Topological edge modes by smart patterning
TL;DR: In this paper, identical coupled mechanical resonators whose collective dynamics are fully determined by the patterns in which they are arranged are shown to have topological properties, and the existence of such patterns is proven using $K$ theory and exemplified using an experimental platform based on magnetically coupled spinners.
Abstract: We study identical coupled mechanical resonators whose collective dynamics are fully determined by the patterns in which they are arranged. In this work, we call a system topological if (1) boundary resonant modes fully fill all existing spectral gaps whenever the system is halved, and (2) if the boundary spectrum cannot be removed or gapped by any boundary condition. We demonstrate that such topological characteristics can be induced solely through patterning, in a manner entirely independent of the structure of the resonators and the details of the couplings. The existence of such patterns is proven using $K$ theory and exemplified using an experimental platform based on magnetically coupled spinners. Topological metamaterials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.
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TL;DR: In this article, a reconfigurable one-dimensional (1D) acoustic array is proposed, in which the resonant frequencies of each element can be independently fine-tuned by a piston.
Abstract: The emergence of a fractal energy spectrum is the quintessence of the interplay between two periodic parameters with incommensurate length scales. crystals can emulate such interplay and also exhibit a topological bulk-boundary correspondence, enabled by their nontrivial topology in virtual dimensions. Here we propose, fabricate and experimentally test a reconfigurable one-dimensional (1D) acoustic array, in which the resonant frequencies of each element can be independently fine-tuned by a piston. We map experimentally the full Hofstadter butterfly spectrum by measuring the acoustic density of states distributed over frequency while varying the long-range order of the array. Furthermore, by adiabatically changing the phason of the array, we map topologically protected fractal boundary states, which are shown to be pumped from one edge to the other. This reconfigurable crystal serves as a model for future extensions to electronics, photonics and mechanics, as well as to quasi-crystalline systems in higher dimensions. Hofstadter’s butterfly is a fractal pattern which pictorially represents the behavior of electrons under an applied magnetic field in a 2D lattice as a pair of butterfly wings. Here, the authors recreate this pattern by measuring the acoustic density of states in a fine-tuned one-dimensional acoustic array.
105 citations
TL;DR: In this article, a topological pump for continuous elastic lattices is proposed, where the cyclic modulation of the stiffness defines a family of lattices whose Bloch eigenmodes accumulate a phase quantified by integer valued Chern numbers.
Abstract: Spatial stiffness modulations defined by the sampling of a two-dimensional surface provide one-dimensional elastic lattices with topological properties that are usually attributed to two-dimensional crystals. The cyclic modulation of the stiffness defines a family of lattices whose Bloch eigenmodes accumulate a phase quantified by integer valued Chern numbers. Nontrivial gaps are spanned by edge modes in finite lattices whose location is determined by the phase of the stiffness modulation. These observations drive the implementation of a topological pump in the form of an array of continuous elastic beams coupled through a distributed stiffness. Adiabatic stiffness modulations along the beams' length lead to the transition of localized states from one boundary, to the bulk and, finally, to the opposite boundary. The first demonstration of topological pumping in a continuous elastic system opens new possibilities for its implementation on elastic substrates supporting surface acoustic waves, or to structural components designed to steer waves or isolate vibrations.
92 citations
TL;DR: The procedure opens many topological gaps in the resonant spectrum and qualitative as well as quantitative assessments of their topological character are supplied and computations of the bulk invariant for the continuum wave equation are performed.
Abstract: 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
81 citations
TL;DR: In this paper, a quasiperiodic arrangement of resonators introduces frequency band gaps in addition to the locally resonant gap and topologically nontrivial gaps with associated edge states.
Abstract: In extending the ideas of topological phases of matter to acoustic and mechanical systems, a quasiperiodic arrangement of resonators introduces frequency band gaps in addition to the locally resonant gap. Here numerical evaluation of the spectrum as a function of the quasiperiodic arrangement reveals a structure reminiscent of the famous Hofstadter butterfly. The onset of the locally resonant band gap and topologically nontrivial gaps with associated edge states is demonstrated numerically and experimentally. These structural designs can induce wave localization and attenuation over multiple frequency bands, for applications in $e.g.$ vibration isolation and energy harvesting.
61 citations
TL;DR: In this paper, the existence of edge states in elastic plates with spatial modulations of stiffness was demonstrated, and it was shown that smooth variations of the phase of the modulation profile along one spatial dimension produces a transition of the edge states from one edge to another.
Abstract: The authors demonstrate the existence of edge states in elastic plates with spatial modulations of stiffness. They show that smooth variations of the phase of the modulation profile along one spatial dimension produces a transition of the edge states from one edge to another, thus implementing a ``topological pump.'' The results on a mechanical substrate open new pathways for the efficient transfer of information through topological protection, for the general investigation of topological phases of matter relying on continuous property modulations, and for the pursuit of topology-based waveguiding relying on slow modulations along a second dimension, spatial or temporal.
58 citations
References
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TL;DR: In this paper, an effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing the operator of the Schr\"odinger equation with a matrix method, and the graph of the spectrum over a wide range of "rational" fields is plotted.
Abstract: An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.
2,656 citations
TL;DR: In this paper, the authors systematically studied topological phases of insulators and superconductors in three dimensions and showed that there exist topologically nontrivial (3D) topologically nonsmooth topological insulators in five out of ten symmetry classes introduced in the context of random matrix theory.
Abstract: We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced ${\mathbb{Z}}_{2}$ topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. In these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a two-dimensional surface, they support a number (which may be an arbitrary nonvanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin-rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral $({p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y})$-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to ``weak pairing'') and topologically trivial (analogous to ``strong pairing'') 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.
2,459 citations
TL;DR: In this paper, the authors constructed representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians.
Abstract: It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a or a topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via 'dimensional reduction' by compactifying one or more spatial dimensions (in 'Kaluza–Klein'-like fashion). For -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -topological insulators in the same class, from which they inherit their topological properties. The eightfold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle–hole symmetries) is a reflection of the eightfold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern–Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization (d=1 spatial dimensions) or to the magnetoelectric polarizability (d=3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).
1,648 citations
TL;DR: In this paper, the authors constructed representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians, using these representatives they demonstrate how topologically insulators (superconductors) in different dimensions and different classes can be related via dimensional reduction by compactifying one or more spatial dimensions.
Abstract: It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a Z or a Z_2 topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via dimensional reduction by compactifying one or more spatial dimensions (in Kaluza-Klein-like fashion). For Z-topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The Z_2-topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent Z-topological insulators in the same class, from which they inherit their topological properties. The 8-fold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle-hole symmetries) is a reflection of the 8-fold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). We derive a relation between the topological invariant that characterizes topological insulators/superconductors with chiral symmetry and the Chern-Simons invariant: it relates the invariant to the electric polarization (d=1), or to the magnetoelectric polarizability (d=3). Finally, we discuss topological field theories describing the space time theory of linear responses, and study how the presence of inversion symmetry modifies the classification.
1,259 citations
TL;DR: In this paper, a topological insulator is characterized by a dichotomy between the interior and the edge of a finite system: the bulk has an energy gap, and the edges sustain excitations traversing this gap.
Abstract: A topological insulator, as originally proposed for electrons governed by quantum mechanics, is characterized by a dichotomy between the interior and the edge of a finite system: The bulk has an energy gap, and the edges sustain excitations traversing this gap. However, it has remained an open question whether the same physics can be observed for systems obeying Newton’s equations of motion. We conducted experiments to characterize the collective behavior of mechanical oscillators exhibiting the phenomenology of the quantum spin Hall effect. The phononic edge modes are shown to be helical, and we demonstrate their topological protection via the stability of the edge states against imperfections. Our results may enable the design of topological acoustic metamaterials that can capitalize on the stability of the surface phonons as reliable wave guides.
887 citations