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A topological insulator, as originally proposed for electrons governed by quantum mechanics, is characterized by a dichotomy between the interior and the edge of a finite system: The bulk has an energy gap, and the edges sustain excitations traversing this gap. However, it has remained an open question whether the same physics can be observed for systems obeying Newton's equations of motion. We conducted experiments to characterize the collective behavior of mechanical oscillators exhibiting the phenomenology of the quantum spin Hall effect. The phononic edge modes are shown to be helical, and we demonstrate their topological protection via the stability of the edge states against imperfections. Our results may enable the design of topological acoustic metamaterials that can capitalize on the stability of the surface phonons as reliable wave guides.

Topological phononic crystals, alike their electronic counterparts, are characterized by a bulk–edge correspondence where the interior of a material dictates the existence of stable surface or boundary modes. In the mechanical setup, such surface modes can be used for various applications such as wave guiding, vibration isolation, or the design of static properties such as stable floppy modes where parts of a system move freely. Here, we provide a classification scheme of topological phonons based on local symmetries. We import and adapt the classification of noninteracting electron systems and embed it into the mechanical setup. Moreover, we provide an extensive set of examples that illustrate our scheme and can be used to generate models in unexplored symmetry classes. Our work unifies the vast recent literature on topological phonons and paves the way to future applications of topological surface modes in mechanical metamaterials.

The current rate of species extinction is rapidly approaching unprecedented highs, and life on Earth presently faces a sixth mass extinction event driven by anthropogenic activity, climate change, and ecological collapse. The field of conservation genetics aims at preserving species by using their levels of genetic diversity, usually measured as neutral genome-wide diversity, as a barometer for evaluating population health and extinction risk. A fundamental assumption is that higher levels of genetic diversity lead to an increase in fitness and long-term survival of a species. Here, we argue against the perceived importance of neutral genetic diversity for the conservation of wild populations and species. We demonstrate that no simple general relationship exists between neutral genetic diversity and the risk of species extinction. Instead, a better understanding of the properties of functional genetic diversity, demographic history, and ecological relationships is necessary for developing and implementing effective conservation genetic strategies.

A neural-network technique can exploit the power of machine learning to mine the exponentially large data sets characterizing the state space of condensed-matter systems. Topological transitions and many-body localization are first on the list. Classifying phases of matter is key to our understanding of many problems in physics. For quantum-mechanical systems in particular, the task can be daunting due to the exponentially large Hilbert space. With modern computing power and access to ever-larger data sets, classification problems are now routinely solved using machine-learning techniques 1 . Here, we propose a neural-network approach to finding phase transitions, based on the performance of a neural network after it is trained with data that are deliberately labelled incorrectly. We demonstrate the success of this method on the topological phase transition in the Kitaev chain 2 , the thermal phase transition in the classical Ising model 3 , and the many-body-localization transition in a disordered quantum spin chain 4 . Our method does not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand. It therefore paves the way to the development of a generic tool for identifying unexplored phase transitions.

A two-dimensional phononic quadrupole topological insulator is demonstrated experimentally using mechanical metamaterials, which has both the one-dimensional edge states and the zero-dimensional corner states predicted by theory. Topological edge and corner states The properties of many materials with topological band structures can be understood in terms of a quantization of the electric polarization. By considering a quantization of higher-order polarizations, a new class of higher-order topological insulators has recently been predicted. Marc Serra-Garcia et al. use a mechanical metamaterial to demonstrate such a system experimentally: a two-dimensional phononic quadrupole topological insulator, which has both one-dimensional states at the edges as well as zero-dimensional states at the corners. As these topological corner states are two dimensions lower than the bulk, they could provide a route to engineering one-dimensional channels along the edges of three-dimensional systems. The modern theory of charge polarization in solids 1 , 2 is based on a generalization of Berry’s phase 3 . The possibility of the quantization of this phase 4 , 5 arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures 6 , 7 , 8 , 9 , 10 . Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic 11 or phononic crystals 12 , 13 . The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments 14 . In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states 14 . However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides 12 , 15 in higher dimensions, and thereby open up a new path for the design of metamaterials 16 , 17 .

We introduce a generalized phase-space representation of qubit systems called the BEADS representation which makes it possible to visualize arbitrary quantum states in an intuitive and an easy to grasp way. Our representation is exact, bijective, and general. It bridges the gap between the highly abstract mathematical description of quantum mechanical phenomena and the mission to convey them to non-specialists in terms of meaningful pictures and tangible models. Several levels of simplifications can be chosen, e.g. when using the BEADS representation in the communication of quantum mechanics to the general public. In particular, this visualization has predictive power in contrast to simple metaphors such as Schrödinger’s cat.

Arguments about how ethnic diversity affects governance typically posit that groups differ from each other in substantively important ways and that these differences make effective governance more difficult. But existing cross-national empirical tests typically use measures of ethnolinguistic fractionalization (ELF) that have no information about substantive differences between groups. This article examines two important ways that groups differ from each other—culturally and economically—and assesses how such differences affect public goods provision. Across 46 countries, the analysis compares existing measures of cultural differences with a new measure that captures economic differences between groups: between-group inequality (BGI). We show that ELF, cultural fractionalization (CF), and BGI measure different things, and that the choice between them has an important impact on our understanding of which countries are most ethnically diverse. Furthermore, empirical tests reveal that BGI has a large, robust, and negative relationship with public goods provision, whereas CF, ELF, and overall inequality do not.

Comparative genomic approaches have been used to identify sites where mutations are under purifying selection and of functional consequence by searching for sequences that are conserved across distantly related species. However, the performance of these approaches has not been rigorously evaluated under population genetic models. Further, short-lived functional elements may not leave a footprint of sequence conservation across many species. We use simulations to study how one measure of conservation, the Genomic Evolutionary Rate Profiling (GERP) score, relates to the strength of selection (Nes). We show that the GERP score is related to the strength of purifying selection. However, changes in selection coefficients or functional elements over time (i.e. functional turnover) can strongly affect the GERP distribution, leading to unexpected relationships between GERP and Nes. Further, we show that for functional elements that have a high turnover rate, adding more species to the analysis does not necessarily increase statistical power. Finally, we use the distribution of GERP scores across the human genome to compare models with and without turnover of sites where mutations under purifying selection. We show that mutations in 4.51% of the noncoding human genome are under purifying selection and that most of this sequence has likely experienced changes in selection coefficients throughout mammalian evolution. Our work reveals limitations to using comparative genomic approaches to identify deleterious mutations. Commonly used GERP score thresholds miss over half of the noncoding sites in the human genome where mutations are under purifying selection.

Symmetries crucially underlie the classification of topological phases of matter. Most materials, both natural as well as architectured, possess crystalline symmetries. Recent theoretical works unveiled that these crystalline symmetries can stabilize fragile Bloch bands that challenge our very notion of topology: Although answering to the most basic definition of topology, one can trivialize these bands through the addition of trivial Bloch bands. Here, we fully characterize the symmetry properties of the response of an acoustic metamaterial to establish the fragile nature of the low-lying Bloch bands. Additionally, we present a spectral signature in the form of spectral flow under twisted boundary conditions.