Observation of D-class topology in an acoustic metamaterial.

Topological materials and metamaterials opened new paradigms to create and manipulate phases of matter with unconventional properties. Topological D-class phases (TDPs) are archetypes of the ten-fold classification of topological phases with particle-hole symmetry. In two dimensions, TDPs support propagating topological edge modes that simulate the elusive Majorana elementary particles. Furthermore, a piercing of π-flux Dirac-solenoids in TDPs stabilizes localized Majorana excitations that can be braided for the purpose of topological quantum computation. Such two-dimensional (2D) TDPs have been a focus in the research frontier, but their experimental realizations are still under debate. Here, with a novel design scheme, we realize 2D TDPs in an acoustic crystal by synthesizing both the particle-hole and fermion-like time reversal symmetries for a wide range of frequencies. The design scheme leverages an enriched unit cell structure with real-valued couplings that emulate the targeted Hamiltonian of TDPs with complex hoppings: A technique that could unlock the realization of all topological classes with passive metamaterials. In our experiments, we realize a pair of TDPs with opposite Chern numbers in two independent sectors that are connected by an intrinsic fermion-like time-reversal symmetry built in the system. We measure the acoustic Majorana-like helical edge modes and visualize their robust topological transport, thus revealing the unprecedented D and DIII class topologies with direct evidence. Our study opens up a new pathway for the experimental realization of two fundamental classes of topological phases and may offer new insights in fundamental physics, materials science, and phononic information processing.

Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations.

Tessellations of the hyperbolic spaces by regular polygons support discrete quantum and classical models with unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our Letter delivers a practical and rigorous solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and construct all but one topological models that deliver the topological gaps predicted by the K theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.

Smart patterning for topological pumping of elastic surface waves.

Topological pumping allows waves to navigate a sample undisturbed by disorders and defects. We demonstrate this phenomenon with elastic surface waves by strategically patterning an elastic surface to create a synthetic dimension. The surface is decorated with arrays of resonating pillars that are connected by spatially slow-varying coupling bridges and support eigenmodes located below the sound cone. We establish a connection between the collective dynamics of the pillars and that of electrons in a magnetic field by developing a tight-binding model and a WKB (Wentzel-Kramers-Brillouin) analysis. This enables us to predict the topological pumping pattern, which we validate through numerical and experimental steering of waves from one edge to the other. Furthermore, we observe the immune nature of the topologically pumped surface waves to disorder and defects. The combination of surface patterning and WKB analysis provides a versatile platform for controlling surface waves and exploring topological matter in higher dimensions.

Revealing topology in metals using experimental protocols inspired by K-theory.

Topological metals are conducting materials with gapless band structures and nontrivial edge-localized resonances. Their discovery has proven elusive because traditional topological classification methods require band gaps to define topological robustness. Inspired by recent theoretical developments that leverage techniques from the field of C∗-algebras to identify topological metals, here, we directly observe topological phenomena in gapless acoustic crystals and realize a general experimental technique to demonstrate their topology. Specifically, we not only observe robust boundary-localized states in a topological acoustic metal, but also re-interpret a composite operator-mathematically derived from the K-theory of the problem-as a new Hamiltonian whose physical implementation allows us to directly observe a topological spectral flow and measure the topological invariants. Our observations and experimental protocols may offer insights for discovering topological behaviour across a wide array of artificial and natural materials that lack bulk band gaps.

Creating synthetic spaces for higher-order topological sound transport.

Modern technological advances allow for the study of systems with additional synthetic dimensions. Higher-order topological insulators in topological states of matters have been pursued in lower physical dimensions by exploiting synthetic dimensions with phase transitions. While synthetic dimensions can be rendered in the photonics and cold atomic gases, little to no work has been succeeded in acoustics because acoustic wave-guides cannot be weakly coupled in a continuous fashion. Here, we formulate the theoretical principles and manufacture acoustic crystals composed of arrays of acoustic cavities strongly coupled through modulated channels to evidence one-dimensional (1D) and two-dimensional (2D) dynamic topological pumpings. In particular, the higher-order topological edge-bulk-edge and corner-bulk-corner transport are physically illustrated in finite-sized acoustic structures. We delineate the generated 2D and four-dimensional (4D) quantum Hall effects by calculating first and second Chern numbers and physically demonstrate robustness against the geometrical imperfections. Synthetic dimensions could provide a powerful way for acoustic topological wave steering and open up a platform to explore any continuous orbit in higher-order topological matter in dimensions four and higher.

Experimental Demonstration of Dynamic Topological Pumping across Incommensurate Bilayered Acoustic Metamaterials.

A Thouless pump can be regarded as a dynamical version of the integer quantum Hall effect. In a finite-size configuration, such a topological pump displays edge modes that emerge dynamically from one bulk band and dive into the opposite bulk band, an effect that can be reproduced with both quantum and classical systems. Here, we report the first unassisted dynamic energy transfer across a metamaterial, via pumping of such topological edge modes. The system is a topological aperiodic acoustic crystal, with a phason that can be fast and periodically driven in adiabatic cycles. When one edge of the metamaterial is excited in a topological forbidden range of frequencies, a microphone placed at the other edge starts to pick up a signal as soon as the pumping process is set in motion. In contrast, the microphone picks no signal when the forbidden range of frequencies is nontopological.

Topological Braiding of Non-Abelian Midgap Defects in Classical Metamaterials.

Nontrivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one- and two-dimensional topological superconductors. Here, we generate such representations with topologically protected domain-wall modes in a classical analog of the Kitaev superconducting chain, with a particle-holelike symmetry and a Z_{2} topological invariant. The midgap modes are found to exhibit distinct fusion channels and rich non-Abelian braiding properties, which are investigated using a T-junction setup. We employ the adiabatic theorem to explicitly calculate the braiding matrices for one and two pairs of these midgap topological defects.

Observation of Topological Edge Modes in a Quasiperiodic Acoustic Waveguide.

Topological boundary and interface modes are generated in an acoustic waveguide by simple quasiperiodic patterning of the walls. The procedure opens many topological gaps in the resonant spectrum and qualitative as well as quantitative assessments of their topological character are supplied. In particular, computations of the bulk invariant for the continuum wave equation are performed. The experimental measurements reproduce the theoretical predictions with high fidelity. In particular, acoustic modes with high Q factors localized in the middle of a breathable waveguide are engineered by a simple patterning of the walls.

Mapping the dispersion of water wave channels.

Large classes of electronic, photonic, and acoustic crystals and quasi-crystals have been predicted to support topological wave-modes. Some of these modes are stabilized by certain symmetries but others occur as pure wave phenomena, hence they can be observed in many other media that support wave propagation. Surface water-waves are mechanical in nature but very different from the elastic waves, hence they can provide a new platform for studying topological wave-modes. Motivated by this perspective, we report theoretical and experimental characterizations of water-wave crystals obtained by periodic patterning of the water surface. In particular, we demonstrate the band structure of the spectra and existence of spectral gaps.

Dynamical Majorana edge modes in a broad class of topological mechanical systems.

Mechanical systems can display topological characteristics similar to that of topological insulators. Here we report a large class of topological mechanical systems related to the BDI symmetry class. These are self-assembled chains of rigid bodies with an inversion centre and no reflection planes. The particle-hole symmetry characteristic to the BDI symmetry class stems from the distinct behaviour of the translational and rotational degrees of freedom under inversion. This and other generic properties led us to the remarkable conclusion that, by adjusting the gyration radius of the bodies, one can always simultaneously open a gap in the phonon spectrum, lock-in all the characteristic symmetries and generate a non-trivial topological invariant. The particle-hole symmetry occurs around a finite frequency, and hence we can witness a dynamical topological Majorana edge mode. Contrasting a floppy mode occurring at zero frequency, a dynamical edge mode can absorb and store mechanical energy, potentially opening new applications of topological mechanics.

Topological criticality in the chiral-symmetric AIII class at strong disorder.

The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.

Quantum plasmonics: optical properties of a nanomatryushka.

Quantum mechanical effects can significantly reduce the plasmon-induced field enhancements around nanoparticles. Here we present a quantum mechanical investigation of the plasmon resonances in a nanomatryushka, which is a concentric core-shell nanoparticle consisting of a solid metallic core encapsulated in a thin metallic shell. We compute the optical response using the time-dependent density functional theory and compare the results with predictions based on the classical electromagnetic theory. We find strong quantum mechanical effects for core-shell spacings below 5 Å, a regime where both the absorption cross section and the local field enhancements differ significantly from the classical predictions. We also show that the workfunction of the metal is a crucial parameter determining the onset and magnitude of quantum effects. For metals with lower workfunctions such as aluminum, the quantum effects are found to be significantly more pronounced than for a noble metal such as gold.

Topological phonon modes in filamentary structures.

This work describes a class of topological phonon modes, that is, mechanical vibrations localized at the edges of special structures that are robust against the deformations of the structures. A class of topological phonons was recently found in two-dimensional structures similar to that of microtubules. The present work introduces another class of topological phonons, this time occurring in quasi-one-dimensional filamentary structures with inversion symmetry. The phenomenon is exemplified using a structure inspired from that of actin microfilaments, present in most live cells. The system discussed here is probably the simplest structure that supports topological phonon modes, a fact that allows detailed analysis in both time and frequency domains. We advance the hypothesis that the topological phonon modes are ubiquitous in the biological world and that living organisms make use of them during various processes.

Entanglement spectrum of a disordered topological Chern insulator.

We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.

Quantum plasmonics: optical properties and tunability of metallic nanorods.

The plasmon resonances in metallic nanorods are investigated using fully quantum mechanical time-dependent density functional theory. The computed optical absorption curves display well-defined longitudinal and transverse plasmon resonances whose energies depend on the aspect ratio of the rods, in excellent agreement with classical electromagnetic modeling. The field enhancements obtained from the quantum mechanical calculations, however, differ significantly from classical predictions for distances shorter than 0.5 nm from the nanoparticle surfaces. These deviations can be understood as arising from the nonlocal screening properties of the conduction electrons at the nanoparticle surface.

Quantum description of the plasmon resonances of a nanoparticle dimer.

Using time-dependent density functional theory, we present a fully quantum mechanical investigation of the plasmon resonances in a nanoparticle dimer as a function of interparticle separation. We show that for dimer separations below 1 nm quantum mechanical effects, such as electron tunneling across the dimer junction and screening, significantly modify the optical response and drastically reduce the electromagnetic field enhancements relative to classical predictions. For larger separations, the dimer plasmons are well described by classical electromagnetic theory.

Topological phonon modes and their role in dynamic instability of microtubules.

Microtubules (MTs) are self-assembled hollow protein tubes playing important functions in live cells. Their building block is a protein called tubulin, which self-assembles in a particulate 2 dimensional lattice. We study the vibrational modes of this lattice and find Dirac points in the phonon spectrum. We discuss a splitting of the Dirac points that leads to phonon bands with nonzero Chern numbers, signaling the existence of topological vibrational modes localized at MTs edges, which we indeed observe after explicit calculations. Since these modes are robust against the large changes occurring at the edges during the dynamic cycle of the MTs, we can build a simple mechanical model to illustrate how they would participate in this phenomenon.

The dielectric response of spherical live cells in suspension: an analytic solution.

We develop a theoretical framework to describe the dielectric response of live cells in suspensions when placed in low external electric fields. The treatment takes into account the presence of the cell's membrane and of the charge movement at the membrane's surfaces. For spherical cells suspended in aqueous solutions, we give an analytic solution for the dielectric function, which is shown to account for the alpha- and beta-plateaus seen in many experimental data. The effect of different physical parameters on the dielectric curves is methodically analyzed.

Tunneling conductance of amine-linked alkyl chains.

The tunneling transport theory developed in ref 9 (Phys. Rev. B 2007, 76, 115102) is applied to molecular devices made of alkyl chains linked to gold electrodes via amine groups. Using the analytic expression of the tunneling conductance derived in our previous work, we identify the key physical quantities that characterize the conductance of these devices. By investigating the transport characteristics of three devices, containing four, six, and eight methyl groups, we extract the dependence of the tunneling conductance on the chain's length, which is an exponential decay law in agreement with recent experimental data.