. We validate our theory with a comprehensive set of experiments on networks trained both in image datasets and on linear manifolds, which result in a remarkable qualitative agreement with the theoretical predictions.

In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. For this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions, and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation of why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

Graphdiyne-based carbon systems generate intriguing layered sp-sp$^2$ organometallic lattices, characterized by flexible acetylenic groups connecting planar carbon units through metal centers. At their thinnest limit, they can result in two-dimensional (2D) organometallic networks exhibiting unique quantum properties and even confining the surface states of the substrate, which is of great importance for fundamental studies. In this work, we present the on-surface synthesis of a highly crystalline 2D organometallic network grown on Ag(111). The electronic structure of this mixed honeycomb-kagome arrangement - investigated by angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy - reveals a strong electronic conjugation within the network, leading to the formation of two intense electronic band-manifolds. In comparison to theoretical density functional theory calculations, we observe that these bands exhibit a well-defined orbital character that can be associated with distinct regions of the sp-sp$^2$ monomers. Moreover, we find that the halogen by-products resulting from the network formation locally affect the pore-confined states, causing a significant energy shift. This work contributes to the understanding of the growth and electronic structure of graphdiyne-like 2D networks, providing insights into the development of novel carbon materials beyond graphene with tailored properties., Comment: 11 pages, 4 figures + supporting information (5 pages, 10 figures)

Hybrid sp-sp2 structures can be efficiently obtained on metal substrates via on-surface synthesis. The choice of both the precursor and of the substrate impacts on the effectiveness of the process and the stability of the formed structures. Here we demonstrate that using anthracene-based molecules as precursor, the formation on Au(111) of polymers hosting sp carbon chains is affected by the steric hindrance between aromatic groups. In particular, by scanning tunneling microscopy and density functional theory calculations we show that the de-metalation of organometallic structures induces a lateral separation of adjacent polymers preventing the formation of ordered domains.

Graphdiynes (GDYs) represent a class of 2D carbon materials based on sp-sp$^2$ hybridization with appealing properties and potential applications. Recent advances have demonstrated the experimental self-assembly of GDYs on metal substrates. Here we focus on $\alpha$- and $\beta$-GDYs on Au(111) and Pt(111), and investigate how their electronic and vibrational properties are affected by the interaction with a metal substrate. We adopt hydrogenated GDY, previously characterized experimentally, as a benchmark for density functional theory simulations, that we apply to show that Au and Pt substrates impose a different degree of distortion on both $\alpha$- and $\beta$-GDY. By comparing the adsorbed and the freestanding structures, we evaluate the effect of the surface interaction on the bandstructure and the simulated Raman spectra. Different charge transfers result in different energy shift of the Dirac cone in semi-metallic $\alpha$-GDY and changes from semiconducting to metallic behavior for $\beta$-GDY. These changes in electronic properties are accompanied by characteristic frequency shifts and modifications of Raman active modes. Our results contribute in the understanding of the metal-interaction effects on GDYs and can open a route to the design of novel 2D materials with tailored properties.

Spin-electronic devices are poised to become part of mainstream microelectronic technology .Downsizing them, however, faces the intrinsic difficulty that as ferromagnets become smaller, it becomes more difficult to stabilize their magnetic moment. Antiferromagnets are much more stable, and thus research on antiferromagnetic spintronics has developed into a fast-growing field. Here, we provide proof of concept data that allows us to expand the area of antiferromagnetic spintronics to the hitherto elusive level of individual molecules. In contrast to all previous work on molecular spintronics, our detection scheme of the molecule's spin state does not rely on a magnetic moment. Instead, we use field-effect transistor devices constituting of an isolated, contacted single-wall carbon nanotube covalently bound to a limited number of molecular antiferromagnets incorporating four Mn(II) or Co(II) ions. Time-dependent quantum transport measurement along the functionalized nanotube show step-like transitions between several distinct current levels, which we attribute to transitions between different antiferromagnetic states of individual molecular complexes grafted on the nanotube. A statistical analysis of the switching events using factorial cumulants indicates that the cobalt complexes switch independently from each other, while a coherent superposition of the antiferromagnetic spin states of the molecules along the nanotube is observed for the manganese complexes. The long coherence time (several seconds at 100 mK) is made possible by the absence of spin and orbital momentum in the relevant states of the manganese complex, while the cobalt complex includes a significant orbital momentum contribution due to the pseudo-octahedral d$^7$ metal centers., Comment: including supplementary information

We present a joint experimental and theoretical characterization of the magnetic properties of coordination clusters with an antiferromagnetic core of four magnetic ions. Two different compounds are analyzed, with Co and Mn ions in the core. While both molecules are antiferromagnetic, they display different sensitivities to external magnetic field, according to the different strength of the intra-molecular magnetic coupling. In particular, the dependence of the magnetization versus field of the two molecules switches with temperatures: at low temperature the magnetization is smaller in \{Mn$_4$\}, while the opposite happens at high temperature. Through a detailed analysis of the electronic and magnetic properties of the two compounds we identify a stronger magnetic interaction between the magnetic ions in \{Mn$_4$\} with respect to \{Co$_4$\}. Moreover \{Co$_4$\} displays not negligible spin-orbit related effects that could affect the spin lifetime in future antiferromagnetic spintronic applications. We highlight the necessity to account for these spin-orbit effects for a reliable description of these compounds., Comment: 8 pages, 5 figures + supplementary Information

The coupling of excitons with atomic vibrations plays a pivotal role on the nonequilibrium optical properties of layered semiconductors. However, addressing the dynamical interaction between excitons and phonons represents a hard task both experimentally and theoretically. By means of time-resolved broadband optical reflectivity combined with state-of-the-art ab-initio calculations of a bismuth triiodide single crystal, we unravel the universal spectral fingerprints of exciton--phonon coupling in layered semiconductors. Furthermore, we microscopically relate a photoinduced coherent energy modulation of the excitonic resonance to coherent optical phonons, thereby tracking the extent of the photoinduced atomic displacement in real-space. Our findings represent a step forward on the road to coherent manipulation of the excitonic properties on ultrafast timescales.

Special point defects in semiconductors have been envisioned as suitable components for quantum-information technology. The identification of new deep centers in silicon that can be easily activated and controlled is a main target of the research in the field. Vacancy-related complexes are suitable to provide deep electronic levels but they are hard to control spatially. With the spirit of investigating solid state devices with intentional vacancy-related defects at controlled position, here we report on the functionalization of silicon vacancies by implanting Ge atoms through single-ion implantation, producing Ge-vacancy (GeV) complexes. We investigate the quantum transport through an array of GeV complexes in a silicon-based transistor. By exploiting a model based on an extended Hubbard Hamiltonian derived from ab-initio results we find anomalous activation energy values of the thermally activated conductance of both quasi-localized and delocalized many-body states, compared to conventional dopants. We identify such states, forming the upper Hubbard band, as responsible of the experimental sub-threshold transport across the transistor. The combination of our model with the single-ion implantation method enables future research for the engineering of GeV complexes towards the creation of spatially controllable individual defects in silicon for applications in quantum information technologies.

Graphdiyne, atomically-thin 2D carbon nanostructure based on sp-sp2 hybridization, is an appealing system potentially showing outstanding mechanical and optoelectronic properties. Surface-catalyzed coupling of halogenated sp-carbon-based molecular precursors represents a promising bottom-up strategy to fabricate extended 2D carbon systems with engineered structure on metallic substrates. Here, we investigate the atomic-scale structure and electronic and vibrational properties of an extended graphdiyne-like sp-sp2 carbon nanonetwork grown on Au(111) by means of on-surface synthesis. The formation of such 2D nanonetwork at its different stages as a function of the annealing temperature after the deposition is monitored by scanning tunneling microscopy (STM), Raman spectroscopy and combined with density functional theory (DFT) calculations. High-resolution STM imaging and the high sensitivity of Raman spectroscopy to the bond nature provide a unique strategy to unravel the atomic-scale properties of sp-sp2 carbon nanostructures. We show that hybridization between the 2D carbon nanonetwork and the underlying substrate states strongly affects its electronic and vibrational properties, modifying substantially the density of states and the Raman spectrum compared to the free standing system. This opens the way to the modulation of the electronic properties with significant prospects in future applications as active nanomaterials for catalysis, photoconversion and carbon-based nanoelectronics.

Chloromethane is the only organochloride detected in space to date. Its recent observation towards the low-mass protostar IRAS 16293-2422 with ALMA offers a prompt for new laboratory studies of CH$_3$Cl and its isotopologues. Here, we report the investigation of the rotational spectrum of monodeuterated chloromethane CH$_2$DCl in the frequency region between 90 and 300 GHz. The measurements have been carried out with a frequency-modulation millimeter-wave spectrometer, arranged to perform saturation spectroscopy. From the analysis of hyperfine-resolved spectra of the two chlorine isotopologues CH$_2$D$^{35}$Cl and CH$_2$D$^{37}$Cl, consistent sets of accurate spectroscopic parameters have been obtained. This work provides reliable spectral predictions which can be used to guide radio-astronomical searches of CH$_2$DCl in the interstellar medium and represents a solid base for future analyses of high-resolution infrared spectra of monodeuterated chloromethane., Comment: 13 pages, 2 figures, 3 tables

The self-assembly of Leucoquinizarin molecule on Au(111) surface is shown to be characterized by the molecules mostly in their keto-enolic tautomeric form, with evidences of their temporary switching to other tautomeric forms. This reveals a metastable chemistry of the assembled molecules, to be considered for their possible employment in the formation of more complex hetero-organic interfaces

We characterize germanium-vacancy GeVn complexes in silicon using first-principles Density Functional Theory calculations with screening-dependent hybrid functionals. We report on the local geometry and electronic excited states of these defects, including charge transition levels corresponding to the addition of one or more electrons to the defect. Our main theoretical result concerns the GeV complex, which we show to give rise to two excited states deep in the gap, at -0.51 and -0.35 eV from the conduction band, consistently with the available spectroscopic data. The adopted theoretical scheme, suitable to compute a reliable estimate of the wavefunction decay, leads us to predict that such states are associated to an electron localization over a length of about 0.45 nm. By combining the electronic properties of the bare silicon vacancy, carrying deep states in the band gap, with the spatial controllability arising from single Ge ion implantation techniques, the GeVn complex emerges as a suitable ingredient for silicon-based room-temperature single-atom devices.

A junction between an n- and p-type semiconductor results in the creation of a depletion region whose properties are at the basis of nowadays electronics. If realized using topological insulators as constituent materials, p-n junctions are expected to manifest several unconventional effects with great potential for applications. Experimentally, all these fascinating properties remained unexplored so far, mainly because prototypical topological PNJs, which can be easily realized and investigated, were not readily available. Here, we report on the creation of topological PNJs which can be as narrow as few tenths of nm showing a built-in potential of 110meV. These junctions are intrinsically obtained by a thermodynamic control of the defects distribution across the crystal. Our results make Bi2Te3 a robust and reliable platform to explore the physics of topological p-n junction.

We discuss expectations for the total and inelastic cross-sections at LHC CM energies $\sqrt{s}\ =\ 7\ TeV$ {and $ 14\ TeV$} obtained in an eikonal minijet model augmented by soft gluon $k_t$-resummation, which we describe in some detail. We present a band of predictions which encompass recent LHC data and suggest that the inelastic cross-section described by two channel eikonal models include only uncorrelated processes. We show that this interpretation of the model is supported by the LHC data., Comment: 32 pages, 6 figures, Version to appear in Physical Review D

The papers review the main theoretical and experimental aspects of the Forward Physics at the Large Hadron Collider.

The objective of this first workshop on Multiple Partonic Interactions (MPI) at the LHC is to raise the profile of MPI studies, summarizing the legacy from the older phenomenology at hadronic colliders and favouring further specific contacts between the theory and experimental communities. The MPI are experiencing a growing popularity and are currently widely invoked to account for observations that would not be explained otherwise: the activity of the Underlying Event, the cross sections for multiple heavy flavour production, the survival probability of large rapidity gaps in hard diffraction, etc. At the same time, the implementation of the MPI effects in the Monte Carlo models is quickly proceeding through an increasing level of sophistication and complexity that in perspective achieves deep general implications for the LHC physics. The ultimate ambition of this workshop is to promote the MPI as unification concept between seemingly heterogeneous research lines and to profit of the complete experimental picture in order to constrain their implementation in the models, evaluating the spin offs on the LHC physics program., Comment: MPI'08 international workshop has been held in October 27-31, 2008, Perugia, Italy - 349 pages

Inclusion of down to zero-momentum gluons and their k_t resummation is shown to quench the too fast rise of the mini jet cross section and thereby obtain realistic total cross-sections., Comment: 6 pages, 2figures, Presented at FLAVIAnet Workshop, Kazimierz, 23-27 July, 2009. Correction of formula (9) with a factor \pi in the denominator and further minor corrections to the text

We present the predictions of a model for proton-proton total cross-section at LHC. It takes into account both hard partonic processes and soft gluon emission effects to describe the proper high energy behavior and to respect the Froissart bound., Comment: 7 pages, 3 figures, presented at MPI08, Perugia, October 27-31, 2008

Proceedings of ISMD08, Comment: Edited by: J. Bartels, K. Borras, G. Gustafson, H. Jung, K. Kutak, S. Levonian, and J. Mnich

A previously successful model for purely hadronic total cross-sections, based on QCD minijets and soft-gluon resummation, is here applied to the total photoproduction cross section. We find that our model in the gamma p case predicts a rise with energy stronger than in the pp -pbarp case., Comment: 5 pages, 3 figures, presented at ISMD08, 15-20 September 2008, DESY

We describe a QCD motivated model for total cross-sections which uses the eikonal representation and incorporates QCD mini-jets to drive the rise with energy of the cross-section, while the impact parameter distribution is obtained through the Fourier transform of the transverse momentum distribution of soft gluons emitted in the parton-parton interactions giving rise to mini-jets in the final state. A singular but integral expression for the running coupling constant in the infrared region is part of this model., Comment: 4 pages, 1 figure, uses slac_one.rtx, 34th ICHEP Conference, Philadelphia 2008

We discuss the effect of infrared soft gluons on the asymptotic behaviour of the total cross-section. We use a singular but integrable expression for the strong coupling constant in the infrared limit and relate its behaviour to the satisfaction of the Froissart bound, giving a specific phenomenological example., Comment: 7 pages, 1 figure, presented at Hadron structure 07, HS07, Modra-Harmonia, September 3-7, 2007

New results are presented for total ${\rm pp}/ {\rm \bar p p}$ cross-sections, in the framework of our QCD based model (GGPS). This is an improved eikonal mini-jet model, where soft gluon radiation tames the fast energy rise normally present in mini-jet models. We discuss the variability in our predictions and provide a handy parametrization of our results for the LHC. We find that our model predictions span the range $\sigma_{tot}^{LHC}=100 ^{+10}_{-13} mb$. While this matches nicely with the range of most other models, it does not agree with recent ones which include a "hard" Pomeron, even though our model does include hard scattering. We compute the survival probability for Large Rapidity Gap (LRG) events at the LHC and at the Tevatron. These events are relevant, for example, for Higgs signal in the $WW$ fusion process. We also explore whether measurements of the total cross-sections at the LHC can help us sharpen the model parameters and hence estimates for these survival probabilities, further., Comment: 15 pages, 3 figures, LaTeX