Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement

Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.

Chiral liquid crystals (LCs) are ubiquitous, useful, and rich systems (1–4). From the first discovery of the liquid crystalline phase to the variety of chiral structures formed by biomolecules (5–9), the twisted structure, breaking both mirror and continuous spatial symmetries, is omnipresent. The unique structure also makes the chiral nematic (cholesteric) LC, an essential material for applications utilizing the tunable, responsive, and periodic modulation of anisotropic properties.The cholesteric is also a popular model system to study the geometry and topology of partially ordered matter. The twisted ground state of the cholesteric is often incompatible with confinement and external fields, exhibiting a large variety of frustrated and metastable director configurations accompanying topological defects. Besides the classic example of cholesterics in a Grandjean−Cano wedge (10, 11), examples include cholesteric droplets (12–16), colloids (17–19), shells (20–22), tori (23, 24), cylinders (25–29), microfabricated structures (30, 31), and films between parallel plates with external fields (32–40). These structures are typically understood using a combination of nematic (achiral) topology (41, 42) and energetic arguments, for example, the highly successful Landau−de Gennes approach (43). However, traditional extensions of the nematic topological approach to cholesterics are known to be conceptually incomplete and difficult to apply in regimes where the system size is comparable to the cholesteric pitch (41, 44).An alternative perspective, chiral topology, can give a deeper understanding of these structures (45–47). In this approach, the key role is played by the twist density, given in terms of the director field n by n⋅∇×n. This choice is not arbitrary; the Frank free energy prefers n⋅∇×n≈−q0=2π/p0 with a helical pitch p0, and, from a geometric perspective, n⋅∇×n≠0 defines a contact structure (48). This allows a number of new integer-valued invariants of chiral textures to be defined (45). A configuration with a single sign of twist is chiral, and two configurations which cannot be connected by a path of chiral configurations are chirally distinct, and hence separated by a chiral energy barrier. Within each chiral class of configuration, additional topological invariants may be defined using methods of contact topology (45–48), such as layer numbers. Changing these chiral topological invariants requires passing through a nonchiral configuration. Cholesterics serve as model systems for the exploration of chirality in ordered media, and the phenomena we describe here—metastability in chiral systems controlled by chiral topological invariants—has applicability to chiral order generally. This, in particular, includes chiral ferromagnets, where, for example, our results on chiral topological invariants apply to highly twisted nontopological Skyrmions (49, 50) (“Skyrmionium”).Our experimental model to explore the chiral topological invariants is the cholesteric phase of lyotropic chromonic LCs (LCLCs). The majority of experimental systems hitherto studied are based on thermotropic LCs with typical elastic and surface-anchoring properties. The aqueous LCLCs exhibiting unusual elastic properties, that is, very small twist modulus K2 and large saddle-splay modulus K24 (51–56), often leading to chiral symmetry breaking of confined achiral LCLCs (53, 54, 56–61), may enable us to access uncharted configurations and defects of topological interests. For instance, in the layer configuration by cholesteric LCLCs doped with chiral molecules, their small K2 provides energetic flexibility to the thickness of the cholesteric layer, that is, the repeating structure where the director n twists by π. The large K24 affords curvature-induced surface interactions in combination with a weak anchoring strength of the lyotropic LCs (62–64).We present a systematic investigation of the director configuration of cholesteric LCLCs confined in cylinders with degenerate planar anchoring, depending on the chiral dopant concentration. We show that the structure of cholesteric configurations is controlled by higher-order chiral topological invariants. We focus on two intriguing phenomena observed in cylindrically confined cholesterics. First, the cylindrical symmetry renders multiple local minima to the energy landscape and induces discontinuous increase of twist angles, that is, a layering transition, upon the dopant concentration increase. Additionally, the director configurations of local minima coexist as metastable domains with point-like defects between them. We demonstrate that a chiral layer number invariant distinguishes these configurations, protects the distinct layer configurations (45), and explains the existence of the topological defect where the invariant changes.     

Activation of indistinguishability-based quantum coherence for enhanced metrological applications with particle statistics imprint

Quantum coherence, an essential feature of quantum mechanics allowing quantum superposition of states, is a resource for quantum information processing. Coherence emerges in a fundamentally different way for nonidentical and identical particles. For the latter, a unique contribution exists linked to indistinguishability that cannot occur for nonidentical particles. Here we experimentally demonstrate this additional contribution to quantum coherence with an optical setup, showing that its amount directly depends on the degree of indistinguishability and exploiting it in a quantum phase discrimination protocol. Furthermore, the designed setup allows for simulating fermionic particles with photons, thus assessing the role of exchange statistics in coherence generation and utilization. Our experiment proves that independent indistinguishable particles can offer a controllable resource of coherence and entanglement for quantum-enhanced metrology.

A quantum system can reside in coherent superpositions of states, which have a role in the interpretation of quantum mechanics (1–4), lead to nonclassicality (5, 6), and imply the intrinsically probabilistic nature of predictions in the quantum realm (7, 8). Besides this fundamental role, quantum coherence is also at the basis of quantum algorithms (9–14) and, from a modern information-theoretic perspective, constitutes a paradigmatic basis-dependent quantum resource (15–17), providing a quantifiable advantage in certain quantum information protocols.For a single quantum particle, coherence manifests itself when the particle is found in a superposition of a reference basis, for instance, the computational basis of the Hilbert space. Formally, any quantum state whose density matrix contains nonzero diagonal elements when expressed in the reference basis is said to display quantum coherence (16). This is the definition of quantum coherence employed in our work. For multiparticle compound systems, the physics underlying the emergence of quantum coherence is richer and strictly connected to the nature of the particles, with fundamental differences for nonidentical and identical particles. A particularly intriguing observation is that the states of identical particle systems can manifest coherence even when no particle resides in superposition states, provided that the wave functions of the particles overlap (18–20). In general, a special contribution to quantum coherence arises thanks to the spatial indistinguishability of identical particles, which cannot exist for nonidentical (or distinguishable) particles (18). Recently, it has been found that the spatial indistinguishability of identical particles can be exploited for entanglement generation (21), applicable even for spacelike-separated quanta (22) and against preparation and dynamical noises (23–26). The presence of entanglement is a signature that the bipartite system as a whole carries coherence even when the individual particles do not, the amount of this coherence being dependent on the degree of indistinguishability. We name this specific contribution to quantumness of compound systems “indistinguishability-based coherence,” in contrast to the more familiar “single-particle superposition-based coherence.” Indistinguishability-based coherence qualifies in principle as an exploitable resource for quantum metrology (18). However, it requires sophisticated control techniques to be harnessed, especially in view of its nonlocal nature. Moreover, a crucial property of identical particles is the exchange statistics, while its experimental study requiring operating both bosons and fermions in the same setup is generally challenging.In the present work, we investigate the operational contribution of quantum coherence stemming from the spatial indistinguishability of identical particles. The main aim of our experiment is to prove that elementary states of two independent spatially indistinguishable particles can give rise to exploitable quantum coherence, with a measurable effect due to particle statistics. By utilizing our recently developed photonic architecture capable of tuning the indistinguishability of two uncorrelated photons (27), we observe the direct connection between the degree of indistinguishability and the amount of generated coherence and show that indistinguishability-based coherence can be concurrent with single-particle superposition-based coherence. In particular, we demonstrate its operational implications, namely, providing a quantifiable advantage in a phase discrimination task (28, 29), as depicted in Fig. 1. Furthermore, we design a setup capable of testing the impact of particle statistics in coherence production and phase discrimination for both bosons and fermions; this is accomplished by compensating for the exchange phase during state preparation, simulating fermionic states with photons, which leads to statistics-dependent efficiency of the quantum task.Open in a separate windowFig. 1.Illustration of the indistinguishability-activated phase discrimination task. A resource state ρin that contains coherence in a computational basis is generated from spatial indistinguishability. The state then enters a black box which implements a phase unitary U^k=eiG^ϕk,k∈{1,…,n} on ρin. The goal is to determine the ϕk actually applied through the output state ρout: indistinguishability-based coherence provides an operational advantage in this task.     

Structures and topological defects in pressure-driven lyotropic chromonic liquid crystals

Lyotropic chromonic liquid crystals are water-based materials composed of self-assembled cylindrical aggregates. Their behavior under flow is poorly understood, and quantitatively resolving the optical retardance of the flowing liquid crystal has so far been limited by the imaging speed of current polarization-resolved imaging techniques. Here, we employ a single-shot quantitative polarization imaging method, termed polarized shearing interference microscopy, to quantify the spatial distribution and the dynamics of the structures emerging in nematic disodium cromoglycate solutions in a microfluidic channel. We show that pure-twist disclination loops nucleate in the bulk flow over a range of shear rates. These loops are elongated in the flow direction and exhibit a constant aspect ratio that is governed by the nonnegligible splay-bend anisotropy at the loop boundary. The size of the loops is set by the balance between nucleation forces and annihilation forces acting on the disclination. The fluctuations of the pure-twist disclination loops reflect the tumbling character of nematic disodium cromoglycate. Our study, including experiment, simulation, and scaling analysis, provides a comprehensive understanding of the structure and dynamics of pressure-driven lyotropic chromonic liquid crystals and might open new routes for using these materials to control assembly and flow of biological systems or particles in microfluidic devices.

Lyotropic chromonic liquid crystals (LCLCs) are aqueous dispersions of organic disk-like molecules that self-assemble into cylindrical aggregates, which form nematic or columnar liquid crystal phases under appropriate conditions of concentration and temperature (1–6). These materials have gained increasing attention in both fundamental and applied research over the past decade, due to their distinct structural properties and biocompatibility (4, 7–14). Used as a replacement for isotropic fluids in microfluidic devices, nematic LCLCs have been employed to control the behavior of bacteria and colloids (13, 15–20).Nematic liquid crystals form topological defects under flow, which gives rise to complex dynamical structures that have been extensively studied in thermotropic liquid crystals (TLCs) and liquid crystal polymers (LCPs) (21–29). In contrast to lyotropic liquid crystals that are dispersed in a solvent and whose phase can be tuned by either concentration or temperature, TLCs do not need a solvent to possess a liquid-crystalline state and their phase depends only on temperature (30). Most TLCs are shear-aligned nematics, in which the director evolves toward an equilibrium out-of-plane polar angle. Defects nucleate beyond a critical Ericksen number due to the irreconcilable alignment of the directors from surface anchoring and shear alignment in the bulk flow (24, 31–33). With an increase in shear rate, the defect type can transition from π-walls (domain walls that separate regions whose director orientation differs by an angle of π) to ordered disclinations and to a disordered chaotic regime (34). Recent efforts have aimed to tune and control the defect structures by understanding the relation between the selection of topological defect types and the flow field in flowing TLCs. Strategies to do so include tuning the geometry of microfluidic channels, inducing defect nucleation through the introduction of isotropic phases or designing inhomogeneities in the surface anchoring (35–39). LCPs are typically tumbling nematics for which α2α3 < 0, where α2 and α3 are the Leslie viscosities. This leads to a nonzero viscous torque for any orientation of the director, which allows the director to rotate in the shear plane (22, 29, 30, 40). The tumbling character of LCPs facilitates the nucleation of singular topological defects (22, 40). Moreover, the molecular rotational relaxation times of LCPs are longer than those of TLCs, and they can exceed the timescales imposed by the shear rate. As a result, the rheological behavior of LCPs is governed not only by spatial gradients of the director field from the Frank elasticity, but also by changes in the molecular order parameter (25, 41–43). With increasing shear rate, topological defects in LCPs have been shown to transition from disclinations to rolling cells and to worm-like patterns (25, 26, 43).Topological defects occurring in the flow of nematic LCLCs have so far received much more limited attention (44, 45). At rest, LCLCs exhibit unique properties distinct from those of TLCs and LCPs (1, 2, 4–6, 44). In particular, LCLCs have significant elastic anisotropy compared to TLCs; the twist Frank elastic constant, K2, is much smaller than the splay and bend Frank elastic constants, K1 and K3. The resulting relative ease with which twist deformations can occur can lead to a spontaneous symmetry breaking and the emergence of chiral structures in static LCLCs under spatial confinement, despite the achiral nature of the molecules (4, 46–51). When driven out of equilibrium by an imposed flow, the average director field of LCLCs has been reported to align predominantly along the shear direction under strong shear but to reorient to an alignment perpendicular to the shear direction below a critical shear rate (52–54). A recent study has revealed a variety of complex textures that emerge in simple shear flow in the nematic LCLC disodium cromoglycate (DSCG) (44). The tumbling nature of this liquid crystal leads to enhanced sensitivity to shear rate. At shear rates γ˙<1 s−1, the director realigns perpendicular to the flow direction adapting a so-called log-rolling state characteristic of tumbling nematics. For 1 s−1<γ˙<10 s−1, polydomain textures form due to the nucleation of pure-twist disclination loops, for which the rotation vector is parallel to the loop normal, and mixed wedge-twist disclination loops, for which the rotation vector is perpendicular to the loop normal (44, 55). Above γ˙>10 s−1, the disclination loops gradually transform into periodic stripes in which the director aligns predominantly along the flow direction (44).Here, we report on the structure and dynamics of topological defects occurring in the pressure-driven flow of nematic DSCG. A quantitative evaluation of such dynamics has so far remained challenging, in particular for fast flow velocities, due to the slow image acquisition rate of current quantitative polarization-resolved imaging techniques. Quantitative polarization imaging traditionally relies on three commonly used techniques: fluorescence confocal polarization microscopy, polarizing optical microscopy, and LC-Polscope imaging. Fluorescence confocal polarization microscopy can provide accurate maps of birefringence and orientation angle, but the fluorescent labeling may perturb the flow properties (56). Polarizing optical microscopy requires a mechanical rotation of the polarizers and multiple measurements, which severely limits the imaging speed. LC-Polscope, an extension of conventional polarization optical microscopy, utilizes liquid crystal universal compensators to replace the compensator used in conventional polarization microscopes (57). This leads to an enhanced imaging speed and better compensation for polarization artifacts of the optical system. The need for multiple measurements to quantify retardance, however, still limits the acquisition rate of LC-Polscopes.We overcome these challenges by using a single-shot quantitative polarization microscopy technique, termed polarized shearing interference microscopy (PSIM). PSIM combines circular polarization light excitation with off-axis shearing interferometry detection. Using a custom polarization retrieval algorithm, we achieve single-shot mapping of the retardance, which allows us to reach imaging speeds that are limited only by the camera frame rate while preserving a large field-of-view and micrometer spatial resolution. We provide a brief discussion of the optical design of PSIM in Materials and Methods; further details of the measurement accuracy and imaging performance of PSIM are reported in ref. 58.Using a combination of experiments, numerical simulations and scaling analysis, we show that in the pressure-driven flow of nematic DSCG solutions in a microfluidic channel, pure-twist disclination loops emerge for a certain range of shear rates. These loops are elongated in the flow with a fixed aspect ratio. We demonstrate that the disclination loops nucleate at the boundary between regions where the director aligns predominantly along the flow direction close to the channel walls and regions where the director aligns predominantly perpendicular to the flow direction in the center of the channel. The large elastic stresses of the director gradient at the boundary are then released by the formation of disclination loops. We show that both the characteristic size and the fluctuations of the pure-twist disclination loops can be tuned by controlling the flow rate.     

Unusual magnetotransport in twisted bilayer graphene

We present transport measurements of bilayer graphene with a 1.38^∘ interlayer twist. As with other devices with twist angles substantially larger than the magic angle of 1.1^∘, we do not observe correlated insulating states or band reorganization. However, we do observe several highly unusual behaviors in magnetotransport. For a large range of densities around half filling of the moiré bands, magnetoresistance is large and quadratic. Over these same densities, the magnetoresistance minima corresponding to gaps between Landau levels split and bend as a function of density and field. We reproduce the same splitting and bending behavior in a simple tight-binding model of Hofstadter’s butterfly on a triangular lattice with anisotropic hopping terms. These features appear to be a generic class of experimental manifestations of Hofstadter’s butterfly and may provide insight into the emergent states of twisted bilayer graphene.

The mesmerizing Hofstadter butterfly spectrum arises when electrons in a two-dimensional periodic potential are immersed in an out-of-plane magnetic field. When the magnetic flux Φ through a unit cell is a rational multiple p / q of the magnetic flux quantum Φ0=h/e, each Bloch band splits into q subbands (1). The carrier densities corresponding to gaps between these subbands follow straight lines when plotted as a function of normalized density n/ns and magnetic field (2). Here, n_s is the density of carriers required to fill the (possibly degenerate) Bloch band. These lines can be described by the Diophantine equation (n/ns)=t(Φ/Φ0)+s for integers s and t. In experiments, they appear as minima or zeros in longitudinal resistivity coinciding with Hall conductivity quantized at σxy=te2/h (3, 4). Hofstadter originally studied magnetosubbands emerging from a single Bloch band on a square lattice. In the following decades, other authors considered different lattices (5–7), the effect of anisotropy (6, 8–10), next-nearest-neighbor hopping (11–15), interactions (16, 17), density wave states (9), and graphene moirés (18, 19).It took considerable ingenuity to realize clean systems with unit cells large enough to allow conventional superconducting magnets to reach Φ/Φ0∼1. The first successful observation of the butterfly in electrical transport measurements was in GaAs/AlGaAs heterostructures with lithographically defined periodic potentials (20–22). These experiments demonstrated the expected quantized Hall conductance in a few of the largest magnetosubband gaps. In 2013, three groups mapped out the full butterfly spectrum in both density and field in heterostructures based on monolayer (23, 24) and bilayer (25) graphene. In all three cases, the authors made use of the 2% lattice mismatch between their graphene and its encapsulating hexagonal boron nitride (hBN) dielectric. With these layers rotationally aligned, the resulting moiré pattern was large enough in area that gated structures studied in available high-field magnets could simultaneously approach normalized carrier densities and magnetic flux ratios of 1. Later work on hBN-aligned bilayer graphene showed that, likely because of electron–electron interactions, the gaps could also follow lines described by fractional s and t (26).In twisted bilayer graphene (TBG), a slight interlayer rotation creates a similar-scale moiré pattern. Unlike with graphene–hBN moirés, in TBG there is a gap between lowest and neighboring moiré subbands (27). As the twist angle approaches the magic angle of 1.1^∘ the isolated moiré bands become flat (28, 29), and strong correlations lead to fascinating insulating (30–37), superconducting (31–33, 35–37), and magnetic (34, 35, 38) states. The strong correlations tend to cause moiré subbands within a fourfold degenerate manifold to move relative to each other as one tunes the density, leading to Landau levels that project only toward higher magnitude of density from charge neutrality and integer filling factors (37, 39). This correlated behavior obscures the single-particle Hofstadter physics that would otherwise be present.In this work, we present measurements from a TBG device twisted to 1.38^∘. When we apply a perpendicular magnetic field, a complicated and beautiful fan diagram emerges. In a broad range of densities on either side of charge neutrality, the device displays large, quadratic magnetoresistance. Within the magnetoresistance regions, each Landau level associated with ν=±8,±12,±16,… appears to split into a pair, and these pairs follow complicated paths in field and density, very different from those predicted by the usual Diophantine equation. Phenomenology similar in all qualitative respects appears in measurements on several regions of this same device with similar twist angles and in two separate devices, one at 1.59^∘ and the other at 1.70^∘ (see SI Appendix for details).We reproduce the unusual features of the Landau levels (LLs) in a simple tight-binding model on a triangular lattice with anisotropy and a small energetic splitting between two species of fermions. At first glance, this is surprising, because that model does not represent the symmetries of the experimental moiré structure. We speculate that the unusual LL features we experimentally observe can generically emerge from spectra of Hofstadter models that include the same ingredients we added to the triangular lattice model. With further theoretical work it may be possible to use our measurements to gain insight into the underlying Hamiltonian of TBG near the magic angle.     

On randomly changing conformity bias in cultural transmission

Humans and nonhuman animals display conformist as well as anticonformist biases in cultural transmission. Whereas many previous mathematical models have incorporated constant conformity coefficients, empirical research suggests that the extent of (anti)conformity in populations can change over time. We incorporate stochastic time-varying conformity coefficients into a widely used conformity model, which assumes a fixed number n of “role models” sampled by each individual. We also allow the number of role models to vary over time (nt). Under anticonformity, nonconvergence can occur in deterministic and stochastic models with different parameter values. Even if strong anticonformity may occur, if conformity or random copying (i.e., neither conformity nor anticonformity) is expected, there is convergence to one of the three equilibria seen in previous deterministic models of conformity. Moreover, this result is robust to stochastic variation in nt. However, dynamic properties of these equilibria may be different from those in deterministic models. For example, with random conformity coefficients, all equilibria can be stochastically locally stable simultaneously. Finally, we study the effect of randomly changing weak selection. Allowing the level of conformity, the number of role models, and selection to vary stochastically may produce a more realistic representation of the wide range of group-level properties that can emerge under (anti)conformist biases. This promises to make interpretation of the effect of conformity on differences between populations, for example those connected by migration, rather difficult. Future research incorporating finite population sizes and migration would contribute added realism to these models.

Cavalli-Sforza and Feldman (1) studied the finite population dynamics of a trait whose transmission from one generation to the next depended on the mean value of that trait in the population. This “group transmission” constrained the within-group variability but could lead to increasing variance in the average trait value between groups. Other analyses of cultural transmission biases have incorporated characteristics of trait variation, such as the quality, and characteristics of transmitters, including success and prestige (2). Another class of transmission biases is couched in terms of the frequencies of the cultural variants in the population (3). These “frequency-dependent” biases include conformity and anticonformity, which occur when a more common variant is adopted at a rate greater or less than its population frequency, respectively (4).Humans have exhibited conformity in mental rotation (5), line discrimination (6), and numerical discrimination tasks (7). Anticonformity has been exhibited by young children performing numerical discrimination (7). Unbiased frequency-dependent transmission, known as random copying (8), has been suggested to account for choices of dog breeds (9), Neolithic pottery motifs, patent citations, and baby names (10, 11). However, baby name distributions appear more consistent with frequency-dependent (8, 12) and/or other (13, 14) biases.In nonhuman animals, conformity has been observed in nine-spined sticklebacks choosing a feeder (15) and great tits solving a puzzle box (16, 17) (but see ref. 18). Fruit flies displayed both conformist and anticonformist bias with respect to mate choice (19) (but these authors used a different definition of anticonformity from that of ref. 4, which we use, and therefore did not consider these behaviors to be anticonformist).Asch (20, 21) used a different definition of conformity from ref. 4, namely “the overriding of personal knowledge or behavioral dispositions by countervailing options observed in others” (ref. 22, p. 34). Aschian conformity (22) has been observed in chimpanzees (23, 24), capuchin monkeys (25, 26) (but see ref. 27), vervet monkeys (28), and great tits (16). It has also been empirically tested in at least 133 studies of humans and, in the United States, has declined from the 1950s to the 1990s (29).Temporal variation may also occur in forms of conformity other than Aschian. In ref. 12, popular US baby names from 1960 to 2010 show a concave turnover function indicative of negative frequency-dependent bias, but male baby names from earlier decades (1880 to 1930) show a convex turnover indicative of positive frequency-dependent or direct bias. However, most previous mathematical models of conformity have incorporated constant, rather than time-dependent, conformity coefficients.Cavalli-Sforza and Feldman (ref. 3, chap. 3) and Boyd and Richerson (ref. 4, chap. 7) studied models of frequency-dependent transmission of a cultural trait with two variants. Boyd and Richerson (4) incorporated conformist and anticonformist bias through a conformity coefficient denoted by D. In their simplest model, if the frequency of variant A is p and that of variant B is 1−p, then the frequency of variant A in the offspring generation, p′, isp′=p+Dp(1−p)(2p−1),[1]where D>0 entails conformity (A increases if its frequency is p>12), D<0 entails anticonformity, D=0 entails random copying, and −2<D<1. In this model, each offspring samples the cultural variants of n=3 members of the parental generation (hereafter, role models). Sampling n>3 role models requires different constraints and, if n>4, there are multiple conformity coefficients (Eq. 19).Many subsequent models have built upon Boyd and Richerson’s (4) simplest model (Eq. 1). These have incorporated individual learning, information inaccuracy due to environmental change (30–34), group selection (35), and other transmission biases, including payoff bias (36), direct bias, and prestige bias (37). Other models, which include a single conformity coefficient and preserve the essential features of Eq. 1, incorporate individual learning, environmental variability (32, 38), group selection (39), and multiple cultural variants (38).In agent-based statistical physics models, the up and down spins of an electron are analogous to cultural variants A and B (40, 41). Individuals are nodes in a network and choose among a series of actions with specified probabilities, such as independently acquiring a spin, or sampling neighboring individuals and adopting the majority or minority spin in the sample. The number of sampled role models can be greater than three (42, 43). (Anti)conformity may occur if all (42–47), or if at least r (40, 48), sampled individuals have the same variant. In contrast, Boyd and Richerson’s (4) general model (Eq. 19) allows, for example, stronger conformity to a 60% majority of role models and weaker conformity or anticonformity to a 95% majority (in humans, this might result from a perceived difference between “up-and-coming” and “overly popular” variants).In Boyd and Richerson’s (4) general model, individuals sample n role models, which is more realistic than restricting n to 3 (as in Eq. 1); individuals may be able to observe more than three members of the previous generation. With n>4, different levels of (anti)conformity may occur for different samples j of n role models with one variant. In addition to the example above with 60 and 95% majorities, other relationships between the level of conformity and the sample j of n are possible. For example, the strength of conformity might increase as the number of role models with the more common variant increases. In a recent exploration of Boyd and Richerson’s (4) general model, we found dynamics that departed significantly from those of Eq. 1 (49). If conformity and anticonformity occur for different majorities j of n role models (i.e., j>n2), polymorphic equilibria may exist that were not possible with Eq. 1. In addition, strong enough anticonformity can produce nonconvergence: With as few as 5 role models, stable cycles in variant frequencies may arise, and with as few as 10 role models, chaos is possible. Such complex dynamics may occur with or without selection.Here, we extend both Boyd and Richerson’s (4) simplest (Eq. 1) and general (Eq. 19) models to allow the conformity coefficient(s) to vary randomly across generations, by sampling them from probability distributions. Although some agent-based models allow individuals to switch between “conformist” and “non-” or “anticonformist” states over time (40, 42, 47, 50, 51), to our knowledge, random temporal variation in the conformity coefficients themselves has not been modeled previously. In reality, the degree to which groups of individuals conform may change over time, as illustrated by the finding that young children anti-conformed while older children conformed in a discrimination task (7); thus, it seems reasonable to expect that different generations may also exhibit different levels of conformity. Indeed, generational changes have occurred for Aschian conformity (29) and possibly in frequency-dependent copying of baby names (12). Our stochastic model may therefore produce more realistic population dynamics than previous deterministic models, and comparisons between the two can suggest when the latter is a reasonable approximation to the former.We also allow the number of role models, nt, to vary over time. Agent-based conformity models have incorporated temporal (43) and individual (43, 45, 46) variation in the number of sampled individuals, whereas here, all members of generation t sample the same number nt of role models. Causes of variation in nt are not explored here, but there could be several. For instance, different generations of animals may sample different numbers of role models due to variation in population density. In humans, changes in the use of social media platforms or their features may cause temporal changes in the number of observed individuals. For example, when Facebook added the feature “People You May Know,” the rate of new Facebook connections in a New Orleans dataset nearly doubled (52).In the stochastic model without selection, regardless of the fluctuation in the conformity coefficient(s), if there is conformity on average, the population converges to one of the three equilibria present in Boyd and Richerson’s (4) model with conformity (D(j)>0 for n2<j<n in Eq. 19). These are p*=1 (fixation on variant A), p*=0 (fixation on variant B), and p*=12 (equal representation of A and B). However, their stability properties may differ from those in the deterministic case. In Boyd and Richerson’s (4) model with random copying, every initial frequency p0 is an equilibrium. Here, with random copying expected and independent conformity coefficients, there is convergence to p*=0,12, or 1. In this case, and in the case with conformity expected, convergence to p*=0,12, or 1 also holds with stochastic variation in the number of role models, nt. With either stochastic or constant weak selection in Boyd and Richerson’s (4) simplest model (Eq. 1) and random copying expected, there is convergence to a fixation state (p*=0 or 1). Finally, with anticonformity in the deterministic model or anticonformity expected in the stochastic model, nonconvergence can occur.     

From hidden order to antiferromagnetism: Electronic structure changes in Fe-doped URu2Si2

In matter, any spontaneous symmetry breaking induces a phase transition characterized by an order parameter, such as the magnetization vector in ferromagnets, or a macroscopic many-electron wave function in superconductors. Phase transitions with unknown order parameter are rare but extremely appealing, as they may lead to novel physics. An emblematic and still unsolved example is the transition of the heavy fermion compound URu2Si2 (URS) into the so-called hidden-order (HO) phase when the temperature drops below T0=17.5 K. Here, we show that the interaction between the heavy fermion and the conduction band states near the Fermi level has a key role in the emergence of the HO phase. Using angle-resolved photoemission spectroscopy, we find that while the Fermi surfaces of the HO and of a neighboring antiferromagnetic (AFM) phase of well-defined order parameter have the same topography, they differ in the size of some, but not all, of their electron pockets. Such a nonrigid change of the electronic structure indicates that a change in the interaction strength between states near the Fermi level is a crucial ingredient for the HO to AFM phase transition.

The transition of URu2Si2 from a high-temperature paramagnetic (PM) phase to the hidden-order (HO) phase below T0 is accompanied by anomalies in specific heat (1–3), electrical resistivity (1, 3), thermal expansion (4), and magnetic susceptibility (2, 3) that are all typical of magnetic ordering. However, the small associated antiferromagnetic (AFM) moment (5) is insufficient to explain the large entropy loss and was shown to be of extrinsic origin (6). Inelastic neutron scattering (INS) experiments revealed gapped magnetic excitations below T0 at commensurate and incommensurate wave vectors (7–9), while an instability and partial gapping of the Fermi surface was observed by angle-resolved photoemission spectroscopy (ARPES) (10–16) and scanning tunneling microscopy/spectroscopy (17, 18). More recently, high-resolution, low-temperature ARPES experiments imaged the Fermi surface reconstruction across the HO transition, unveiling the nesting vectors between Fermi sheets associated with the gapped magnetic excitations seen in INS experiments (14, 19) and quantitatively explaining, from the changes in Fermi surface size and quasiparticle mass, the large entropy loss in the HO phase (19). Nonetheless, the nature of the HO parameter is still hotly debated (20–23).The HO phase is furthermore unstable above a temperature-dependent critical pressure of about 0.7 GPa at T=0, at which it undergoes a first-order transition into a large moment AFM phase where the value of the magnetic moment per U atom exhibits a sharp increase, by a factor of 10 to 50 (6, 24–30). When the system crosses the HO → AFM phase boundary, the characteristic magnetic excitations of the HO phase are either suppressed or modified (8, 31), while resistivity and specific heat measurements suggest that the partial gapping of the Fermi surface is enhanced (24, 27).As the AFM phase has a well-defined order parameter, studying the evolution of the electronic structure across the HO/AFM transition would help develop an understanding of the HO state. So far, the experimental determination of the Fermi surface by Shubnikov de Haas (SdH) oscillations only showed minor changes across the HO → AFM phase boundary (32). Here, we take advantage of the HO/AFM transition induced by chemical pressure in URu2Si2, through the partial substitution of Ru with Fe (33–37), to directly probe its electronic structure in the AFM phase using ARPES. As we shall see, our results reveal that changes in the Ru 4d–U 5f hybridization across the HO/AFM phase boundary seem essential for a better understanding of the HO state.     

Label-free probe of HIV-1 TAT peptide binding to mimetic membranes

The transacting activator of transduction (TAT) protein plays a key role in the progression of AIDS. Studies have shown that a +8 charged sequence of amino acids in the protein, called the TAT peptide, enables the TAT protein to penetrate cell membranes. To probe mechanisms of binding and translocation of the TAT peptide into the cell, investigators have used phospholipid liposomes as cell membrane mimics. We have used the method of surface potential sensitive second harmonic generation (SHG), which is a label-free and interface-selective method, to study the binding of TAT to anionic 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-1′-rac-glycerol (POPG) and neutral 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) liposomes. It is the SHG sensitivity to the electrostatic field generated by a charged interface that enabled us to obtain the interfacial electrostatic potential. SHG together with the Poisson–Boltzmann equation yielded the dependence of the surface potential on the density of adsorbed TAT. We obtained the dissociation constants K_d for TAT binding to POPC and POPG liposomes and the maximum number of TATs that can bind to a given liposome surface. For POPC K_d was found to be 7.5 ± 2 μM, and for POPG K_d was 29.0 ± 4.0 μM. As TAT was added to the liposome solution the POPC surface potential changed from 0 mV to +37 mV, and for POPG it changed from −57 mV to −37 mV. A numerical calculation of K_d, which included all terms obtained from application of the Poisson–Boltzmann equation to the TAT liposome SHG data, was shown to be in good agreement with an approximated solution.The HIV type 1 (HIV-1) transacting activator of transduction (TAT) is an important regulatory protein for viral gene expression (1–3). It has been established that the TAT protein has a key role in the progression of AIDS and is a potential target for anti-HIV vaccines (4). For the TAT protein to carry out its biological functions, it needs to be readily imported into the cell. Studies on the cellular internalization of TAT have led to the discovery of the TAT peptide, a highly cationic 11-aa region (protein transduction domain) of the 86-aa full-length protein that is responsible for the TAT protein translocating across phospholipid membranes (5–8). The TAT peptide is a member of a class of peptides called cell-penetrating peptides (CPPs) that have generated great interest for drug delivery applications (ref. 9 and references therein). The exact mechanism by which the TAT peptide enters cells is not fully understood, but it is likely to involve a combination of energy-independent penetration and endocytosis pathways (8, 10). The first step in the process is high-affinity binding of the peptide to phospholipids and other components on the cell surface such as proteins and glycosaminoglycans (1, 9).The binding of the TAT peptide to liposomes has been investigated using a variety of techniques, each of which has its own advantages and limitations. Among the techniques are isothermal titration calorimetry (9, 11), fluorescence spectroscopy (12, 13), FRET (12, 14), single-molecule fluorescence microscopy (15, 16), and solid-state NMR (17). Second harmonic generation (SHG), as an interface-selective technique (18–24), does not require a label, and because SHG is sensitive to the interface potential, it is an attractive method to selectively probe the binding of the highly charged (+8) TAT peptide to liposome surfaces. Although coherent SHG is forbidden in centrosymmetric and isotropic bulk media for reasons of symmetry, it can be generated by a centrosymmetric structure, e.g., a sphere, provided that the object is centrosymmetric over roughly the length scale of the optical coherence, which is a function of the particle size, the wavelength of the incident light, and the refractive indexes at ω and 2ω (25–30). As a second-order nonlinear optical technique SHG has symmetry restrictions such that coherent SHG is not generated by the randomly oriented molecules in the bulk liquid, but can be generated coherently by the much smaller population of oriented interfacial species bound to a particle or planar surfaces. As a consequence the SHG signal from the interface is not overwhelmed by SHG from the much larger populations in the bulk media (25–28).The total second harmonic electric field, E_2ω, originating from a charged interface in contact with water can be expressed as (31–33)E2ω∝∑iχc,i(2)EωEω+∑jχinc,j(2)EωEω+χH2O(3)EωEωΦ,[1]where χc,i(2) represents the second-order susceptibility of the species i present at the interface; χinc,j(2) represents the incoherent contribution of the second-order susceptibility, arising from density and orientational fluctuations of the species j present in solution, often referred to as hyper-Rayleigh scattering; χH2O(3) is the third-order susceptibility originating chiefly from the polarization of the bulk water molecules polarized by the charged interface; Φ is the potential at the interface that is created by the surface charge; and E_ω is the electric field of the incident light at the fundamental frequency ω. The second-order susceptibility, χc,i(2), can be written as the product of the number of molecules, N, at the surface and the orientational ensemble average of the hyperpolarizability α_i⁽²⁾ of surface species i, yielding χc,i(2)=N〈αi(2)〉 (18). The bracket ?? indicates an orientational average over the interfacial molecules. The third term in Eq. 1 depicts a third-order process by which a second harmonic field is generated by a charged interface. This term is the focus of our work. The SHG signal is dependent on the surface potential created by the electrostatic field of the surface charges, often called the χ⁽³⁾ contribution to the SHG signal. The χ⁽³⁾ method has been used to extract the surface charge density of charged planar surfaces and microparticle surfaces, e.g., liposomes, polymer beads, and oil droplets in water (21, 25, 34–39).In this work, the χ⁽³⁾ SHG method is used to explore a biomedically relevant process. The binding of the highly cationic HIV-1 TAT peptide to liposome membranes changes the surface potential, thereby enabling the use of the χ⁽³⁾ method to study the binding process in a label-free manner. Two kinds of liposomes, neutral 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) and anionic 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-1′-rac-glycerol (POPG), were investigated. The chemical structures of TAT, POPC, and POPG lipids are shown in Scheme 1.Open in a separate windowScheme 1.Chemical structures of HIV-1 TAT (47–57) peptide and the POPC and POPG lipids.     

Nature of dynamic gradients,glass formation,and collective effects in ultrathin freestanding films

Molecular, polymeric, colloidal, and other classes of liquids can exhibit very large, spatially heterogeneous alterations of their dynamics and glass transition temperature when confined to nanoscale domains. Considerable progress has been made in understanding the related problem of near-interface relaxation and diffusion in thick films. However, the origin of “nanoconfinement effects” on the glassy dynamics of thin films, where gradients from different interfaces interact and genuine collective finite size effects may emerge, remains a longstanding open question. Here, we combine molecular dynamics simulations, probing 5 decades of relaxation, and the Elastically Cooperative Nonlinear Langevin Equation (ECNLE) theory, addressing 14 decades in timescale, to establish a microscopic and mechanistic understanding of the key features of altered dynamics in freestanding films spanning the full range from ultrathin to thick films. Simulations and theory are in qualitative and near-quantitative agreement without use of any adjustable parameters. For films of intermediate thickness, the dynamical behavior is well predicted to leading order using a simple linear superposition of thick-film exponential barrier gradients, including a remarkable suppression and flattening of various dynamical gradients in thin films. However, in sufficiently thin films the superposition approximation breaks down due to the emergence of genuine finite size confinement effects. ECNLE theory extended to treat thin films captures the phenomenology found in simulation, without invocation of any critical-like phenomena, on the basis of interface-nucleated gradients of local caging constraints, combined with interfacial and finite size-induced alterations of the collective elastic component of the structural relaxation process.

Spatially heterogeneous dynamics in glass-forming liquids confined to nanoscale domains (1–7) play a major role in determining the properties of molecular, polymeric, colloidal, and other glass-forming materials (8), including thin films of polymers (9, 10) and small molecules (11–15), small-molecule liquids in porous media (2, 4, 16, 17), semicrystalline polymers (18, 19), polymer nanocomposites (20–22), ionomers (23–25), self-assembled block and layered (26–33) copolymers, and vapor-deposited ultrastable molecular glasses (34–36). Intense interest in this problem over the last 30 y has also been motivated by the expectation that its understanding could reveal key insights concerning the mechanism of the bulk glass transition.Considerable progress has been made for near-interface altered dynamics in thick films, as recently critically reviewed (1). Large amplitude gradients of the structural relaxation time, τ(z,T), converge to the bulk value, τ_bulk(T), in an intriguing double-exponential manner with distance, z, from a solid or vapor interface (13, 37–42). This implies that the corresponding effective activation barrier, F_total(z,T,H) (where H is film thickness), varies exponentially with z, as does the glass transition temperature, T_g (37). Thus the fractional reduction in activation barrier, ε(z,H), obeys the equation ε(z,H)≡1−Ftotal(z,T,H)/Ftotal,bulk(T)=ε0⁡exp(−z/ξF), where F_total,bulk(T) is the bulk temperature-dependent barrier and ξ_F a length scale of modest magnitude. Although the gradient of reduction in absolute activation barriers becomes stronger with cooling, the amplitude of the fractional reduction of the barrier gradient, quantified by ε₀, and the range ξ_F of this gradient, exhibit a weak or absent temperature dependence at the lowest temperatures accessed by simulations (typically with the strength of temperature dependence of ξ_F decreasing rather than increasing on cooling), which extend to relaxation timescales of order 10⁵ ps. This finding raises questions regarding the relevance of critical-phenomena–like ideas for nanoconfinement effects (1). Partially due to this temperature invariance, coarse-grained and all-atom simulations (1, 37, 42, 43) have found a striking empirical fractional power law decoupling relation between τ(z,T) and τ_bulk(T):τ(T,z)τbulk(T)∝(τbulk(T))−ε(z).[1]Recent theoretical analysis suggests (44) that this behavior is consistent with a number of experimental data sets as well (45, 46). Eq. 1 also corresponds to a remarkable factorization of the temperature and spatial location dependences of the barrier:Ftotal(z,T)=[1−ε(z)]Ftotal,bulk(T).[2]This finding indicates that the activation barrier for near-interface relaxation can be factored into two contributions: a z-dependent, but T-independent, “decoupling exponent,” ε(z), and a temperature-dependent, but position-insensitive, bulk activation barrier, F_total,bulk(T). Eq. 2 further emphasizes that ε(z) is equivalent to an effective fractional barrier reduction factor (for a vapor interface), 1−Ftotal(z,T,H)/Ftotal,bulk(T), that can be extracted from relaxation data.In contrast, the origin of “nanoconfinement effects” in thin films, and how much of the rich thick-film physics survives when dynamic gradients from two interfaces overlap, is not well understood. The distinct theoretical efforts for aspects of the thick-film phenomenology (44, 47–50) mostly assume an additive summation of one-interface effects in thin films, thereby ignoring possibly crucial cooperative and whole film finite size confinement effects. If the latter involve phase-transition–like physics as per recent speculations (14, 51), one can ask the following: do new length scales emerge that might be truncated by finite film size? Alternatively, does ultrathin film phenomenology arise from a combination of two-interface superposition of the thick-film gradient physics and noncritical cooperative effects, perhaps in a property-, temperature-, and/or thickness-dependent manner?Here, we answer these questions and establish a mechanistic understanding of thin-film dynamics for the simplest and most universal case: a symmetric freestanding film with two vapor interfaces. We focus on small molecules (modeled theoretically as spheres) and low to medium molecular weight unentangled polymers, which empirically exhibit quite similar alterations in dynamics under “nanoconfinement.” We do not address anomalous phenomena [e.g., much longer gradient ranges (29), sporadic observation of two distinct glass transition temperatures (52, 53)] that are sometimes reported in experiments with very high molecular weight polymers and which may be associated with poorly understood chain connectivity effects that are distinct from general glass formation physics (54–56).We employ a combination of molecular dynamics simulations with a zero-parameter extension to thin films of the Elastically Cooperative Nonlinear Langevin Equation (ECNLE) theory (57, 58). This theory has previously been shown to predict well both bulk activated relaxation over up to 14 decades (44–46) and the full single-gradient phenomenology in thick films (1). Here, we extend this theory to treat films of finite thickness, accounting for coupled interface and geometric confinement effects. We compare predictions of ECNLE theory to our previously reported (37, 43) and new simulations, which focus on translational dynamics of films comprised of a standard Kremer–Grest-like bead-spring polymer model (see SI Appendix). These simulations cover a wide range of film thicknesses (H, from 4 to over 90 segment diameters σ) and extend to low temperatures where the bulk alpha time is ∼0.1 μs (10⁵ Lennard Jones time units τ_LJ).The generalized ECNLE theory is found to be in agreement with simulation for all levels of nanoconfinement. We emphasize that this theory does not a priori assume any of the empirically established behaviors discovered using simulation (e.g., fractional power law decoupling, double-exponential barrier gradient, gradient flattening) but rather predicts these phenomena based upon interfacial modifications of the two coupled contributions to the underlying activation barrier– local caging constraints and a long-ranged collective elastic field. It is notable that this strong agreement is found despite the fact the dynamical ideas are approximate, and a simple hard sphere fluid model is employed in contrast to the bead-spring polymers employed in simulation. The basic unit of length in simulation (bead size σ) and theory (hard sphere diameter d) are expected to be proportional to within a prefactor of order unity, which we neglect in making comparisons.As an empirical matter, we find from simulation that many features of thin-film behavior can be described to leading order by a linear superposition of the thick-film gradients in activation barrier, that is:ε(z,H)=1−Ftotal(z,T,H)/Ftotal,bulk(T)≈ε0[exp(−z/ξF)+exp(−(H−z)/ξF)],[3]where the intrinsic decay length ξ_F is unaltered from its thick-film value and where ε₀ is a constant that, in the hypothesis of literal gradient additivity, is invariant to temperature and film thickness. We employ this functional form [originally suggested by Binder and coworkers (59)], which is based on a simple superposition of the two single-interface gradients, as a null hypothesis throughout this study: this form is what one expects if no new finite-size physics enters the thin-film problem relative to the thick film.However, we find that the superposition approximation progressively breaks down, and eventually entirely fails, in ultrathin films as a consequence of the emergence of a finite size confinement effect. The ECNLE theory predicts that this failure is not tied to a phase-transition–like mechanism but rather is a consequence of two key coupled physical effects: 1) transfer of surface-induced reduction of local caging constraints into the film, and 2) interfacial truncation and nonadditive modifications of the collective elastic contribution to the activation barrier.     

Nonequilibrium topological spin textures in momentum space

Nonequilibrium quantum dynamics of many-body systems is the frontier of condensed matter physics; recent advances in various time-resolved spectroscopic techniques continue to reveal rich phenomena. Angle-resolved photoemission spectroscopy (ARPES) as one powerful technique can resolve electronic energy, momentum, and spin along the time axis after excitation. However, dynamics of spin textures in momentum space remains mostly unexplored. Here, we demonstrate theoretically that the photoexcited surface state of genuine or magnetically doped topological insulators shows intriguing topological spin textures (i.e., tornado-like patterns) in the spin-resolved ARPES. We systematically reveal its origin as a unique nonequilibrium photoinduced topological winding phenomenon. As all intrinsic and extrinsic topological helicity factors of both material and light are embedded in a robust and delicate manner, the tornado patterns not only allow a remarkable tomography of such important system information, but also enable various unique dichroic topological switchings of the momentum-space spin texture. These results open a direction of nonequilibrium topological spin states in quantum materials.

The recent decade has witnessed significant advances in the detection means of ultrafast light-induced phenomena (1, 2) in terms of time-resolved spectroscopic techniques, including angle-resolved photoemission spectroscopy (ARPES) (3–5), terahertz pump-probe scanning–tunneling microscopy, optical conductivity measurement (6–9), etc. Unprecedented precise access into the inherently time-dependent phenomena is beneficial and important to both the fundamental interest in nonequilibrium physics and the practical connection to ultrafast manipulation of novel quantum degrees of freedom toward application (10–12). To this end, a robust low-dimensional nontrivial system would be a versatile playground for such surface-sensitive pump-probe–type investigations. The protected surface state of the topological insulator fits into this role for its long-enough mean free path and lifetime and also, for excluding the insulating and spin-degenerate bulk influence (13–15). The tunable exchange gap from controlling magnetic doping further allows for demonstrating both massless and massive Dirac physics (16–19).However, nonequilibrium spin dynamics is usually studied in time domain or real space only (20, 21). For the surface state, it has been focused on the equilibrium spin-orbit coupling features (22, 23) and the photo-driven steady-state or highly pumped charge current responses (24–29). The nonequilibrium phenomena of light–matter interaction in this system remain largely buried partially due to the little appreciated spin-channel physics. In fact, such information connects well to the state-of-the-art experimental reach; for example, spin-resolved angle-resolved photoemission spectroscopy (SARPES) has been established in equilibrium and as well, extended to time-dependent measurement well below picosecond resolution (5, 22, 23, 30–34). As an example of the new front of nonequilibrium quantum dynamics of topological matters, we draw attention to this highly informative time-dependent signal in an optical pump-probe experiment upon the surface state.In particular, we simulate the irradiation of a terahertz short laser pulse, which can be either linearly polarized (LP) or circularly polarized (CP) (35), to pump across the exchange gap and then, detect the SARPES signal after a controllable delay time with a probe pulse. Apart from possible resonant transition, virtual excitation at the early stage of time evolution is a purely quantum mechanical effect and can turn the system into a many-particle coherent nonequilibrium state. Surprisingly, the SARPES signal exhibits robust and topological tornado-like spiral structures in the two-dimensional (2D) momentum k space, which can be characterized by topological indices. This happens in both the normal and in-plane spin channels and embeds a delicate relation to three helicity factors determining the pumped system: intrinsic helicity of the surface state χ=±1, sign of the Dirac mass ν=sgn(m), and extrinsic helicity τ=0,±1 for LP and right or left CP lights, respectively. Depending on these, the tornado-like responses can dichotomously change characteristic winding senses and even dichroically switch between topological and trivial as a ℤ2-like topological optical activity.     

Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues ρ(λ) has a sharp maximum near (but not at) λ=0 and decreases monotonically with |λ| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for ρ(λ), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of ρ(λ) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T=0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

The investigation of the potential energy surface (PES) V(r1(t)…rN(t)) of a liquid (made up of N particles with positions r1(t)…rN(t) at a time instant t) and the corresponding instantaneous normal modes (INMs) of the (Hessian) matrix of curvatures has been a focus of liquid and glass science since the appearance of Goldstein’s seminal article (1) on the relation between the PES and the liquid dynamics in the viscous regime above the glass transition (2–27).The PES has been shown to form a rather ragged landscape in configuration space (8, 28, 29) characterized by its stationary points. In a glass these points are minima and are called “inherent structures.” The PES is believed to contain important information on the liquid–glass transformation mechanism. For the latter a complete understanding is still missing (28, 30, 31). The existing molecular theory of the liquid–glass transformation is mode-coupling theory (MCT) (32, 33) and its mean-field Potts spin version (28, 34). MCT predicts a sharp transition at a temperature TMCT>Tg, where T_g is the temperature of structural arrest (glass transition temperature). MCT completely misses the heterogeneous activated relaxation processes (dynamical heterogeneities), which are evidently present around and below T_MCT and which are related to the unstable (negative-λ) part of the INM spectrum (28, 30).Near and above T_MCT, apparently, there occurs a fundamental change in the PES. Numerical studies of model liquids have shown that minima present below T_MCT change into saddles, which then explains the absence of activated processes above T_MCT (2–24). Very recently, it was shown that T_MCT is related to a localization–delocalization transition of the unstable INM modes (25, 26).The INM spectrum is obtained in molecular dynamic simulations by diagonalizing the Hessian matrix of the interaction potential, taken at a certain time instant t:Hijαβ(t)=∂2∂xi(α)∂xj(β)V{r1(t)…rN(t)},[1]with ri=(xi(1),xi(2),xi(3)). For large positive values of the eigenvalues λ_j (j=1…N, N being the number of particles in the system) they are related to the square of vibrational frequencies λj=ωj2, and one can consider the Hessian as the counterpart of the dynamical matrix of a solid. In this high-frequency regime one can identify the spectrum with the density of vibrational states (DOS) of the liquid viag(ω)=2 ωρ(λ(ω))=13N∑jδ(ω−ωj) .[2]For small and negative values of λ this identification is not possible. For the unstable part of the spectrum (λ<0) it has become common practice to call the imaginary number λ=iω˜ and define the corresponding DOS asg(ω˜)≡2 ω˜ρ(λ(ω˜)).[3]This function is plotted on the negative ω axis and the stable g(ω), according to [2], on the positive axis. However, the (as we shall see, very interesting) details of the spectrum ρ(λ) near λ = 0 become almost completely hidden by multiplying the spectrum with |ω|. In fact, it has been demonstrated by Sastry et al. (6) and Taraskin and Elliott (7) already 2 decades ago that the INM spectrum of liquids, if plotted as ρ(λ) and not as g(ω) according to [2] and [3], exhibits a characteristic cusp-like maximum at λ = 0. The shape of the spectrum changes strongly with temperature. This is what we find as well in our simulation and what we want to explore further in our present contribution.In the present contribution we demonstrate that the strong change of the spectrum with temperature can be rather well explained in terms of a model, in which the instantaneous harmonic spectrum of the liquid is interpreted to be that of an elastic medium, in which the local shear moduli exhibit strong spatial fluctuations, which includes a large number of negative values. Because these fluctuations are just a snapshot of thermal fluctuations, we assume that they are obeying Gaussian statistics, the variance of which is proportional to the temperature.Evidence for a characteristic change in the liquid configurations in the temperature range above T_g has been obtained in recent simulation studies of the low-frequency vibrational spectrum of glasses, which have been rapidly quenched from a certain parental temperature T*. If T* is decreased from high temperatures toward T_MCT, the low-frequency exponent of the vibrational DOS of the daughter glass (quenched from T* to T = 0) changed from Debye-like g(ω)∝ω2 to g(ω)∝ωs with s > 2. In our numerical investigation of the INM spectra we show a correlation of some details of the low-eigenvalue features of these spectra with the low-frequency properties of the daughter glasses obtained by quenching from the parental temperatures.The stochastic Helmholtz equations (Eq. 7) of an elastic model with spatially fluctuating shear moduli can be readily solved for the averaged Green’s functions by field theoretical techniques (35–37). Via a saddle point approximation with respect to the resulting effective field theory one arrives at a mean-field theory (self-consistent Born approximation [SCBA]) for the self-energy of the averaged Green’s functions. The SCBA predicts a stable spectrum below a threshold value of the variance. Restricted to this stable regime, this theory, called heterogeneous elasticity theory (HET), was rather successful in explaining several low-frequency anomalies in the vibrational spectrum of glasses, including the so-called boson peak, which is an enhancement at finite frequencies over the Debye behavior of the DOS g(ω)∝ω2 (37–41). We now explore the unstable regime of this theory and compare it to the INM spectrum of our simulated soft-sphere liquid.^*We start Results by presenting a comparison of the simulated spectra of the soft-sphere liquid with those obtained by the unstable version of HET-SCBA theory. We then concentrate on some specific features of the INM spectra, namely, the low-eigenvalue slopes and the shift of the spectral maximum from λ = 0. Both features are accounted for by HET-SCBA. In particular, we find an interesting law for the difference between the slopes of the unstable and the stable parts of the spectrum, which behaves as T−2/3, which, again, is accounted for by HET-SCBA.In the end we compare the shift of the spectral maximum with the low-frequency exponent of the DOS of the corresponding daughter glasses and find an empirical correlation. We discuss these results in connection with the saddle to minimum transformation near T_MCT.     

Fragile charge order in the nonsuperconducting ground state of the underdoped high-temperature superconductors

The normal state in the hole underdoped copper oxide superconductors has proven to be a source of mystery for decades. The measurement of a small Fermi surface by quantum oscillations on suppression of superconductivity by high applied magnetic fields, together with complementary spectroscopic measurements in the hole underdoped copper oxide superconductors, point to a nodal electron pocket from charge order in YBa₂Cu₃O6+δ. Here, we report quantum oscillation measurements in the closely related stoichiometric material YBa₂Cu₄O₈, which reveals similar Fermi surface properties to YBa₂Cu₃O6+δ, despite the nonobservation of charge order signatures in the same spectroscopic techniques, such as X-ray diffraction, that revealed signatures of charge order in YBa₂Cu₃O6+δ. Fermi surface reconstruction in YBa₂Cu₄O₈ is suggested to occur from magnetic field enhancement of charge order that is rendered fragile in zero magnetic fields because of its potential unconventional nature and/or its occurrence as a subsidiary to more robust underlying electronic correlations.The normal state of the underdoped copper oxide superconductors has proven to be even more perplexing than the d-wave superconducting state in these materials. At high temperatures in zero magnetic fields, the normal state of the underdoped cuprates comprises an unconventional Fermi surface of truncated “Fermi arcs” in momentum space, which is referred to as the pseudogap state (1). At low temperatures in high magnetic fields, quantum oscillations reveal the nonsuperconducting ground state in various families of underdoped hole-doped copper oxide superconductors to comprise small Fermi surface pockets (2–15). These small Fermi pockets in YBa₂Cu₃O6+δ have been identified as nodal electron pockets (2, 3, 11, 16, 17) originating from Fermi surface reconstruction associated with charge order measured by X-ray diffraction (18–20), ultrasound (21), nuclear magnetic resonance (22), and optical reflectometry (23). However, various aspects of the underlying charge order and the associated Fermi surface reconstruction remain obscure. A central question pertains to the origin of this charge order, curious features of which include a short correlation length in zero magnetic field that grows with increasing magnetic field and decreasing temperature (20). It is crucial to understand the nature of this ground-state order that is related to the high-temperature pseudogap state and delicately balanced with the superconducting ground state. Here, we shed light on the nature of this state by performing extended magnetic field, temperature, and tilt angle-resolved quantum oscillation experiments in the stoichiometric copper oxide superconductor YBa₂Cu₄O₈ (24). This material with double CuO chains has fixed oxygen stoichiometry, making it a model system to study. YBa₂Cu₄O₈ avoids disorder associated with the fractional oxygen stoichiometry in the YBa₂Cu₃O6+δ chains, which has been shown by microwave conductivity to be the dominant source of weak-limit (Born) scattering (25).Intriguingly, we find magnetic field- and angle-dependent signatures of quantum oscillations in YBa₂Cu₄O₈ (13, 14) that are very similar to those in YBa₂Cu₃O6+δ, indicating a similar nodal Fermi surface that arises from Fermi surface reconstruction by charge order with orthogonal wave vectors (16). However, the same X-ray diffraction measurements that show a Bragg peak characteristic of charge order in YBa₂Cu₃O6+δ for a range of hole dopings from 0.084≤p≤0.164 (19, 20, 26) have, thus far, not revealed a Bragg peak in the case of YBa₂Cu₄O₈ (19). We suggest that charge order enhanced by applied magnetic fields reconstructs the Fermi surface in YBa₂Cu₄O₈, whereas charge order is revealed even in zero magnetic fields in YBa₂Cu₃O6+δ because of pinning by increased disorder from oxygen vacancies.     

Dynamic rupture initiation and propagation in a fluid-injection laboratory setup with diagnostics across multiple temporal scales

Fluids are known to trigger a broad range of slip events, from slow, creeping transients to dynamic earthquake ruptures. Yet, the detailed mechanics underlying these processes and the conditions leading to different rupture behaviors are not well understood. Here, we use a laboratory earthquake setup, capable of injecting pressurized fluids, to compare the rupture behavior for different rates of fluid injection, slow (megapascals per hour) versus fast (megapascals per second). We find that for the fast injection rates, dynamic ruptures are triggered at lower pressure levels and over spatial scales much smaller than the quasistatic theoretical estimates of nucleation sizes, suggesting that such fast injection rates constitute dynamic loading. In contrast, the relatively slow injection rates result in gradual nucleation processes, with the fluid spreading along the interface and causing stress changes consistent with gradually accelerating slow slip. The resulting dynamic ruptures propagating over wetted interfaces exhibit dynamic stress drops almost twice as large as those over the dry interfaces. These results suggest the need to take into account the rate of the pore-pressure increase when considering nucleation processes and motivate further investigation on how friction properties depend on the presence of fluids.

The close connection between fluids and faulting has been revealed by a large number of observations, both in tectonic settings and during human activities, such as wastewater disposal associated with oil and gas extraction, geothermal energy production, and CO₂ sequestration (1–11). On and around tectonic faults, fluids also naturally exist and are added at depths due to rock-dehydration reactions (12–15) Fluid-induced slip behavior can range from earthquakes to slow, creeping motion. It has long been thought that creeping and seismogenic fault zones have little to no spatial overlap. Nonetheless, growing evidence suggests that the same fault areas can exhibit both slow and dynamic slip (16–19). The existence of large-scale slow slip in potentially seismogenic areas has been revealed by the presence of transient slow-slip events in subduction zones (16, 18) and proposed by studies investigating the physics of foreshocks (20–22).Numerical and laboratory modeling has shown that such complex fault behavior can result from the interaction of fluid-related effects with the rate-and-state frictional properties (9, 14, 19, 23, 24); other proposed rheological explanations for complexities in fault stability include combinations of brittle and viscous rheology (25) and friction-to-flow transitions (26). The interaction of frictional sliding and fluids results in a number of coupled and competing mechanisms. The fault shear resistance τres is typically described by a friction model that linearly relates it to the effective normal stress σ^n via a friction coefficient f:τres=f σ^n=f (σn−p),[1]where σn is the normal stress acting across the fault and p is the pore pressure. Clearly, increasing pore pressure p would reduce the fault frictional resistance, promoting the insurgence of slip. However, such slip need not be fast enough to radiate seismic waves, as would be characteristic of an earthquake, but can be slow and aseismic. In fact, the critical spatial scale h* for the slipping zone to reach in order to initiate an unstable, dynamic event is inversely proportional to the effective normal stress (27, 28) and hence increases with increasing pore pressure, promoting stable slip. This stabilizing effect of increasing fluid pressure holds for both linear slip-weakening and rate-and-state friction; it occurs because lower effective normal stress results in lower fault weakening during slip for the same friction properties. For example, the general form for two-dimensional (2D) theoretical estimates of this so-called nucleation size, h*, on rate-and-state faults with steady-state, velocity-weakening friction is given by:h*=(μ*DRS)/[F(a,b)(σn−p)],[2]where μ*=μ/(1−ν) for modes I and II, and μ*=μ for mode III (29); DRS is the characteristic slip distance; and F(a, b) is a function of the rate-and-state friction parameters a and b. The function F(a, b) depends on the specific assumptions made to obtain the estimate: FRR(a,b)=4(b−a)/π (ref. 27, equation 40) for a linearized stability analysis of steady sliding, or FRA(a,b)=[π(b−a)2]/2b, with a/b>1/2 for quasistatic crack-like expansion of the nucleation zone (ref. 30, equation 42).Hence, an increase in pore pressure induces a reduction in the effective normal stress, which both promotes slip due to lower frictional resistance and increases the critical length scale h*, potentially resulting in slow, stable fault slip instead of fast, dynamic rupture. Indeed, recent field and laboratory observations suggest that fluid injection triggers slow slip first (4, 9, 11, 31). Numerical modeling based on these effects, either by themselves or with an additional stabilizing effect of shear-layer dilatancy and the associated drop in fluid pressure, have been successful in capturing a number of properties of slow-slip events observed on natural faults and in field fluid-injection experiments (14, 24, 32–34). However, understanding the dependence of the fault response on the specifics of pore-pressure increase remains elusive. Several studies suggest that the nucleation size can depend on the loading rate (35–38), which would imply that the nucleation size should also depend on the rate of friction strength change and hence on the rate of change of the pore fluid pressure. The dependence of the nucleation size on evolving pore fluid pressure has also been theoretically investigated (39). However, the commonly used estimates of the nucleation size (Eq. 2) have been developed for faults under spatially and temporally uniform effective stress, which is clearly not the case for fluid-injection scenarios. In addition, the friction properties themselves may change in the presence of fluids (40–42). The interaction between shear and fluid effects can be further affected by fault-gauge dilation/compaction (40, 43–45) and thermal pressurization of pore fluids (42, 46–48).Recent laboratory investigations have been quite instrumental in uncovering the fundamentals of the fluid-faulting interactions (31, 45, 49–57). Several studies have indicated that fluid-pressurization rate, rather than injection volume, controls slip, slip rate, and stress drop (31, 49, 57). Rapid fluid injection may produce pressure heterogeneities, influencing the onset of slip. The degree of heterogeneity depends on the balance between the hydraulic diffusion rate and the fluid-injection rate, with higher injection rates promoting the transition from drained to locally undrained conditions (31). Fluid pressurization can also interact with friction properties and produce dynamic slip along rate-strengthening faults (50, 51).In this study, we investigate the relation between the rate of pressure increase on the fault and spontaneous rupture nucleation due to fluid injection by laboratory experiments in a setup that builds on and significantly develops the previous generations of laboratory earthquake setup of Rosakis and coworkers (58, 59). The previous versions of the setup have been used to study key features of dynamic ruptures, including sub-Rayleigh to supershear transition (60); rupture directionality and limiting speeds due to bimaterial effects (61); pulse-like versus crack-like behavior (62); opening of thrust faults (63); and friction evolution (64). A recent innovation in the diagnostics, featuring ultrahigh-speed photography in conjunction with digital image correlation (DIC) (65), has enabled the quantification of the full-field behavior of dynamic ruptures (66–68), as well as the characterization of the local evolution of dynamic friction (64, 69). In these prior studies, earthquake ruptures were triggered by the local pressure release due to an electrical discharge. This nucleation procedure produced only dynamic ruptures, due to the nearly instantaneous normal stress reduction.To study fault slip triggered by fluid injection, we have developed a laboratory setup featuring a hydraulic circuit capable of injecting pressurized fluid onto the fault plane of a specimen and a set of experimental diagnostics that enables us to detect both slow and fast fault slip and stress changes. The range of fluid-pressure time histories produced by this setup results in both quasistatic and dynamic rupture nucleation; the diagnostics allows us to capture the nucleation processes, as well as the resulting dynamic rupture propagation. In particular, here, we explore two injection techniques: procedure 1, a gradual, and procedure 2, a sharp fluid-pressure ramp-up. An array of strain gauges, placed on the specimen’s surface along the fault, can capture the strain (translated into stress) time histories over a wide range of temporal scales, spanning from microseconds to tens of minutes. Once dynamic ruptures nucleate, an ultrahigh-speed camera records images of the propagating ruptures, which are turned into maps of full-field displacements, velocities, and stresses by a tailored DIC) analysis. One advantage of using a specimen made of an analog material, such as poly(methyl meth-acrylate) (PMMA) used in this study, is its transparency, which allows us to look at the interface through the bulk and observe fluid diffusion over the interface. Another important advantage of using PMMA is that its much lower shear modulus results in much smaller nucleation sizes h* than those for rocks, allowing the experiments to produce both slow and fast slip in samples of manageable sizes.We start by describing the laboratory setup and the diagnostics monitoring the pressure evolution and the slip behavior. We then present and discuss the different slip responses measured as a result of slow versus fast fluid injection and interpret our measurements by using the rate-and-state friction framework and a pressure-diffusion model.     

Thermodynamic controls on rates of iron oxide reduction by extracellular electron shuttles

Anaerobic microbial respiration in suboxic and anoxic environments often involves particulate ferric iron (oxyhydr-)oxides as terminal electron acceptors. To ensure efficient respiration, a widespread strategy among iron-reducing microorganisms is the use of extracellular electron shuttles (EES) that transfer two electrons from the microbial cell to the iron oxide surface. Yet, a fundamental understanding of how EES–oxide redox thermodynamics affect rates of iron oxide reduction remains elusive. Attempts to rationalize these rates for different EES, solution pH, and iron oxides on the basis of the underlying reaction free energy of the two-electron transfer were unsuccessful. Here, we demonstrate that broadly varying reduction rates determined in this work for different iron oxides and EES at varying solution chemistry as well as previously published data can be reconciled when these rates are instead related to the free energy of the less exergonic (or even endergonic) first of the two electron transfers from the fully, two-electron reduced EES to ferric iron oxide. We show how free energy relationships aid in identifying controls on microbial iron oxide reduction by EES, thereby advancing a more fundamental understanding of anaerobic respiration using iron oxides.

The use of iron oxides as terminal electron acceptors in anaerobic microbial respiration is central to biogeochemical element cycling and pollutant transformations in many suboxic and anoxic environments (1–6). To ensure efficient electron transfer to solid-phase ferric iron, Fe(III), at circumneutral pH, metal-reducing microorganisms from diverse phylae use dissolved extracellular electron shuttle (EES), including quinones (7–9), flavins (10–16), and phenazines (17–19), to transfer two electrons per EES molecule from the respiratory chain proteins in the outer membrane of the microbial cell to the iron oxide (17, 20, 21). The oxidized EES can diffuse back to the cell surface for rereduction, thereby completing the catalytic redox cycle involving the EES.The electron transfer from the reduced EES to Fe(III) is considered a key step in overall microbial Fe(III) respiration. Several lines of evidence suggest that the free energy of the electron transfer reaction, ΔrG, controls Fe(III) reduction rates (15, 17, 22, 23). For instance, microbial Fe(III) oxide reduction by dissolved model quinones as EES was accelerated only for quinones with standard two-electron reduction potentials, EH,1,20, that fell into a relatively narrow range of −180±80 mV at pH 7 (24). Furthermore, in abiotic experiments, Fe(III) reduction rates by EES decreased with increasing ΔrG that resulted from increasing either EH,1,20 of the EES (25, 26), the concentration of Fe(II) in the system (27), or solution pH (25, 26, 28). However, substantial efforts to relate Fe(III) reduction rates for different EES species, iron oxides, and pH to the EH,1,20 averaged over both electrons transferred from the EES to the iron oxides were only partially successful (25, 28). Reaction free energies of complex redox processes involving the transfer of multiple electrons can readily be calculated using differences in the reduction potentials averaged over all electrons transferred, and this approach is well established in biogeochemistry and microbial ecology. For kinetic considerations, however, the use of averaged reduction potentials is inappropriate.Herein, we posit that rates of Fe(III) reduction by EES instead relate to the ΔrG of the less exergonic first one-electron transfer from the two-electron reduced EES species to the iron oxide, following the general notion that reaction rates scale with reaction free energies (29). Our hypothesis is based on the fact that, at circumneutral to acidic pH and for many EES, the reduction potential of the first electron transferred to the fully oxidized EES to form the one-electron reduced intermediate semiquinone species, EH,1, is lower than the reduction potential of the second electron transferred to the semiquinone to form the fully two-electron reduced EES species, EH,2 [i.e., EH,1<EH,2 (30–33)]. This difference in one-electron reduction potentials implies that the two-electron reduced EES (i.e., the hydroquinone) is the weaker one-electron reductant for Fe(III) as compared to the semiquinone species. We therefore expect that rates of iron oxide reduction relate to the ΔrG of the first electron transferred from the hydroquinone to Fe(III). The ΔrG of this first electron transfer may even be endergonic provided that the two-electron transfer is exergonic.We verified our hypothesis in abiotic model systems by demonstrating that reduction rates of two geochemically important crystalline iron oxides, goethite and hematite, by two-electron reduced quinone- and flavin-based EES over a wide pH range, and therefore thermodynamic driving force for Fe(III) reduction, correlate with the ΔrG of the first electron transferred from the fully reduced EES to Fe(III). We further show that rates of goethite and hematite reduction by EES reported in the literature are in excellent agreement with our rate data when comparing rates on the basis of the thermodynamics of the less exergonic first of the two electron transfers.     

Metastability and discrete spectrum of long-range systems

Long-lived quasi-stationary states (QSSs) are a signature characteristic of long-range interacting systems both in the classical and in the quantum realms. Often, they emerge after a sudden quench of the Hamiltonian internal parameters and present a macroscopic lifetime, which increases with the system size. Despite their ubiquity, the fundamental mechanism at their root remains unknown. Here, we show that the spectrum of systems with power-law decaying couplings remains discrete up to the thermodynamic limit. As a consequence, several traditional results on the chaotic nature of the spectrum in many-body quantum systems are not satisfied in the presence of long-range interactions. In particular, the existence of QSSs may be traced back to the finiteness of Poincaré recurrence times. This picture justifies and extends known results on the anomalous magnetization dynamics in the quantum Ising model with power-law decaying couplings. The comparison between the discrete spectrum of long-range systems and more conventional examples of pure point spectra in the disordered case is also discussed.

Equilibration is at the roots of thermodynamics and has been verified under general conditions in a wide range of physical systems. The current scientific literature has focused on several aspects of this problem, starting from quantum quenches and relaxation (1, 2) and arriving at thermalization of integrable and quasi-integrable systems (3–5), typicality as a foundation of quantum statistical mechanics (6–8) and many others.Despite the ubiquity of equilibration, or possibly due to it, the known examples of diverging equilibration times and recurrent behavior have attracted wide attention in modern physics. Dynamical protocols, where the system initially relaxes into a long-lived state and then undergoes actual equilibration on a longer timescale, are commonly referred to as metastable. The observation of various dynamical regimes, separated by distinct timescales, is the most common evidence of metastability and is found in several classical systems and, especially, glasses (9–11). In closed quantum systems ordinary examples of metastability appear in the presence of localized states (2, 12, 13), whose energy eigenvalues are separated from the rest of the spectrum. An analogous spectral structure justifies the observation of metastability also in open quantum systems (14).Diverging equilibration times in the thermodynamic limit are also a notorious characteristic of long-range interacting systems. A physical system is said to be long range when the two-body interaction potential decays as a power law of the distance r between its microscopic components: V(r)∼r−α in the large distance (r→∞) limit. If one focuses on the thermodynamic behavior, two main regimes appear as a function of α. For α>d, where d is the spatial dimension of the system, textbook thermodynamics is well defined and long-range interactions alter the universal scaling only close to critical points (15). We refer to this regime (α>d) as weak long-range interactions.Conversely, for α<d the thermodynamic quantities become nonadditive, leading to apparently paradoxical predictions such as ensemble inequivalence or negative specific heat and susceptibilities (16). In the out-of-equilibrium realm, the most striking property of strong long-range systems is the appearance of quasi-stationary states (QSSs), i.e., metastable configurations whose lifetime scales superlinearly with the system size. QSSs have been mainly studied in classical systems, such as the Hamiltonian mean-field model (17), where an ensemble of plane rotators is subject to a fully connected flat interaction (α=0). There, QSSs are often described in terms of the magnetization dynamics, which, after a sudden quench from an appropriate set of initial conditions, stabilize to a different value with respect to their equilibrium expectation. Then, actual equilibration occurs only after a macroscopic timescale τ∝Nβ with β>0 (16). Apart from this peculiar case, QSSs are characteristic of long-range interactions (18), ranging from gravitational (19) to electromagnetic systems (20).The progress in experiments with atomic molecular and optical systems has largely broadened the interest in long-range physics, due to the possibility of realizing nonlocal interactions via several different means, such as dipolar systems (21–23), cold atoms excited into Rydberg states (24), and trapped ions (25). In the context of metastable dynamics and QSSs a crucial role is played by cold atoms confined in optical resonators, where the photons are stored within the cavity for a sufficiently long time and mediate interactions whose range extends over the entire cavity volume (26). At the semiclassical level, a strict relation between the dynamics of cold atoms into cavity systems and the one of the Hamiltonian mean-field model has been demonstrated (27), designating these devices as optimal candidates for the observation of slow or absent equilibration.Given this broad physical interest, as well as the universal presence of QSSs in long-range interacting systems, it is surprising that the general mechanism at the root of their existence has still to be identified. Indeed, while most results concerning QSSs in classical systems derive from numerical simulations (16), first evidences of their appearance in the quantum realm have been rooted in an analytic approach, which was, however, restricted to a 1/2-spin Hamiltonian with specific boundaries of the quench protocol (28).In this paper, we will prove that the absence of equilibration of long-range quantum systems is directly connected to the persistence of finite Poincaré recurrence times also in the thermodynamic limit. Hence, the physics of macroscopic long-range systems cannot be described by the “traditional” thermodynamic limit procedure. This is in agreement with well-known observations of properties, which are common to thermodynamically large long-range systems and finite local ones, such as the impossibility to fully disregard boundary over bulk phenomena (29, 30), the existence of concave entropy regions (31), or the presence of a macroscopic energy gap between the ground state and the first excited state (32, 33).All of the above features are consequences of the spectral properties of long-range many-body systems, whose spectrum does not become continuous in the thermodynamic limit. Indeed, the eigenvalues of a long-range coupling matrix can be shown to remain discrete even in the infinite components limit, forming a pure point spectrum (34) similar to the one observed in celebrated examples of disordered systems (35). However, at variance with most disordered systems, the spectrum of strong long-range interactions possesses no continuous subspace in the thermodynamic limit, in analogy with the case of the Anderson model at infinite disorder strength (36–38).This paper is organized as follows: In H Theorem and Kinematical Chaos we outline the general picture for equilibration in closed integrable quantum systems. In Spectrum of Long-Range Systems a proof of the spectral discreteness of long-range couplings in the thermodynamic limit is presented in the case of the tight-binding chain. Then, in Vanishing Recurrence Time in the N → ∞ Limit, the connection between this result and the vanishing of the Poincaré recurrence times for critical quenches in a generic quantum system is explored. In QSSs in Spin Systems, the above picture is employed to justify the observation of diverging equilibration times in a long-range Ising model, quenched across its quantum critical point (28). In Lack of Equilibration and Relation with Disorder lack of equilibration is shown in an ensemble of long-range coupled spin waves for a generic (noncritical) quench. Finally, in Discussion the conclusive remarks are reported.