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How many different words exist on the Internet, in all published books, or in a language as a whole? How are these numbers growing, in time and with database size? Is the finiteness of the vocabulary of any practical significance? The recent availability of large databases opens up possibilities of making new analyses, but deeper and systematic insights and predictive power must come from models that go beyond data alone. In this paper, we obtain a simple stochastic model that reveals the basic mechanisms that govern vocabulary growth and provides new insight into the questions above. The motivation and validation of this model come from a careful analysis of the Google Ngram database, a collection of several millions of books over the last five centuries.

Statistics of word usages share remarkable similarities with other social, physical, and biological systems. The most well-known similarity is the widespread appearance of fat-tailed distributions, e.g., Zipf’s law, which shows that words in a text span a wide range of frequencies. Revisiting some of the statistical features of word frequencies in our larger database, we observe the appearance of two scaling regimes in the Zipf’s law analysis and also in the functional dependence of the number of different words on the database size. Our two-parameter fitting curve explains within 50% accuracy the number of different words in texts that contain from 1000 to 100 000 000 000 words. These findings lead us to propose a model that describes the growth of the vocabulary as a simple stochastic process. Hypothetical, arbitrarily large texts that share the crucial properties observed in our database can be generated from the model. Interpreting the empirical observations with the help of this model, we conclude that (i) the vocabulary is virtually infinite, (ii) there exists a core vocabulary of the size of approximately 10 000 words, and (iii) the composition of the core vocabulary changes at a constant rate of 30 words per year following an exponential decay.

Our findings have a direct impact on the study of language dynamics on historical time scales and on applications in computer science such as search engines. They have additional significance as a remarkable example of how simple models are able to capture the main statistical features that emerge from the interaction of millions of individuals (or components). The simplicity and generality of our model make us confident that it will also find applications in different complex systems showing similar statistical behavior.

The number of neurons that form the cerebral cortex (part of the gray matter) of a human brain is immense: some 8×10¹⁰ on average. The spatiotemporal patterning in the activity of these neurons, as observed in brain imaging studies, is believed to be linked to cognitive functions such as learning, recognition, and speech. But the precise linkage between pattern and function remains elusive and is one of the most hotly debated topics in neuroscience today. The problem is being tackled at many structural levels (from networks of individual neurons to neuronal populations or continuum) and using many scientific approaches (such as electrical recordings and functional magnetic resonance imaging). In this paper, we put forward the idea that many aspects of the spatial patterning observed in imaging studies are the result of spontaneous pattern formation generated from the electrical synaptic coupling, also known as gap junctions, between neurons. Such communication channels have been observed experimentally.

We pursue this idea theoretically, in the context of the wake-to-coma transition induced by general anesthesia. Our model, many ingredients of which have been proposed and investigated before, describes the cortex as an excitable, continuum medium of interconnected neurons that is poised at a special point where a spatial (Turing) instability and a temporal (Hopf) instability can coexist. We suggest that normal functioning in the awake brain requires a delicate balance between two competing pressures: the need to form spatial patterns versus the need to generate global rhythms in the cortex. Small changes in model parameters, which reflect altered neurophysiological conditions, can instigate a rebalancing of these two instabilities. Extreme imbalance in either direction is almost certainly pathological: Frozen activity patterns prevent information transfer, while synchronous oscillations across the whole cortex correspond to seizure.

Our study of the model reveals some interesting findings. If the gap-junction-based inhibitory coupling between neurons is sufficiently strong, spatiotemporally patterned neural activity emerges spontaneously with a temporal rhythm that is controlled by the responsiveness of inhibitory chemical synapses. In particular, the model predicts that introduction of anesthetics to the awake brain—in the form of an increased influence of the inhibitory chemical synapses—leads to the emergence of a turbulent, chaotic pattern of slow-wave activity, the result of a subtle rebalancing between Turing and Hopf influences. Gap-junction-facilitated interneuronal communication, acting across the cortex, is crucial to the emergence of this pattern.

Significantly, the slow-wave oscillation is the defining feature of scalp-measured electroencephalography, both under general anesthesia and during the natural nonREM (nonrapid-eye-movement) sleep. Our proposed dynamic mechanism for the slow-wave oscillation contrasts with conventional explanations for anesthesia-induced slow oscillations that require cyclic modulation of ion-channel conductances in single neurons. We suggest that a Turing-Hopf interaction may also underpin the slow-wave stage of natural sleep.

Should I open a Facebook account or not? Which of the two presidential candidates should I vote for in an American federal election? Will I get infected or not in a disease epidemic? Under what conditions does my choice or opinion become the popular one? The answer to each of these questions depends not only on the individual in question, but also crucially on their social or physical contacts with other individuals. Formulating this observation mathematically in terms of both local-contact-based binary decision-making processes and the concept of social network structures has led to many simple paradigmatic models, such as the voter model and the susceptible-infected model, that scientists study in order to understand how behaviors, opinions, and infectious diseases spread among human populations.

While these models are simple, analytical methods for tackling them are few and often not accurate, or achieve high accuracy at the cost of computational complexity, because dealing with interactions among many entities in a large system is known to be difficult, in general. In this paper, we present a low-complexity approximation approach, called pair approximation, and demonstrate that for certain classes of local decision rules, it achieves accuracy that is equivalent to that of a recently developed high-accuracy, high-complexity approach.

This new low-complexity approach should find broad utility in the theoretical studies of social phenomena at the population level. To facilitate the spread of its use, we have made a computational code available by download that implements the approach on the Octave or Matlab software platforms.

Recently, a novel class of materials called topological insulators (TIs) was predicted and experimentally discovered. A key characteristic of these materials is that current flow at the boundaries of the material—on the surface of a three-dimensional TI or the edge of a two-dimensional TI—should be protected against backscattering. This characteristic suggests that TIs may be an interesting platform for possible applications, such as low-resistance interconnects in computer chips. In real samples, deviations from this “perfect” behavior have been observed but not yet investigated on a local scale. In a study of the two-dimensional version of a TI, the quantum spin Hall (QSH) system, we find that current flow in the one-dimensional edge channels of the QSH state is affected by small puddles of electrons that lift the protection against backscattering.

Using a technique called scanning gate microscopy, we locally perturb the QSH edge states and monitor the effect on current flow through those edges. We identify individual well-localized sites that control the current flow in the one-dimensional edge states. These scattering sites occur with a separation of around 1  μm and are present even in the absence of our external perturbation. Thus, they help us understand scattering observed in earlier devices, where ballistic transport occurred only on length scales up to a few microns. In the regions between the inherent scattering sites, the edge states are rather robust against perturbations.

Our experiments provide a first spatially resolved study of the QSH state. Similar future studies might provide a more complete picture of scattering mechanisms in TI systems.

The origin of superconductivity in bulk SrTiO₃ (strontium titanate) is a mystery several decades old. Pure SrTiO₃ is an insulator even at zero temperature. But, it becomes a superconductor upon the addition of small amounts of mobile electrons (n doping) achieved by either dilute substitution of strontium by lanthanum, or titanium by niobium, or removal of a very small fraction of oxygen atoms. Is there a threshold in the mobile-electron density for the emergence of the superconductivity? If yes, does the doped system preceding the superconductor behave as a normal metal? Ultimately, does the standard BCS theory for superconductivity hold for SrTiO₃? In this experimental paper, we find that SrTiO₃ is in fact a superconductor with the lowest mobile-charge density currently known in all superconductors—in other words, the most dilute superconductor—and is also a new and interesting candidate for non-BCS-type unconventional superconductors.

Our experimental approach is to employ a probe that is extremely sensitive to features of tiny Fermi surfaces: the so-called Nernst effect—the generation of a transverse electric field under the application of a longitudinal temperature gradient and a perpendicular magnetic field. As the strength of the magnetic field is varied, the Nernst signal changes and is effectively a map of the Fermi surface of the material probed. From this map, the velocity, the effective mass, and the density of mobile electrons can all be quantitatively determined. We have thus been able to obtain the following results: (a) SrTiO₃ remains superconducting at an extremely low doping of mobile electrons—2 times lower than previously thought—that corresponds to the removal of one in 10⁵ oxygen atoms. (b) The normal state of this superconductor has a well-defined Fermi surface, indicating that even minute doping is enough to put the system on the metallic side of the metal-insulator transition. This is a consequence of the anomalously large dielectric coefficient of SrTiO₃. (c) The mobile electrons in the superconducting SrTiO₃ are too slow and too far apart compared to those in conventional superconductors. This poses a serious challenge for the standard BCS pairing scenario, which relies on phonon-induced electron-electron attraction.

These results, which beg for a fundamental theoretical explanation, add new motivations and new information for solving the decade-old puzzle. The work should also be important to the currently very active research on the oxide interface between SrTiO₃ and another oxide insulator, lanthanum aluminate (LaAlO₃).

Electrophoresis, the motion of charged particles in solution driven by an applied electric field, was discovered in 1807 and has, by now, been fairly well understood and broadly applied, as in DNA electrophoresis and electronic ink devices. It may seem that there is not much room left for new fundamental experiments in this rather mature field. In this paper, we present just such an experiment with optically trapped single latex spheres of micron size in a nonpolar solvent not typical in the traditional context of electrophoresis.

When a charged particle, such as the latex sphere we use, is dispersed in a solution, its charge attracts from the solution a cloud of ions of the opposite charge (what the experts call a “diffuse double layer”). When driven to move by an applied electric field, its motion is a result of three forces: the force coming from the electric field, a frictional force from the solution fluid, and a so-called electrophoretic retardation force that comes from the ion cloud that is driven to move by the same field, but in the opposite direction. The two centuries of development in electrophoresis notwithstanding, direct experimental measurements of the retardation force are very scarce. Measurements of this force through a continuous, controlled depletion of the ion cloud have not been done before. Our work fills this fundamental gap.

The central new idea underlying our experiment is that of stripping off the ion cloud in a continuous fashion to the limit of a “naked” charged latex particle by adding charged micelles formed by surfactant molecules to the solution. These micelles, which are nanometers in size, act as the counterions that form the ion cloud. Depleting them from the charged latex particles requires only the application of a very small electric field, because the charged micelles are not replaced as quickly by spontaneous generation as the atomic or molecular counterions present in traditional electrophoresis experiments. By optically trapping the latex particle and measuring its excursion in the trap at the application of an additional small ac field—an already established method—as the ion cloud is gradually reduced and finally fully stripped, we can then determine the electrophoretic retardation force and its dependence on the ion cloud.

While our experimental measurements do not change the theory of electrophoresis, they make an important addition to our fundamental understanding of electrophoresis. Already, they explain the previously unexplained particle acceleration seen in electronic ink devices. And our method should become useful for studying the electrophoretic retardation effect in colloidal systems.

In our social life, law-breaking is penalized. Violation of well-established physical laws—if demonstrated by sound experimental evidences or founded upon rigorous theoretical predictions—usually causes excitement as a harbinger of new physics. In this experimental paper, we report evidences of violation of one of the basic laws that govern electrical circuits, Ohm’s law, in nanoscale charge-transport networks formed at the interface between two oxides, LaAlO₃ and SrTiO₃.

This particular two-dimensional interface system has attracted intense interest from condensed matter physicists, because it displays a rich range of properties, not least, superconductivity and magnetism. It is then fundamentally interesting to ask the following question: Will we see new physics if the interface is shrunk into very narrow nanowires or a network of such nanowires? Using a sharp conductive probe like an “Etch-a-Sketch” toy to induce or erase conducting nanowires at the LaAlO₃/SrTiO₃ interface, we have created a number of nanoscale networks for charge transport and investigated how current and voltage are related to each other in such networks. Our findings are quite extraordinary: Ohm’s law is violated in two different respects. First, while Ohm’s law states that the resistance of a wire should be proportional to its length, we have observed instead a length-independent resistance, whose value is of the order of the resistance quantum h/e². Second, while a voltage is only established along the path of a current according to Ohm’s law, we have observed “nonlocal” resistances—voltages that are separated from current paths by as much as 10 micrometers.

Precisely what microscopic mechanisms are operating behind these fascinating findings is not yet well understood. But we hope that the findings will be the kindling and match that are needed to ignite a fire.

Physicists working on unconventional superconductivity believe that magnetism plays an important role in how the normally mutually repelling electrons pair up to give rise to the superconductivity. The question of precisely what that role is remains open still, even for cuprates, the class of copper-based unconventional superconductors discovered several decades ago. In cuprates, the emergence of superconductivity is associated with the suppression of certain magnetic order in the materials. The recently discovered iron-based superconductors have added richness, further challenges, and urgency to the question of magnetism and electron pairing: In some iron-based superconductors, spin-density wave (SDW)—a type of magnetic order associated with the Fe atoms—can even coexist with superconductivity. Such coexistence raises a number of immediate questions: Is it encoded in the electronic structure? How is the electron pairing affected by the coexistence? In this paper, we address these questions with several important discoveries made with high-resolution angle-resolved photoemission spectroscopy experiments on NaFe_0.9825Co_0.0175As, an iron-based compound known to show the coexistence.

We have mapped out the electronic structure of NaFe_0.9825Co_0.0175As extensively in the three-dimensional momentum space and for a range of temperatures. Our effort has uncovered a few highly interesting properties. Both the signature of spin-density wave and that of superconductivity appear in the same electronic band structure but are located in different parts of that structure. This observation clearly indicates that the coexistence is an intrinsic property of the material and also explains at the same time why the magnetic order and superconductivity are not mutually exclusive. Moreover, the material’s superconducting gap, a measure of the underlying electron pairing, varies on the Fermi surface in a highly anisotropic, yet nodeless (i.e., nonzero everywhere) manner. This gap anisotropy is in direct contrast to the gap isotropy observed in a related superconducting, but SDW-free, compound, NaFe_0.955Co_0.045As, and is therefore most likely directly and intimately related to the coexistence.

Our findings add a number of important pieces to the fundamental magnetism-superconductivity puzzle. The absence of a zero superconducting gap on the Fermi surface will, in particular, put strong experimental constraints on the theories of iron-based superconductors that are being developed.

In semiconductor-based electronics, impurities in silicon either have to be reduced as much as possible or play a supporting role as dopants in devices. But such impurities have been given a new life in physics by the realization that they could be used to work as the “qubits” of quantum computation or as trapped atoms or molecules for studying atomic and molecular physics. Indeed, a single atom of phosphorous or bismuth—called “donors” due to the fact that they have one more valence electron than silicon—doped into silicon has been shown to behave like an enlarged hydrogen atom with a similar series of quantum states of discrete energies. Like hydrogen, these donor impurities in silicon have their antihydrogen-like analogues, which are “acceptors” that have one valence electron less than silicon. Boron and aluminum are two examples of such acceptors.

For the donor impurities, which have been much better studied than the acceptor impurities, both long excited-state lifetimes and long coherence times of their spins have been observed—qualities that make them a desirable class of candidates for qubits and quantum coherent control. For the acceptor impurities, however, similar quantities were not known and were expected to be much shorter based on the atomic spectra of the acceptors. In this paper, we show experimentally that the expectation is incorrect: The acceptor impurities in silicon have relaxation times that are in the several tens of picoseconds—of the same order of magnitude as those of the donor impurities.

We have shown that quantum coherent control of the acceptor impurities is also possible. This possibility not only expands the range of systems for silicon-based quantum device applications, but also opens a new playground for trapped-atom experiments where quantum chemistry involving atoms and antiatoms can be explored.

Computer simulations of dynamics at atomistic or molecular resolutions in a large material system are most often performed under the imagined conditions of a fixed number of atoms or molecules, a fixed volume, and a fixed temperature—the so-called canonical-ensemble setup. This setup treats all particles on an equal footing and is straightforward to implement. But, for its all-equal concept, it pays an unavoidable computational cost with little gain in cases where the system in question separates into regions with distinct physical features and where the simulator is only interested in the physical properties of one particular region rather than all. A case in point is that of a large protein molecule in water, where a solvating shell of water molecules with different molecular organization and dynamics forms next to the protein—a region separate from, but exchanging molecules with, the rest of the bulk region. Another setup, the grand-canonical-ensemble approach, deals with such cases naturally at a conceptual level: Rather than maintaining a fixed number of particles, the approach allows a simulated system to exchange them with an external reservoir and thus has the inherent flexibility to choose a region to become the focus of the simulation. In practice, however, this approach is very difficult and computationally expensive to execute because of the technical difficulty associated with adding or removing particles into a simulated system. A practical simulation method that is free of the difficulties and is computationally efficient would be most desirable. In this paper, we present just such an original method.

The seed of our method is a simulation method—the so-called adaptive-resolution-simulation (AdResS) method—that was developed originally for the purpose of performing simulations that can change the spatiotemporal resolution from an atomistic to a much lower, coarse-grained one. In an AdResS simulation, the system simulated has a natural separation of the region of molecules described by the desired atomistic resolution from a region containing coarse-grained particles, and the numbers of the particles in the two regions change on the fly. Our new realization is that this feature can be translated into a setup similar to a grand-canonical ensemble: While the atomistic region represents the system of interest, the region of coarse-grained particles can be interpreted as the external particle reservoir. We have proved the validity of this idea by mathematical arguments and challenging numerical tests and have demonstrated the high computational efficiency of our new method for simulations of the grand-canonical type. Indeed, some properties unique to this type of simulation, such as chemical potential, can be computed at a much lower cost than that required by the current popular methods of molecular-dynamics simulations.

Many interesting and important problems in natural science require molecular-dynamics simulations of size and duration that are beyond the capabilities of many existing methods: The structure and solvation of large biomolecules in water in biochemistry, the chemical potential for the optimal insertion of additives in polymer melts in materials science, and the role of hydrogen quantum spatial delocalization in hydrogen-bonded complex liquids in physics are just a few examples. We expect that with our new method these problems may now be studied at costs that are no longer very high or prohibitive.

Absorption of an x-ray photon of proper wavelength by a molecule can take the molecule to a highly excited electronic state accompanied by molecular vibration. Molecules in these states are very reactive chemically. Such highly excited and reactive molecules are ubiquitous and important in diverse fields such as plasma physics, astrochemistry, radiation physics, and photochemistry. But, how much do we actually understand the quantum dynamics of these unstable species? The answer is: not yet much, even in the case of one of the simplest molecules, the nitrogen molecule. In this work, we apply state-of-the-art resonant photoemission spectroscopy to the study of the electron and nuclear dynamics of highly excited nitrogen molecules and map out, with an unprecedented resolution and degree of comprehensiveness, the vibrational wave functions and their ultrafast evolution with time.

A highly electronically excited nitrogen molecule is very unstable and lives over an extremely short time scale of a few femtoseconds. Its decay yields a molecular ion through the so-called Auger process, where a high-energy electron is ejected when another electron falls into the hole left by the initial excited electron. The Auger-electron emission spectra naturally contain fundamental information about the extremely short-lived intermediate states, and not least, also on what the vibrational motion of the resulting molecular ion is like—knowledge that has not been available so far. It is such spectra that we measure experimentally and analyze.

Indeed, taking advantage of the high spectral brightness of the synchrotron radiation at the PLEIADES beam line at the SOLEIL Synchrotron, Paris-Saclay campus, we are able to selectively excite nitrogen molecules, control the spatial forms of the vibrational wave functions of the excited molecules, and follow the time evolution of the vibrational wave functions of the final molecular ions. Adding to our toolbox first-principles calculations of the evolution of the vibrational wave functions, that correctly account for the coupling between the nuclear and electronic motion in highly excited, and thus deformed, molecules, we have succeeded in not only mapping out for the first time the actual shapes of the vibrational wave functions of highly excited nitrogen molecules, but also identifying a few ionic states that have not been known before.

This approach is not only limited to the nitrogen molecule: It can be extended to studies of excited ionic states of even larger molecules, and can also be easily transposed to neutral molecular states by detecting the radiative decay instead of the electron emission by Auger decay.

Are there unifying scientific concepts that define or distinguish physical states of matter? Symmetry has long been known to be such a concept, as illustrated by the contrast between the formlessness of liquid water and the hexagonal form of snowflakes, which evidences the breaking of the rotational symmetry of space. Topology, the mathematical framework that describes global characteristics of geometrical structures, has been recognized as another, as epitomized by the quantum Hall effect. Only very recently, theoretical physicists imagined, by combining symmetry and topology, a whole new world of exotic states of matter: symmetry-protected topological states of electronic materials.

The “topology” in this new context is associated with the twisting of electronic bands, and the symmetry-topology interplay is manifested in the form of unusual states that necessarily transform an insulator into a conducting material through its surface. Theoretical predictions based on the simplest form of these ideas, of topological insulators of noninteracting electrons, have received spectacular experimental confirmation. Do analogous phases arise more generally, when particles strongly interact with each other and when there is no notion of a band structure? Indeed, we find that many new phases appear with interactions. Here we determine the defining properties for a broad class of them.

We focus on symmetry-protected topological states in three-dimensional systems whose excitations in the bulk are characterized quasiparticles with an energy gap that behave like bosons. Remarkably, we find that interactions lead to a plethora of novel states that exist on the two-dimensional surface of such a material. The most fundamentally salient feature of these states is that they have symmetry properties that are impossible to realize in a purely two-dimensional electronic system. These states are not just a dream of theorists: They may potentially be realized in frustrated spin systems in solid materials or in systems of ultracold bosonic atoms confined to an optical lattice.

As exotic states of matter, topological insulators have, for a number of years now, commanded the fundamental interest, and fueled the scientific creativity, of a broad swath of condensed matter physicists. But, identifying these remarkable states of matter unambiguously has been fiendishly difficult. One interesting approach—a powerful one if viable—relies on the answer to the following question: Is it possible—and if yes, how—to detect unambiguously a topological insulator from its bulk properties, as opposed to from its surface properties? In this theoretical paper, we answer this question in the context of 2-dimensional (2D) topological insulators with substantial electronic interactions (thus termed “correlated topological insulators”). Our approach exploits a unique response of these systems to localized magnetic defects engineered through injections of magnetic fluxes.

In a 2D topological insulator, insertion of a magnetic flux of the size of half a flux quantum—a π flux—gives rise to two types of states localized around the π flux: a so-called Kramers pair of spin fluxons carrying spin ±1/2 and a pair of charge fluxons carrying charge ±e. The existence of these states and their quantum numbers are intimately tied to the properties of the topological insulator, in particular, its topological invariant. Previous work on π fluxes was limited to 2D topological insulators in which electronic interactions can be safely ignored. We have shown, with a paradigmatic model for correlated topological insulators, that electronic interactions remove charge fluxons from the low-energy excitation spectrum of the systems but leave spin fluxons detectable via a Curie-law-type contribution to the bulk magnetic susceptibility—a property routinely measured in condensed matter physics labs. Combining these types of theoretical predictions, which are enabled by the powerful quantum Monte Carlo computational methods and easily accessible bulk-measurement techniques, leads to a simple, yet highly effective tool for identifying 2D correlated topological insulators.

Another, but no less significant, usefulness of this approach, which we have also demonstrated, is that the creation of spin fluxons in response to magnetic fluxes opens a new route to artificially designing and simulating quantum spin models within the bulk gap of topological insulators—creative stimuli to new theoretical and experimental efforts.

In many situations in life, frustration is counterproductive and best avoided. In physics, however, frustration—the presence of competing forces that cannot be simultaneously satisfied—is often the source of exotic physical properties of materials systems. The exotic spin-liquid behavior in frustrated magnets provides just such an example. Defying the conventional expectation that at low temperatures individual microscopic spins in a material should become ordered in their orientations, spins in a frustrated magnet are thought to show no apparent orientational order. Rather, their orientations fluctuate among a large number of configurations where the fluctuations, though order-destroying, are actually highly correlated. Adding more kindling to the fire of interest in this so-called spin-liquid state are the tantalizing possibilities that spin liquids can host a range of very exotic phenomena, not least, topological phases and artificial gauge fields. In this article, we report a combined theoretical and experimental study of a frustrated magnet (Ho₂Ti₂O₇) that points to the presence of a low-temperature spin-liquid state in which the correlations in the spin fluctuations are topological in nature.

Ho₂Ti₂O₇ is a well-known frustrated magnet. In this material, two competing forces frustrate each other: While the physical interaction between the individual microscopic spins requires the spins to align in the same direction, the geometry of the crystalline lattice the spins are located in makes it impossible for this requirement to be met for all spins. The result is a local spin-lattice structure that resembles the local coordinated oxygen framework in water, ice, and a large number of low-energy, but energetically equivalent states of globally correlated spin configurations. These configurations can be classified into different sectors by a topological invariant. We are able to show that experimental techniques such as neutron scattering and magnetic susceptibility measurements, which probe spin-spin correlations, can be used to gauge the topological properties. Indeed, we have succeeded in identifying an abnormal temperature dependence in the spin-spin correlations as the emergence of correlated fluctuations among the many topological sectors at very low temperatures, indicative of a spin-liquid state characterized by topological fluctuations.

We believe that our findings are general, in that they should apply to a vast family of models and materials for frustrated magnets.

Suppose that Alice and Bob share some 1000-sided dice having special properties: When Alice rolls her “fair” die, it is equally likely to land on any of the 1000 sides. However, when Bob rolls his equally “fair” die, he always rolls the same side as Alice, even if they are far apart and roll at the same time. Even though Alice and Bob’s individual outcomes are random, when considered together, they are correlated and always agree. This rather odd behavior is the essence of quantum entanglement and can be replicated by photons when trying to measure their direction of propagation.

Quantum mechanics tells us that a photon is equally likely to be traveling in many different directions, but, upon actually measuring this direction (rolling the dice), only one direction is found. Strangely, measurements of a second photon entangled with the first, also equally likely to propagate in many directions, will always give the same outcome as the first photon. Unfortunately, this behavior is difficult to observe with state-of-the-art detectors. In our paper, we report that, by combining a technique called compressive sensing with standard imaging arrays, we are able to efficiently observe these complex quantum correlations of pairs of photons produced in a nonlinear optical device.

Characterizing high-dimensional entanglement is challenging because the amount of data required grows exponentially with the number of dimensions. Interesting signals within this parameter space also tend to be sparse, so measuring one is like hunting for needles in a haystack with an unknown number of needles. Furthermore, most entanglement measures require a tomographic procedure to recover a wave function or density matrix with complex coefficients that cannot be directly measured. This procedure requires a complicated series of measurements and postprocessing.

To overcome these limitations, we apply ideas from two fields, information theory and compressive sensing. Information theory provides a way for us to characterize an entangled system directly from measurements by means of a quantity called the classical mutual information. A comparison of classical mutual information in the position and momentum bases allows us to show that the correlations are nonclassical. To handle the high dimensionality, we adapt compressive-sensing single-pixel cameras to change sparsity from a problem into a resource. Compressive sensing is a measurement technique that effectively compresses a signal during measurement, greatly reducing the requisite amount of data. In our system, measurement time is reduced from a year to a few hours.

Does a moving car have a definitive position before we make a measurement of it? Anyone who answers this question with a “no” will be considered delusional: That the act of observation of an object and the object’s “properties” are independent is philosophically the most fundamental tenet of classical physics as we know it. When one replaces the moving car with a quantum particle, however, the answer is indeed a mind-boggling “no.” According to quantum mechanics, “unperformed experiments have no conceivable results.” Why that is so is still one of the ongoing and most fundamental debates about quantum mechanics.

Some of the contestants for a fundamental explanation of what physical “reality” is in the quantum realm have been the so-called hidden-variable theories, built on the assumption that more fundamental mechanisms that are consistent with our notion of classical physical reality are behind the apparent quantum-mechanical puzzles but are just not observable. Such hidden-variable theories have been challenged by many renowned scientists from many different angles for decades. In the 1960s, Simon Kochen and Ernst Specker proved a theorem, known as the KS theorem, that excludes a type of hidden-variable theory known as noncontextual in which the results of measurements reveal preexisting properties and are independent of other compatible measurements. So far, however, the KS theorem has remained a purely theoretical and abstract construct. In this paper, we report the first implementation of the KS theorem in two different single-photon experiments, each also illustrating one possible application in quantum-information processing.

The proof of the KS theorem has one basic ingredient: a set of measurements or tests that return only either a yes or no answer on their own but have the peculiarity that there is no way for them to return yes or no answers in a way that is in agreement with the assumptions of noncontextual hidden-variable theories and the foundations of quantum mechanics. Our experiments have realized a complete set of KS tests by employing the polarization, orbital angular momentum, and path of single photons. But, what may also be promising is the potential of the KS tests for quantum computing and information processing: We have demonstrated that the KS tests can be used for solving an algorithmic task with an unbeatable quantum advantage (compared to the classical protocol) and for producing correlations that, independent of the initial state of the system, are maximally contextual, as they lead to the maximum possible violation of an inequality satisfied by any noncontextual hidden-variable theory.

The electrical behavior of a compound discovered in 1969, samarium hexaboride (SmB₆), has been a 40-year mystery in condensed matter physics. The material was thought to be a Kondo lattice, an insulator in which screening of its localized spins associated with the ion lattice by the itinerant electrons was expected to lead to a temperature-dependent evolution of its electronic structure and “freezing” of the electrons at low temperatures. Yet, a finite electrical conductance was observed at very low temperatures. A theoretical solution to this mystery was proposed two years ago in the form of the novel concept of topological Kondo insulators, materials or physical states where nontrivial surface electrical conduction coexists with the Kondo effect. However, clear evidence for the evolution of the material’s electronic structure with temperature has not been obtained, nor has there been experimental confirmation of the predicted surface conduction. In this paper, we report experimental results that not only show for the first time the systematic Kondo-type evolution with temperature of the electronic structure of SmB₆ but also support the prediction of SmB₆ as a topological Kondo insulator.

Our experimental results are obtained by the use of point-contact spectroscopy. Soft point-contact junctions between SmB₆ and silver are fabricated on a SmB₆ single crystal. Conductance through such a junction as the function of the bias voltage across it—a conductance spectrum—probes directly the electronic structure of the bulk crystal. An asymmetric conductance spectrum observed in our junctions is a clear indication of the existence of the expected Kondo screening effect in SmB₆. An analysis of the spectra confirms that, by its electronic structure, the SmB₆ single crystal is indeed a bulk insulator, a conclusion that strongly suggests that the finite low-temperature conductance observed earlier should be attributed to the conduction of surface states, in other words, that SmB₆ is a topological Kondo insulator.

The experimental findings reported here provide a simple but clear example for the existing understanding of the correlation between the electronic structure and the screening effect in Kondo lattices. In addition, the implication of the presence of conducting surface states in SmB₆ should also add new incentive to the study of topological insulators and their applications.

When a neuron in our brain fires or a muscle cell gets activated in an exercise, inorganic ions such as sodium and potassium flow across the cell membrane to generate and propagate an electric signal. Such flows are exquisitely regulated by a variety of molecular machineries. Some of these machineries are the so-called voltage-gated ion channels, transmembrane proteins that open and close—in response to the electric voltage across the membrane—to allow or stop (i.e., “gate”) ion flows. Much has been learned, over the past 50 years or so, about the chemistry, structure, and dynamics of the ion channels that underlie their physiological function. In this paper, we report a new experimental finding that opens another dimension to understanding ion-channel dynamics: Voltage-gated potassium channels show a nonlinear viscoelastic dynamics that is similar to how Silly Putty responds to mechanical forces that deform it.

Our experiments focus on a cousin of voltage-gated potassium channels in neurons, a membrane protein that is called KvAP. After reconstituting the channel-protein molecules into an artificial membrane that models biological cell membranes, we control the gating of the channels through an ac electric field applied across the membrane and study the gating dynamics by monitoring the ionic current. We find an essential nonlinearity in how the current responds to the driving voltage: The current saturates and even decreases for increasing voltage. Using a simple mechanical model for the gating-related conformational motion of the ion channel, we can explain this nonlinearity with a Silly-Putty-like internal molecular viscosity for the ion channel’s conformational motion. In essence, the harder and faster one squeezes the molecule, the stiffer it becomes. Whether, and how, this nonlinearity is related to the physiological function of voltage-gated ion channels remains to be seen.

Each motorized mechanical machine is ultimately driven by an engine. The physically most insightful understanding of the workings of engines comes from the realization that engines operate in a thermodynamic cycle, during which mechanical work is generated. In a molecular parallel, enzymes endogenous to biological cells are the engines and machines of the cells, without which biological life would not be possible. When in operation, a biological enzyme evidently also goes through a cycle. But, by what essential physical parameters is the cycle defined? In this paper, we propose, based on the nanomechanical measurements we have performed on an enzyme, that a fundamental representation of the cycle of the enzyme is provided by the strain-stress response of the enzyme.

The enzyme we studied is guanylate kinase, an important player in a cell’s energy metabolism. It is known that its enzymatic function is closely associated with its structural (mechanical) deformation. By fixing two ends of the enzyme molecules to a substrate and a gold nanoparticle, driving the gold nanoparticle with an oscillating force of varying frequencies and strengths, and then measuring the extent of the molecular mechanical deformation, we were able to show that the molecular mechanical deformation responds to the applied force in a frequency-dependent way. If, in its functional cycle, the enzyme changes its conformation at different rates in the forward and reverse parts of the cycle—an assumption that is very realistic—a nonequilibrium mechanical cycle must then open, corresponding to the functional cycle. This is the essence of our proposal.

This picture relates the functional properties of an enzyme to its mechanical properties. Assuming that it applies to a broad class of enzymes including molecular motors, we are able to predict several functional properties of real enzymes, for example, the maximum possible rate of an enzyme, in terms of the mechanical properties of the enzyme. Although the current accuracy for determining the mechanical properties of a molecular enzyme is not yet satisfactory, our predictions, when applied to molecular motors, show reasonable order-of-magnitude consistency with the existing data on the functional properties of the motors. It should be worthwhile to explore further this mechanical-functional perspective to gain more fundamental understanding of the workings of biological molecular machines.

In the world of electronics, the quest for smaller and faster devices is a never-ending one. In that quest, one of the challenges is the problem of heating in such devices. The field of spintronics emerged as an answer to that challenge, which exploits the spins of electrons in addition to their charges in electronic transport. Spin-polarized currents—currents of electrons whose spins point in the same direction—and efficient spin filters are the foremost must-haves in spintronics. The so-called helical modes, which transport opposite spins in opposite directions, are the physical basis for spin filters. These modes have their origin in the spin-orbit interaction (SOI), a relativistic effect that couples the spin and orbital (motional) degrees of freedom of an electron together. For the modes to be observed and operative, pristine materials are needed to avoid mode mixing caused by impurities. The high purity of graphene makes the material an ideal candidate in this respect. However, its intrinsic SOI is known to be extremely small.

In the present work, we address this challenge and propose a novel way to generate a giant, effective SOI in graphene nanoribbons. The essential idea is to enhance the SOI intrinsic in a nanoribbon by applying to it spatially varying magnetic fields that can be produced by nanomagnets. As our work demonstrates, large values of SOI become possible and result in helical modes of nearly perfect spin polarization. Moreover, the high purity of graphene nanoribbons and their considerable sub-band splittings allow for great control of the number of helical transport channels at temperatures higher than what could be achieved with semiconducting nanowires. All these together make graphene nanoribbons the most promising family of candidates for spintronics effects in the temperature range of a few kelvins.

The potential of our proposal goes beyond the context of spintronics. The search for Majorana fermions, exotic particles that are their own antiparticles, is currently a white-hot topic in physics. Our proposal offers another material avenue for the search: Bringing a nanoribbon with helical modes into proximity to an s-wave superconductor would lead to a topological state that supports Majorana fermions.

Enabled partly by the breathtaking developments in computer technology, simulations on modern computers of the dynamics of small and large atomic or molecular systems are ubiquitous and indispensable in physics, chemistry, and biology, for the testing of new theories or the modeling of experiments that can’t be done in a lab. But, such simulations suffer inevitably some inaccuracies or artifacts. One source of artifacts comes from approximating the continuous passage of time with a digitized (or discretized) series of time slices. To develop a physical understanding of this source of artifacts and to systematically control or correct such artifacts in simulations is an ongoing quest of the computational physics community. In this paper, we achieve this goal in the context of simulations of Langevin dynamics—a workhorse theory for describing molecular dynamics. We do this through a new cross-topic perspective that brings together recent fundamental developments in nonequilibrium statistical mechanics and computer simulations.

Langevin dynamics, first developed about a century ago by French physicist Paul Langevin, is a simple theory for describing motions—at molecular scales or beyond—of effective particles that are not necessarily the constituent atoms or molecules themselves. It lumps faster and shorter-scale motions than those described into an effective friction and a series of random kicks on the effective particles. We used this theory to model the behavior of simple model systems, and confirmed what had already been shown mathematically by others: that computer simulations based on the time slices of the dynamics deviated significantly from what the exact theory predicted.

Our new insight is a physical interpretation of this otherwise-cryptic result: that the use of time slices has an effect analogous to doing additional work on the system—which we call “shadow work” because the system behaves as if there is an additional nonequilibrium driving force, one that is not explicitly modeled. This insight, confirmed by simulations of a toy model of one effective particle, leads to further understanding and methodological progress. Simulations of a more complex system consisting of hundreds or thousands of water molecules permitted a measurement of how big the error is. With an appropriate algorithm, we are able to isolate this shadow work from the physically meaningful work, quantify it, and, in turn, systematically correct for it, using recently developed theorems from the field of nonequilibrium statistical mechanics.

We are extremely optimistic about future developments in this new direction. We anticipate that other insights from nonequilibrium statistical mechanics can be exploited to the gain of computer simulations, and we also believe that other methods that simulate molecular dynamics with related but more complicated schemes than the straightforward Langevin dynamics may benefit from this perspective, too.

When a crystalline solid such as silicon is exposed to a laser beam of very high intensity, what happens to the crystal? It melts, even when the laser beam is an extremely short pulse of femtosecond duration, but not in the same way as it would if it were heated up slowly. Such laser-induced nonthermal melting processes have opened up not only a new window to the electronic and atomic dynamics in solid crystals that take place on unprecedented short time scales of femtoseconds but also new possibilities of laser-controlled material processing. In this theoretical paper, we present the first accurate quantum-mechanical simulation of the motions of hundreds of atoms in silicon crystal after femtosecond-laser excitation and reveal how the atomic nuclei vibrate under the influence of laser-excited electrons as the laser intensity is tuned across the threshold for the nonthermal melting of silicon.

In a crystal in thermal equilibrium with its surroundings, the constituent atomic nuclei vibrate—in concerted fashions—around their average positions that form the crystal lattice. Both the lattice constants and the mean-square amplitudes of the atomic vibrations (or, equivalently, the numbers of frequency-distinct phonons) depend not only on the temperature but also on the interactions that hold the nuclei together, to which the electrons contribute. When a femtosecond-laser pulse that can strongly excite some of the electrons hits the crystal, a substantial fraction of the electrons becomes heated almost instantaneously. This nonequilibrium heating alters the forces between the atomic nuclei. If the laser intensity is above a threshold, the altered forces can no longer hold the atoms together and the crystal eventually melts via an ultrafast process. If the laser intensity is below the threshold, the altered forces define new average atomic positions and new phonon modes. In either case, however, the positions and momenta of the nuclei cannot adjust to the new values immediately. What, then, happens to the nuclei immediately after the ultrafast laser excitation?

To answer this question accurately, we have taken into account the effect of the hot-electron plasma generated by the laser pulse in our quantum-mechanical description of the atomic motion. A newly developed computer algorithm has allowed us to simulate the motions of hundreds of silicon atoms. What we have discovered is the “squeezing” of the thermal-equilibrium phonons at below-the-threshold laser intensities: The motions of the nuclei switch synchronously between vibrations of amplitudes that are either smaller or bigger than the equilibrium amplitudes, corresponding to either squeezing in the real space or squeezing in the momentum space, respectively. Moreover, we have determined a correlation between the appearance of the squeezed thermal phonons below the melting threshold and the nonthermal melting: The former necessarily and sufficiently heralds the occurrence of the latter.

The conclusions we have drawn about silicon should apply generally to more atomic solids and add an important contribution to our understanding of the responses of solid materials to strong and ultrafast electronic excitations.

In a conventional metal or semiconductor, electrons that are responsible for the electrical properties move without much regard to the local electronic configuration at each atomic site. “Strongly correlated” materials, however, provide a fundamentally interesting contrast. In these materials, electrons that are mobile move by hopping from one atom to the next, and they are very sensitive to the local electronic environment of the atoms they hop from and to. Each single electron has a complex influence on its neighbors. Such strong electronic correlations are known to operate in unconventional high-temperature cuprate superconductors. However, whether electronic correlations play any important role in the physics of a new class of high-temperature superconductors containing iron layers is still an open, hotly debated question. In this experimental paper, we show that substituting cobalt for iron in these superconductors, which is similar to adding one electron to each iron atom, decreases strongly the electronic correlations.

Using angle-resolved photoemission spectroscopy, we have determined the electronic band structure of BaCo₂As₂, the cobalt-based nonsuperconducting analog of an iron-based superconductor. From the electronic band structure, we are able to extract the strength of the electronic correlations in this material, which is significantly weaker than its iron-based superconducting cousin. To understand the microscopic mechanism responsible for this substitution-induced weakening of the electronic correlations, we have performed theoretical calculations of the electronic structure not only of BaCo₂As₂, which compares well with the experimental results, but also of a hypothetical compound, a BaFe₂As₂ crystal with one extra electron added on each iron site. The results of the calculations for the two materials are almost identical, suggesting that electronic band filling, when combined with a particular type of correlation between the electronic spins—the so-called Hund’s-rule coupling—tunes the electronic correlations in this material.

Our results suggest that the electronic correlations we have uncovered cannot be neglected in the iron-based superconductors and may even be intimately related to the mechanism by which electron pairs form in the superconducting state.

The huge success of modern electronics is an indisputable testimony to the power of the concept of electronic circuits. At the heart of the concept are modularity based on elementary building blocks and limitless possibilities to combine those blocks into large networks for new functionality. A thorough scientific understanding of how the elements work individually and in combination together with the ease of fabrication has made the concept a technologically powerful one. Can the concept of superconducting circuits, in which electric voltages and currents are governed by the laws of quantum physics, similarly find its power in quantum-information processing? Quantum behavior in such circuits is fragile, and experiments are only now starting to probe larger circuit networks. While the energy levels of such large quantum circuits can be sensitively probed by experiments, their theoretical prediction poses a significant challenge. In our work, we address this challenge by leveraging the symmetry properties of certain superconducting circuits and achieving dramatic simplifications in the computation of energy levels.

The basic building block of a superconducting circuit is a Josephson junction—a superconducting wire interrupted by an insulating link. A circuit containing an array of more than 40 Josephson junctions, the so-called fluxonium device, is already experimentally realized. In pace with this experimental development and in anticipation of further experimental advances to come, we have developed a theory that enables, for the first time, quantitative modeling of energy spectra of large circuits. What gives the theory its capability is our realization and employment of the power of the symmetry properties of the circuits. Symmetries such as the approximate invariance of the energy spectra under permutations of individual junction variables divide the energy spectrum of a fluxonium device into subspectra corresponding to nearly degenerate states. This conceptual simplification also allows us to obtain new, otherwise-difficult-to-obtain predictions of the energies of the collective excitations in the circuit under perturbation.

Experiments have shown that coherence times of fluxonium devices are on par with those of much simpler circuits. Our findings should encourage further research that explores quantum coherence in superconducting circuit networks of increasing complexity, and the approach of harnessing the symmetries of circuit networks will be an important ingredient for quantitative theory in the future.