The progress in the field of topological insulators is impetuous, being sustained by a suite of exciting results on three fronts: experiment, theory and numerical simulation. Many of these ...predictions have already been confirmed. This review deals with strongly disordered topological insulators and covers some recent applications of a well-established analytic theory based on the methods of non-commutative geometry and developed for the integer quantum Hall effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chem number, and to discuss the physical properties protected by these invariants. The non-commutative spin-Chem number is defined and the analytic theory, together with the explicit calculation of the topological invariants in the presence of strong disorder, is used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.
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.
The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the ...concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.
Consider a finite triangulation of a surface
M
of genus
g
and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the ...algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if
P
is any of its spectral projections, the Booleanization of the fundamental group
π
1
(
M
)
can be embedded inside the group of invertible elements of the corner algebra
P
CAR
P
. As a consequence,
P
decomposes in
4
g
lower projections. Furthermore, a projective representation of
Z
2
4
g
is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group
(
2
X
,
Δ
)
, where
X
is the set of fermion sites and
Δ
is the symmetric difference of its sub-sets.
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.
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.
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 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.
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.
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