Publisher Correction: Enhanced strength and ductility in a high-entropy alloy via ordered oxygen complexes.

Change history: In this Letter, owing to a production error, all the data points (except the two points for O-2 and N-2, respectively) were missing in Fig. 1b. The figure has been corrected online.

Realization of Gapped and Ungapped Photonic Topological Anderson Insulators.

It is commonly believed that topologically nontrivial one-dimensional systems support edge states rather than bulk states at zero energy. In this work, we find an unanticipated case of topological Anderson insulator (TAI) phase where two bulk modes are degenerate at zero energy, in addition to degenerate edge modes. We term this "ungapped TAI" to distinguish it from the previously known gapped TAIs. Our experimental realization of both gapped and ungapped TAIs relies on coupled photonic resonators, in which the disorder in coupling is judiciously engineered by adjusting the spacing between the resonators. By measuring the local density of states both in the bulk and at the edges, we demonstrate the existence of these two types of TAIs, together forming a TAI plateau in the phase diagram. Our experimental findings are well supported by theoretical analysis. In the ungapped TAI phase, we observe stable coexistence of topological edge states and localized bulk states at zero energy, highlighting the distinction between TAIs and traditional topological insulators.

Enhanced strength and ductility in a high-entropy alloy via ordered oxygen complexes.

Oxygen, one of the most abundant elements on Earth, often forms an undesired interstitial impurity or ceramic phase (such as an oxide particle) in metallic materials. Even when it adds strength, oxygen doping renders metals brittle1-3. Here we show that oxygen can take the form of ordered oxygen complexes, a state in between oxide particles and frequently occurring random interstitials. Unlike traditional interstitial strengthening4,5, such ordered interstitial complexes lead to unprecedented enhancement in both strength and ductility in compositionally complex solid solutions, the so-called high-entropy alloys (HEAs)6-10. The tensile strength is enhanced (by 48.5 ± 1.8 per cent) and ductility is substantially improved (by 95.2 ± 8.1 per cent) when doping a model TiZrHfNb HEA with 2.0 atomic per cent oxygen, thus breaking the long-standing strength-ductility trade-off11. The oxygen complexes are ordered nanoscale regions within the HEA characterized by (O, Zr, Ti)-rich atomic complexes whose formation is promoted by the existence of chemical short-range ordering among some of the substitutional matrix elements in the HEAs. Carbon has been reported to improve strength and ductility simultaneously in face-centred cubic HEAs12, by lowering the stacking fault energy and increasing the lattice friction stress. By contrast, the ordered interstitial complexes described here change the dislocation shear mode from planar slip to wavy slip, and promote double cross-slip and thus dislocation multiplication through the formation of Frank-Read sources (a mechanism explaining the generation of multiple dislocations) during deformation. This ordered interstitial complex-mediated strain-hardening mechanism should be particularly useful in Ti-, Zr- and Hf-containing alloys, in which interstitial elements are highly undesirable owing to their embrittlement effects, and in alloys where tuning the stacking fault energy and exploiting athermal transformations13 do not lead to property enhancement. These results provide insight into the role of interstitial solid solutions and associated ordering strengthening mechanisms in metallic materials.

Exact New Mobility Edges between Critical and Localized States.

The disorder systems host three types of fundamental quantum states, known as the extended, localized, and critical states, of which the critical states remain being much less explored. Here we propose a class of exactly solvable models which host a novel type of exact mobility edges (MEs) separating localized states from robust critical states, and propose experimental realization. Here the robustness refers to the stability against both single-particle perturbation and interactions in the few-body regime. The exactly solvable one-dimensional models are featured by a quasiperiodic mosaic type of both hopping terms and on-site potentials. The analytic results enable us to unambiguously obtain the critical states which otherwise require arduous numerical verification including the careful finite size scalings. The critical states and new MEs are shown to be robust, illustrating a generic mechanism unveiled here that the critical states are protected by zeros of quasiperiodic hopping terms in the thermodynamic limit. Further, we propose a novel experimental scheme to realize the exactly solvable model and the new MEs in an incommensurate Rydberg Raman superarray. This Letter may pave a way to precisely explore the critical states and new ME physics with experimental feasibility.

Tuning Anomalous Floquet Topological Bands with Ultracold Atoms.

The Floquet engineering opens the way to create new topological states without counterparts in static systems. Here, we report the experimental realization and characterization of new anomalous topological states with high-precision Floquet engineering for ultracold atoms trapped in a shaking optical Raman lattice. The Floquet band topology is manipulated by tuning the driving-induced band crossings referred to as band inversion surfaces (BISs), whose configurations fully characterize the topology of the underlying states. We uncover various exotic anomalous topological states by measuring the configurations of BISs that correspond to the bulk Floquet topology. In particular, we identify an unprecedented anomalous Floquet valley-Hall state that possesses anomalous helical-like edge modes protected by valleys and a chiral state with high Chern number.

Berry Curvature and Bulk-Boundary Correspondence from Transport Measurement for Photonic Chern Bands.

Berry curvature is a fundamental element to characterize topological quantum physics, while a full measurement of Berry curvature in momentum space was not reported for topological states. Here we achieve two-dimensional Berry curvature reconstruction in a photonic quantum anomalous Hall system via Hall transport measurement of a momentum-resolved wave packet. Integrating measured Berry curvature over the two-dimensional Brillouin zone, we obtain Chern numbers corresponding to -1 and 0. Further, we identify bulk-boundary correspondence by measuring topology-linked chiral edge states at the boundary. The full topological characterization of photonic Chern bands from Berry curvature, Chern number, and edge transport measurements enables our photonic system to serve as a versatile platform for further in-depth study of novel topological physics.

Ultrastrong steel via minimal lattice misfit and high-density nanoprecipitation.

Next-generation high-performance structural materials are required for lightweight design strategies and advanced energy applications. Maraging steels, combining a martensite matrix with nanoprecipitates, are a class of high-strength materials with the potential for matching these demands. Their outstanding strength originates from semi-coherent precipitates, which unavoidably exhibit a heterogeneous distribution that creates large coherency strains, which in turn may promote crack initiation under load. Here we report a counterintuitive strategy for the design of ultrastrong steel alloys by high-density nanoprecipitation with minimal lattice misfit. We found that these highly dispersed, fully coherent precipitates (that is, the crystal lattice of the precipitates is almost the same as that of the surrounding matrix), showing very low lattice misfit with the matrix and high anti-phase boundary energy, strengthen alloys without sacrificing ductility. Such low lattice misfit (0.03 ± 0.04 per cent) decreases the nucleation barrier for precipitation, thus enabling and stabilizing nanoprecipitates with an extremely high number density (more than 1024 per cubic metre) and small size (about 2.7 ± 0.2 nanometres). The minimized elastic misfit strain around the particles does not contribute much to the dislocation interaction, which is typically needed for strength increase. Instead, our strengthening mechanism exploits the chemical ordering effect that creates backstresses (the forces opposing deformation) when precipitates are cut by dislocations. We create a class of steels, strengthened by Ni(Al,Fe) precipitates, with a strength of up to 2.2 gigapascals and good ductility (about 8.2 per cent). The chemical composition of the precipitates enables a substantial reduction in cost compared to conventional maraging steels owing to the replacement of the essential but high-cost alloying elements cobalt and titanium with inexpensive and lightweight aluminium. Strengthening of this class of steel alloy is based on minimal lattice misfit to achieve maximal precipitate dispersion and high cutting stress (the stress required for dislocations to cut through coherent precipitates and thus produce plastic deformation), and we envisage that this lattice misfit design concept may be applied to many other metallic alloys.

Topological Spin Texture of Chiral Edge States in Photonic Two-Dimensional Quantum Walks.

Topological insulators host topology-linked boundary states, whose spin and charge degrees of freedom could be exploited to design topological devices with enhanced functionality. We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk. Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-spin texture has nontrivial winding. This topological spin winding is protected by a chiral-like symmetry emerging for the low-energy Hamiltonian. Our experiment confirms that two-dimensional anomalous Floquet topological systems exhibit topological spin texture on the boundary, which could inspire novel topology-based spintronic phenomena and devices.

The role of anthropogenic habitats in freshwater mussel conservation.

Anthropogenic freshwater habitats may provide undervalued prospects for long-term conservation as part of species conservation planning. This fundamental, but overlooked, issue requires attention considering the pace that humans have been altering natural freshwater ecosystems and the accelerated levels of biodiversity decline in recent decades. We compiled 709 records of freshwater mussels (Bivalvia, Unionida) inhabiting a broad variety of anthropogenic habitat types (from small ponds to large reservoirs and canals) and reviewed their importance as refuges for this faunal group. Most records came from Europe and North America, with a clear dominance of canals and reservoirs. The dataset covered 228 species, including 34 threatened species on the IUCN Red List. We discuss the conservation importance and provide guidance on how these anthropogenic habitats could be managed to provide optimal conservation value to freshwater mussels. This review also shows that some of these habitats may function as ecological traps owing to conflicting management practices or because they act as a sink for some populations. Therefore, anthropogenic habitats should not be seen as a panacea to resolve conservation problems. More information is necessary to better understand the trade-offs between human use and the conservation of freshwater mussels (and other biota) within anthropogenic habitats, given the low number of quantitative studies and the strong biogeographic knowledge bias that persists.

Many-Body Critical Phase: Extended and Nonthermal.

The transition between ergodic and many-body localization (MBL) phases lies at the heart of understanding quantum thermalization of many-body systems. Here, we predict a many-body critical (MBC) phase with finite-size scaling analysis in the one-dimensional extended Aubry-André-Harper-Hubbard model, which is different from both the ergodic phase and MBL phase, implying that the quantum system hosts three different fundamental phases in the thermodynamic limit. The level statistics in the MBC phase are well characterized by the so-called critical statistics, and the wave functions exhibit deep multifractal behavior only in the critical region. We further study the half-chain entanglement entropy and thermalization properties and show that the former, in the MBC phase, manifest a volume law scaling, while the many-body states violate the eigenstate thermalization hypothesis. The results are confirmed by the state-of-the-art numerical calculations with system size up to L=22. This work unveils a novel many-body phase which is extended but nonthermal.

Unified Theory to Characterize Floquet Topological Phases by Quench Dynamics.

The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory, based on quantum quenches, to characterize generic d-dimensional Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d-1)-dimensional momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.

Realization and Detection of Nonergodic Critical Phases in an Optical Raman Lattice.

The critical phases, being delocalized but nonergodic, are fundamental phases different from both the many-body localization and ergodic extended quantum phases, and have so far not been realized in experiment. Here we propose an incommensurate topological insulating model of AIII symmetry class to realize such critical phases through an optical Raman lattice scheme, which possesses a one-dimensional (1D) spin-orbit coupling and an incommensurate Zeeman potential. We show the existence of both noninteracting and many-body critical phases, which can coexist with the topological phase, and show that the critical-localization transition coincides with the topological phase boundary in noninteracting regime. The dynamical detection of the critical phases is proposed and studied in detail based on the available experimental techniques. Finally, we demonstrate how the proposed critical phases can be achieved within the current ultracold atom experiments. This work paves the way to observe the novel critical phases.

Emergence of the Unconventional Type-II Nambu-Goldstone Modes with Topological Origin in Bose Superfluids.

The Nambu-Goldstone (NG) modes in a nonrelativistic system can be classified into two types from their characteristic features: being of either an odd (type I) or an even (type II) power energy-momentum dispersion. Conventionally, the type-II NG modes may universally arise from spontaneous breaking of noncommutative symmetry pairs. Here, we predict a novel type of quadratically dispersed NG modes that emerges in mixed s and p band Bose superfluids in an optical lattice and, unlike the conventional type-II NG modes, cannot be solely interpreted with the celebrated symmetry-based argument. Instead, we show that the existence of such modes has a profound connection to the topological transition on projective complex order-parameter space. The detection scheme is also proposed. Our Letter reveals a new universal mechanism for emergence of type-II NG modes, which bridges intrinsically the Landau symmetry-breaking and topological theories.

Localization Driven Superradiant Instability.

The prominent Dicke superradiant phase arises from coupling an ensemble of atoms to a cavity optical field when an external optical pumping exceeds a threshold strength. Here we report a prediction of the superradiant instability driven by Anderson localization, realized with a hybrid system of the Dicke and Aubry-André (DAA) model for bosons trapped in a one-dimensional (1D) quasiperiodic optical lattice and coupled to a cavity. Our central finding is that for bosons condensed in a localized phase given by the DAA model, the resonant superradiant scattering is induced, for which the critical optical pumping of the superradiant phase transition approaches zero, giving an instability driven by the Anderson localization. The superradiant phase for the DAA model with or without a mobility edge is investigated, showing that the localization driven superradiant instability is in sharp contrast to the superradiance as widely observed for a Bose-Einstein condensate in extended states, and should be insensitive to the temperature of the system. This study unveils a novel effect of localization on the Dicke superradiance, and is well accessible based on the current experiments.

One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges.

The mobility edges (MEs) in energy that separate extended and localized states are a central concept in understanding the localization physics. In one-dimensional (1D) quasiperiodic systems, while MEs may exist for certain cases, the analytic results that allow for an exact understanding are rare. Here we uncover a class of exactly solvable 1D models with MEs in the spectra, where quasiperiodic on-site potentials are inlaid in the lattice with equally spaced sites. The analytical solutions provide the exact results not only for the MEs, but also for the localization and extended features of all states in the spectra, as derived through computing the Lyapunov exponents from Avila's global theory and also numerically verified by calculating the fractal dimension. We further propose a novel scheme with experimental feasibility to realize our model based on an optical Raman lattice, which paves the way for experimental exploration of the predicted exact ME physics.

Quantum Simulation for Three-Dimensional Chiral Topological Insulator.

Quantum simulation, as a state-of-the-art technique, provides a powerful way to explore topological quantum phases beyond natural limits. Nevertheless, it is usually hard to simulate both the bulk and surface topological physics at the same time to reveal their correspondence. Here we build up a quantum simulator using nitrogen-vacancy center to investigate a three-dimensional (3D) chiral topological insulator, and demonstrate the study of both the bulk and surface topological physics by quantum quenches. First, a dynamical bulk-surface correspondence in momentum space is observed, showing that the bulk topology of the 3D phase uniquely corresponds to the nontrivial quench dynamics emerging on 2D momentum hypersurfaces called band inversion surfaces (BISs). This is the momentum-space counterpart of the bulk-boundary correspondence in real space. Further, the symmetry protection of the 3D chiral phase is uncovered by measuring dynamical spin textures on BISs, which exhibit perfect (broken) topology when the chiral symmetry is preserved (broken). Finally, we measure the topological charges to characterize directly the bulk topology and identify an emergent dynamical topological transition when varying the quenches from deep to shallow regimes. This work demonstrates how a full study of topological phases can be achieved in quantum simulators.

Observing Topological Charges and Dynamical Bulk-Surface Correspondence with Ultracold Atoms.

Quantum dynamics induced in quenching a d-dimensional topological phase across a phase transition may exhibit a nontrivial dynamical topological pattern on the (d-1)D momentum subspace, called band inversion surfaces (BISs), which have a one-to-one correspondence to the bulk topology of the postquench phase. Here we report the experimental observation of such dynamical bulk-surface correspondence through measuring the topological charges in a 2D quantum anomalous Hall model realized in an optical Raman lattice. The system can be quenched with respect to every spin axis by suddenly varying the two-photon detuning or phases of the Raman couplings, in which the topological charges and BISs are measured dynamically by the time-averaged spin textures. We observe that the total charges in the region enclosed by BISs define a dynamical topological invariant, which equals the Chern number of the postquench band and also characterizes the topological pattern of a dynamical field emerging on the BISs, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically.

Highly Controllable and Robust 2D Spin-Orbit Coupling for Quantum Gases.

We report the realization of a robust and highly controllable two-dimensional (2D) spin-orbit (SO) coupling with a topological nontrivial band structure. By applying a retro-reflected 2D optical lattice, phase tunable Raman couplings are formed into the antisymmetric Raman lattice structure, and generate the 2D SO coupling with precise inversion and C_{4} symmetries, leading to considerably enlarged topological regions. The lifetime of the 2D SO coupled Bose-Einstein condensate reaches several seconds, which enables exploring fine-tuning interaction effects. These essential advantages of the present new realization open the door to explore exotic quantum many-body effects and nonequilibrium dynamics with novel topology.

Uncover Topology by Quantum Quench Dynamics.

Topological quantum states are characterized by nonlocal invariants. We present a new dynamical approach for ultracold-atom systems to uncover their band topology, and we provide solid evidence to demonstrate its experimental advantages. After quenching a two-dimensional (2D) Chern band, realized in an ultracold ^{87}Rb gas from a trivial to a topological parameter regime, we observe an emerging ring structure in the spin dynamics during the unitary evolution, which uniquely corresponds to the Chern number for the postquench band. By extracting 2D bulk topology from the 1D ring pattern, our scheme displays simplicity and is insensitive to perturbations. This insensitivity enables a high-precision determination of the full phase diagram for the system's band topology.

Ion Irradiation-Enhanced Raman Scattering on Nanoporous Copper.

Nanoporous copper (NPC) is the potential affordable surface-enhanced Raman scattering (SERS) substrate in practical use, although restricted by a relatively small enhancement factor. In this report, Cu ion irradiation is applied to effectively increase the enhancement factor of NPC. Two levels of surface roughness in NPC after ion irradiation are proposed to account for the improved SERS effect by careful characterization of microstructures. This study provides a new strategy to acquire a higher Raman enhancement factor in NPC, which perhaps can be extended to other SERS substrate systems.