Cover's function counting theorem is a milestone in the theory of artificial neural networks. It provides an answer to the fundamental question of determining how many binary assignments (dichotomies) of $p$ points in $n$ dimensions can be linearly realized. Regrettably, it has proved hard to extend the same approach to more advanced problems than the classification of points. In particular, an emerging necessity is to find methods to deal with geometrically structured data, and specifically with non-point-like patterns. A prominent case is that of invariant recognition, whereby identification of a stimulus is insensitive to irrelevant transformations on the inputs (such as rotations or changes in perspective in an image). An object is thus represented by an extended perceptual manifold, consisting of inputs that are classified similarly. Here, we develop a function counting theory for structured data of this kind, by extending Cover's combinatorial technique, and we derive analytical expressions for the average number of dichotomies of generically correlated sets of patterns. As an application, we obtain a closed formula for the capacity of a binary classifier trained to distinguish general polytopes of any dimension. These results extend our theoretical understanding of the role of data structure in machine learning, and provide useful quantitative tools for the analysis of generalization, feature extraction, and invariant object recognition by neural networks.

We investigate a mesoscopic spin current for strongly interacting Fermi gases through a quantum point contact. Under the situation in which spin polarizations in the left and right reservoirs are the same in magnitude but opposite in sign, we calculate the contribution of quasiparticles to the current by means of the linear response theory and many-body $T$-matrix approximation. For a small spin-bias regime, the current in the vicinity of the superfluid transition temperature is strongly suppressed due to the formation of pseudogaps. For a large spin-bias regime where the gases become highly polarized, on the other hand, the current is affected by the enhancement of a minority density of states due to Fermi polarons. We also discuss the broadening of a quasiparticle peak associated with an attractive polaron at a large momentum, which is relevant to the enhancement.

Machine-learning and neural-network approaches have gained huge attention in the context of quantum science and technology in recent years. One of the most essential tasks for the future development of quantum technologies is the verification of nonclassical resources. Here, we present an artificial neural-network approach for the identification of nonclassical states of light based on recorded measurement statistics. In particular, we implement and train a network which is capable of recognizing nonclassical states based on the click statistics recorded with multiplexed detectors. We use simulated data for training and testing the network, and we show that it is capable of identifying some nonclassical states even if they were not used in the training phase. Especially, in the case of small sample sizes, our approach can be more sensitive in identifying nonclassicality than established criteria, which suggests possible applications in presorting of experimental data and online applications.

The current filamentation instability is a key phenomenon underpinning various processes in astrophysics, laboratory laser-plasma, and beam-plasma experiments. Here we show that the ultrafast dynamics of this instability can be explored in the context of relativistic laser-solid interactions through deflectometry by low-emittance, highly relativistic electron bunches from a laser wakefield accelerator. We present experimental measurements of the femtosecond timescale generation of strong magnetic-field fluctuations, with a measured line-integrated $B$ field of $2.70\pm 0.39\mathrm{kT}\mu \mathrm{m}$. Three-dimensional, fully relativistic particle-in-cell simulations demonstrate that such fluctuations originate from the current filamentation instability arising at submicron scales around the irradiated target surface, and that they grow to amplitudes strong enough to broaden the angular distribution of the probe electron bunch a few tens of femtoseconds after the laser pulse maximum. Our results open a branch of physics experiments investigating the femtosecond dynamics of laser-driven plasma instabilities by means of synchronized, wakefield-accelerated electron beams.

Locality plays a fundamental role in quantum computation but also severely restricts our ability to store and process quantum information. We argue that this restriction may be unwarranted and re-examine quantum error correcting codes. We proceed to introduce new defects on the surface code called wormholes. These novel defects entangle two spatially separated sectors of the lattice. When anyonic excitations enter the mouth of a wormhole, they emerge through the other mouth. Wormholes thus serve to connect two spatially separated sectors of a flat two-dimensional lattice. We show that these defects are capable of encoding logical qubits and can be used to perform all gates in the Clifford group.

Split-pulse x-ray photon correlation spectroscopy has been proposed as one of the unique capabilities made possible with x-ray free electron lasers. It enables characterization of atomic-scale structural dynamics that dictates the macroscopic properties of various disordered material systems. Central to the experimental concept are x-ray optics that are capable of splitting an individual coherent femtosecond x-ray pulse into two distinct pulses, introduce an adjustable time delay between them, and then recombine the two pulses at the sample position such that they generate two coherent scattering patterns in rapid succession. Recent developments in such optics showed that, while true “amplitude-splitting” optics at hard x-ray wavelengths remains a technical challenge, wavefront and wavelength splitting are both feasible, able to deliver two micron-sized focused beams to the sample with sufficient relative stability. Here we show, however, that the conventional approach to speckle visibility spectroscopy using these beam-splitting techniques can be problematic, even leading to a decoupling of speckle visibility and material dynamics. In response, we discuss the details of the experimental approaches and data analysis protocols for addressing issues caused by subtle beam dissimilarities for both wavefront- and wavelength-splitting setups. We also show that in some scattering geometries, the $Q$-space mismatch can be resolved by using two beams of slightly different incidence angles and slightly different wavelengths at the same time. Instead of measuring the visibility of weak speckle patterns, the time correlation in sample structure is encoded in the “side band” of the spatial autocorrelation of the summed speckle patterns and can be retrieved straightforwardly from the experimental data. We demonstrate this with a numerical simulation.

We study signatures of the energy landscape's evolution through the crystal-to-glass transition by compressing two dimensional (2D) finite aggregates of oil droplets. Droplets of two distinct sizes are used to compose small aggregates in an aqueous environment. Aggregates range from perfectly ordered monodisperse single crystals to disordered bidisperse glasses. The aggregates are compressed between two parallel boundaries, with one acting as a force sensor. The compression force provides a signature of the aggregate composition and gives insight into the energy landscape. In particular, crystals dissipate all the stored energy through single catastrophic fracture events whereas the glassy aggregates break step by step. Remarkably, the yielding properties of the 2D aggregates are strongly impacted by even a small amount of disorder.

During the past decade, a number of attempts to formulate a continuum description of complex states of matter have been proposed to circumvent more cumbersome many-body and simulation methods. Typically, these have been quantum systems (e.g., electrons) and the resulting phenomenologies collectively often called “quantum hydrodynamics.” However, there is extensive work from the past based in nonequilibrium statistical mechanics on the microscopic origins of macroscopic continuum dynamics that has not been exploited in this context. Although formally exact, its original target was the derivation of Navier-Stokes hydrodynamics for slowly varying states in space and time. The objective here is to revisit that work for the present interest in complex quantum states—possible strong degeneracy, strong coupling, and all space-time scales. The result is an exact representation of generalized hydrodynamics suitable for introducing controlled approximations for diverse specific cases and for critiquing existing work.

We show that in the mathematical framework of the quantum theory, the classical pigeonhole principle can be violated more directly than previously suggested, i.e., in a setting closer to the traditional statement of the principle. We describe how the counterfactual reasoning of the paradox may be operationally grounded in the analysis of the tiny footprints left in the environment by the pigeons. After identifying the drawbacks of recent experiments of the quantum pigeonhole effect, we argue that a definitive experimental violation of the pigeonhole principle is still needed and propose such an implementation using modern quantum computing hardware: a superconducting circuit with transmon qubits.

The ${\mathrm{Cu}}_2{\mathrm{ZnSnS}}_4$ kesterite is currently among the most promising inorganic, nontoxic, earth-abundant materials for a new generation of solar cells. Interfacial defects and secondary phases present in the kesterite active layer are, however, detrimental to the performance of the device. They are typically probed with techniques that are destructive or limited to the surface, and x-ray diffraction cannot reliably distinguish small amounts of zinc sulfide or copper tin sulfide from kesterite. Conversely, resonant ptychographic tomography, which relies on electron density contrast, overcomes these limitations. Here, we demonstrate how this technique can enable localization and quantification of secondary phases, along with measurements of adherence at the interfacial layers, on complete and functioning devices. In our experiment, we utilize an x-ray energy value far from absorption edges as well as three single energies corresponding to the absorption edges of Cu, Zn, and Sn, to gain elemental sensitivity to these elements and enhance contrast between phases with similar electron density. As a result, we image and identify in the active layer grains of a secondary phase, namely, zinc sulfide, which is not easily discriminated by other standard characterization techniques. In addition, we are able to observe Cu diffused from the active layer into the CdS buffer layer as well as Cu in the form of copper sulfide at their interface. Other relevant morphological features are best resolved off-resonance at the optimal energy for the synchrotron beamline with ∼20 nm resolution.

Vector-borne epidemics are progressively becoming a global burden, especially those related to flaviviruses, and the effects of different factors such as climate change or the increase of human mobility can sensibly increase the population at risk worldwide. Such outbreaks are the result of the combination of different factors including crossed contagions between humans and vectors, their demographic distribution and human mobility among others. The current availability of information about all those ingredients demands their incorporation into current mathematical models for vector-borne disease transmission. Here, relying on a Markovian formulation of the metapopulation dynamics, we propose a framework that explicitly includes human-vector interactions, mobility, and demography. The analysis of the framework allows us not only to derive an expression of the epidemic threshold capturing the conditions for the onset of the epidemics but also to highlight some unseen features of vector-borne epidemics, such as abrupt changes in the unfolding patterns of the disease for small variations of the degree of mobility. Finally, driven by these insights, we obtain a prevalence indicator to rank populations according to their risk of being affected by a vector-borne disease. We illustrate the utility of this indicator by reproducing the spatial distribution Dengue cases reported in the city of Santiago de Cali (Colombia) from 2015 to 2016.

Recently, the very first large-gap Kane-Mele quantum spin Hall insulator was predicted to be monolayer jacutingaite (${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$), a naturally occurring exfoliable mineral discovered in Brazil in 2008. The stacking of quantum spin Hall monolayers into a van-der-Waals layered crystal typically leads to a (0;001) weak topological phase, which does not protect the existence of surface states on the (001) surface. Unexpectedly, recent angle-resolved photoemission spectroscopy experiments revealed the presence of surface states dispersing over large areas of the 001-surface Brillouin zone of jacutingaite single crystals. The 001-surface states have been shown to be topologically protected by a mirror Chern number $C_M=-2$, associated with a nodal line gapped by spin-orbit interactions. Here, we extend the two-dimensional Kane-Mele model to bulk jacutingaite and unveil the microscopic origin of the gapped nodal line and the emerging crystalline topological order. By using maximally localized Wannier functions, we identify a large nontrivial second nearest-layer hopping term that breaks the standard paradigm of weak topological insulators. Complemented by this term, the predictions of the Kane-Mele model are in remarkable agreement with recent experiments and first-principles simulations, providing an appealing conceptual framework also relevant for other layered materials made of stacked honeycomb lattices.

In topology, a torus remains invariant under certain nontrivial transformations known as modular transformations. In the context of topologically ordered quantum states of matter supported on a torus geometry in real space, these transformations encode the braiding statistics and fusion rules of emergent anyonic excitations and thus serve as a diagnostic of topological order. Moreover, modular transformations of higher genus surfaces, e.g., a torus with multiple handles, can enhance the computational power of a topological state, in many cases providing a universal fault-tolerant set of gates for quantum computation. However, due to the intrusive nature of modular transformations, which abstractly involve global operations, physical implementations of them in local systems have remained elusive. Here, we show that by engineering an effectively folded manifold corresponding to a multilayer topological system, modular transformations can be applied in a single shot by independent local unitaries, providing a novel class of transversal logical gates for fault-tolerant quantum computation. Specifically, we demonstrate that multilayer topological states with appropriate boundary conditions and twist defects allow modular transformations to be effectively implemented by a finite sequence of local SWAP gates between the layers. We further provide methods to directly measure the modular matrices, and thus the fractional statistics of anyonic excitations, providing a novel way to directly measure topological order. A more general theory of transversal gates and the deep connection to anyon symmetry transformation and symmetry-enriched topological orders are also discussed.

The concept of Floquet engineering is to subject a quantum system to time-periodic driving in such a way that it acquires interesting properties. It has been employed, for instance, for the realization of artificial magnetic fluxes in optical lattices and, typically, it is based on two approximations. First, the driving frequency is assumed to be low enough to suppress resonant excitations to high-lying states above some energy gap separating a low-energy subspace from excited states. Second, the driving frequency is still assumed to be large compared to the energy scales of the low-energy subspace, so that also resonant excitations within this space are negligible. Eventually, however, deviations from both approximations will lead to unwanted heating on a time scale $\tau$. Using the example of a one-dimensional system of repulsively interacting bosons in a shaken optical lattice, we investigate the optimal frequency (window) that maximizes $\tau$. As a main result, we find that, when increasing the lattice depth, $\tau$ increases faster than the experimentally relevant timescale given by the tunneling time $\hslash /J$, so that Floquet heating becomes suppressed.

We present a systematic investigation of the inverse spin-Hall effect (ISHE) in the $\pi$-conjugated polymer poly(3,4-ethylenedioxythiophene):poly(4-styrenesulfonate) (PEDOT:PSS). Using a number of reference experiments, we are able to identify and isolate side effects which obscure the small but finite inverse spin-Hall effect in the polymer. We employ a sample geometry in which the contact areas and the area of spin current injection are laterally separated, which allows us to distinguish the ISHE from thermovoltages induced by nonreciprocal magnetostatic spin waves (MSSW) and from the ISHE induced by spin pumping through the polymer into the contacts. With an additional control experiment, we can even quantify the Nernst effect, which also needs to be taken into account. With these results, we can unambiguously show that the ISHE is present in this material, albeit at a level which requires a dedicated sample design and careful consideration of various artifacts.

Multi-Weyl semimetals feature band crossings with the dispersion that is, in general, linear in only one direction, and as a consequence their band structure is characterized by the monopole charge $n$ which can be greater than one. We show that a single screw dislocation defect oriented in the direction connecting the nodal points, which acts as an effective pseudomagnetic flux tube, can serve as a direct probe of the monopole charge $n\ge 1$ characterizing the bulk band structure of a multi-Weyl semimetal. To this end, as a proof of principle, we propose a rather simple mesoscopic setup in which the monopole charge leaves a direct imprint on the conductance measured in the plane perpendicular to the dislocation. In particular, the ratio of the positions of the neighboring maxima in the conductance as a function of the gate voltage can serve to deduce the monopole charge, while the value of the effective pseudomagnetic flux can be extracted from the position of a conductance maximum. We expect that these findings will prompt further studies on the role of multiple dislocations, as well as other topological lattice defects, such as grain boundaries and disclinations, in topological nodal materials.

Topology plays an increasing role in physics beyond the realm of topological insulators in condensed mater. From geophysical fluids to active matter, acoustics or photonics, a growing family of systems presents topologically protected chiral edge modes. The number of such modes should coincide with the bulk topological invariant (e.g., Chern number) defined for a sample without boundary, in agreement with the bulk-edge correspondence. However, this is not always the case when dealing with continuous media where there is no small scale cutoff. The number of edge modes actually depends on the boundary condition, even when the bulk is properly regularized, showing an apparent paradox where the bulk-edge correspondence is violated. In this paper, we solve this paradox by showing that the anomaly is due to ghost edge modes hidden in the asymptotic part of the spectrum, which have a signature at finite frequency both in the local density of states and in a channel geometry. We provide a general formalism based on scattering theory to detect all edge modes properly, so that the bulk-edge correspondence is restored in a broader sense, implying in particular that chiral edge modes are not necessarily topological, and conversely. Our approach is illustrated through the odd-viscous shallow-water model and the massive Dirac Hamiltonian.

Hall viscosity, also known as the Lorentz shear modulus, has been proposed as a topological property of a quantum Hall fluid. Using a recent formulation of the composite fermion theory on the torus, we evaluate the Hall viscosities for a large number of fractional quantum Hall states at filling factors of the form $\nu =n/(2pn\pm 1)$, where $n$ and $p$ are integers, from the explicit wave functions for these states. The calculated Hall viscosities ${\eta }^A$ agree with the expression ${\eta }^A=(\hslash /4)\mathcal{S}\rho$, where $\rho$ is the density and $\mathcal{S}=2p\pm n$ is the “shift” in the spherical geometry. We discuss the role of modular covariance of the wave functions, projection of the center-of-mass momentum, and also of the lowest-Landau-level projection. Finally, we show that the Hall viscosity for $\nu =\frac{n}{2pn+1}$ may be derived analytically from the microscopic wave functions, provided that the overall normalization factor satisfies a certain behavior in the thermodynamic limit. This derivation should be applicable to a class of states in the parton construction, which are products of integer quantum Hall states with magnetic fields pointing in the same direction.

A given neural network in the brain is involved in many different tasks. This implies that, when considering a specific task, the network's connectivity contains a component which is related to the task and another component which can be considered random. Understanding the interplay between the structured and random components and their effect on network dynamics and functionality is an important open question. Recent studies addressed the coexistence of random and structured connectivity but considered the two parts to be uncorrelated. This constraint limits the dynamics and leaves the random connectivity nonfunctional. Algorithms that train networks to perform specific tasks typically generate correlations between structure and random connectivity. Here we study nonlinear networks with correlated structured and random components, assuming the structure to have a low rank. We develop an analytic framework to establish the precise effect of the correlations on the eigenvalue spectrum of the joint connectivity. We find that the spectrum consists of a bulk and multiple outliers, whose location is predicted by our theory. Using mean-field theory, we show that these outliers directly determine both the fixed points of the system and their stability. Taken together, our analysis elucidates how correlations allow structured and random connectivity to synergistically extend the range of computations available to networks.

Superconductivity owes its properties to the phase of the electron pair condensate that breaks the $\text{U}\left(1\right)$ symmetry. In the most traditional ground state, the phase is uniform and rigid. The normal state can be unstable towards special inhomogeneous superconducting states: the Abrikosov vortex state and the Fulde-Ferrell-Larkin-Ovchinnikov state. Here we show that the phase-uniform superconducting state can go into a fundamentally different and more ordered nonuniform ground state, which we refer to as a phase crystal. This state breaks translational invariance through formation of a spatially periodic modulation of the phase, manifested by unusual superflow patterns and circulating currents, that also break time-reversal symmetry. We list the general conditions needed for realization of phase crystals. Using microscopic theory, we then derive an analytic expression for the superfluid density tensor for the case of a nonuniform environment in a semi-infinite superconductor. We demonstrate how the surface quasiparticle states enter the superfluid density and identify phase crystallization as the main player in several previous numerical observations in unconventional superconductors, and predict the existence of a similar phenomenon in superconductor-ferromagnetic structures. This analytic approach provides a unifying aspect for the exploration of boundary-induced quasiparticles and collective excitations in superconductors. More generally, we trace the origin of phase crystallization to nonlocal properties of the gradient energy, which implies the existence of similar pattern-forming instabilities in many other contexts.