Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals
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.
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TL;DR: In this paper, the authors consider a non-Hermitian extension of the Aubry-Andr\'e-Harper model, and show that the dynamical localization-delocalization transition is discontinuous, not only in the diffusion exponent, but also in the speed $v$ of ballistic transport.
Abstract: The Aubry-Andr\'e-Harper model provides a paradigmatic example of aperiodic order in a one-dimensional lattice displaying a delocalization-localization phase transition at a finite critical value ${V}_{c}$ of the quasiperiodic potential amplitude $V$. In terms of the dynamical behavior of the system, the phase transition is discontinuous when one measures the quantum diffusion exponent $\ensuremath{\delta}$ of wave-packet spreading, with $\ensuremath{\delta}=1$ in the delocalized phase $Vl{V}_{c}$ (ballistic transport), $\ensuremath{\delta}\ensuremath{\simeq}1/2$ at the critical point $V={V}_{c}$ (diffusive transport), and $\ensuremath{\delta}=0$ in the localized phase $Vg{V}_{c}$ (dynamical localization). However, the phase transition turns out to be smooth when one measures, as a dynamical variable, the speed $v(V)$ of excitation transport in the lattice, which is a continuous function of potential amplitude $V$ and vanishes as the localized phase is approached. Here we consider a non-Hermitian extension of the Aubry-Andr\'e-Harper model, in which hopping along the lattice is asymmetric, and show that the dynamical localization-delocalization transition is discontinuous, not only in the diffusion exponent $\ensuremath{\delta}$, but also in the speed $v$ of ballistic transport. This means that even very close to the spectral phase transition point, rather counterintuitively, ballistic transport with a finite speed is allowed in the lattice. Also, we show that the ballistic velocity can increase as $V$ is increased above zero, i.e., surprisingly, disorder in the lattice can result in an enhancement of transport.
71 citations
Cites background from "Observation of Hofstadter butterfly..."
...displaying a long-range periodicity intermediate between ordinary periodic crystals and disordered systems, provide fascinating models to study unusual transport phenomena in a wide variety of classical and quantum systems, ranging from condensed-matter systems to ultracold atoms, photonic and acoustic systems [1–7]....
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TL;DR: In this article, the switching behavior in one dimensional perovskite memristors is exploited to design security primitives for key generation and device authentication, and a prototype of a 1'kb propyl pyridinium lead iodide (PrPyr[PbI3]) weak memristor PUF with a differential write-back strategy is presented.
Abstract: Physical Unclonable Functions (PUFs) address the inherent limitations of conventional hardware security solutions in edge-computing devices. Despite impressive demonstrations with silicon circuits and crossbars of oxide memristors, realizing efficient roots of trust for resource-constrained hardware remains a significant challenge. Hybrid organic electronic materials with a rich reservoir of exotic switching physics offer an attractive, inexpensive alternative to design efficient cryptographic hardware, but have not been investigated till date. Here, we report a breakthrough security primitive exploiting the switching physics of one dimensional halide perovskite memristors as excellent sources of entropy for secure key generation and device authentication. Measurements of a prototypical 1 kb propyl pyridinium lead iodide (PrPyr[PbI3]) weak memristor PUF with a differential write-back strategy reveals near ideal uniformity, uniqueness and reliability without additional area and power overheads. Cycle-to-cycle write variability enables reconfigurability, while in-memory computing empowers a strong recurrent PUF construction to thwart machine learning attacks. Despite the impressive demonstrations with silicon and oxide memristors, realizing efficient roots of trust for resource-constrained hardware remains a challenge. Here, the authors exploit switching behavior in one dimensional perovskite memristors to design security primitives for key generation and device authentication.
65 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: It is shown that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges and remarkably, topological phases that cannot be realized in any crystalline insulators are found.
Abstract: The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding ∼10^{4} T magnetic fields to a trivial band structure. In this Letter, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane-Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a higher order TI phase where corner modes are pumped by flux. We show that a model of twisted bilayer graphene realizes this phase. Our results rely primarily on the magnetic translation group that exists at rational values of the flux. The advent of Moire lattices renders our work relevant experimentally. Due to the enlarged Moire unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.
58 citations
TL;DR: In this article, the authors investigate non-Hermitian elastic lattices characterized by non-local feedback control interactions and show that proportional control interactions produce complex dispersion relations characterized by gain and loss in opposite propagation directions.
Abstract: We investigate non-Hermitian elastic lattices characterized by non-local feedback control interactions. In one-dimensional lattices, we show that the proportional control interactions produce complex dispersion relations characterized by gain and loss in opposite propagation directions. Depending on the non-local nature of the control interactions, the resulting non-reciprocity occurs in multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is also investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional dependent wave amplification. Moreover, the non-Hermitian skin effect manifests as modes localized at the boundaries of finite lattice strips, whose combined effect in two directions leads to the presence of bulk modes localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and open new possibilities for the design of metamaterials with novel functionalities related to selective wave filtering, amplification and localization. The results also suggest that feedback interactions may be a useful strategy to investigate topological phases of non-Hermitian systems.
58 citations
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TL;DR: In this article, the Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential, where the Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap.
Abstract: The Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential $U$. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap. Explicit expressions have been obtained for the Hall conductance for both large and small $\frac{U}{\ensuremath{\hbar}{\ensuremath{\omega}}_{c}}$.
4,811 citations
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: Band structure engineering in a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally aligned hexagonal boron nitride substrate is demonstrated, resulting in an unexpectedly large band gap at charge neutrality.
Abstract: van der Waals heterostructures constitute a new class of artificial materials formed by stacking atomically thin planar crystals. We demonstrated band structure engineering in a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally aligned hexagonal boron nitride substrate. The spatially varying interlayer atomic registry results in both a local breaking of the carbon sublattice symmetry and a long-range moire superlattice potential in the graphene. In our samples, this interplay between short- and long-wavelength effects resulted in a band structure described by isolated superlattice minibands and an unexpectedly large band gap at charge neutrality. This picture is confirmed by our observation of fractional quantum Hall states at ± 5 3 filling and features associated with the Hofstadter butterfly at ultrahigh magnetic fields.
1,454 citations
TL;DR: It is demonstrated that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum.
Abstract: Moire superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic potential modulation on a length scale ideally suited to studying the fractal features of the Hofstadter energy spectrum in large magnetic fields. In 1976 Douglas Hofstadter predicted that electrons in a lattice subjected to electrostatic and magnetic fields would show a characteristic energy spectrum determined by the interplay between two quantizing fields. The expected spectrum would feature a repeating butterfly-shaped motif, known as Hofstadter's butterfly. The experimental realization of the phenomenon has proved difficult because of the problem of producing a sufficiently disorder-free superlattice where the length scales for magnetic and electric field can truly compete with each other. Now that goal has been achieved — twice. Two groups working independently produced superlattices by placing ultraclean graphene (Ponomarenko et al.) or bilayer graphene (Kim et al.) on a hexagonal boron nitride substrate and crystallographically aligning the films at a precise angle to produce moire pattern superstructures. Electronic transport measurements on the moire superlattices provide clear evidence for Hofstadter's spectrum. The demonstrated experimental access to a fractal spectrum offers opportunities for the study of complex chaotic effects in a tunable quantum system. Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum1. Known as Hofstadter’s butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields1,2,3,4,5,6,7,8,9,10, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely11,12,13,14,15,16,17. Here we demonstrate that moire superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.
1,438 citations
TL;DR: Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures.
Abstract: Placing graphene on a boron nitride substrate and accurately aligning their crystallographic axes, to form a moire superlattice, leads to profound changes in the graphene’s electronic spectrum. In 1976 Douglas Hofstadter predicted that electrons in a lattice subjected to electrostatic and magnetic fields would show a characteristic energy spectrum determined by the interplay between two quantizing fields. The expected spectrum would feature a repeating butterfly-shaped motif, known as Hofstadter's butterfly. The experimental realization of the phenomenon has proved difficult because of the problem of producing a sufficiently disorder-free superlattice where the length scales for magnetic and electric field can truly compete with each other. Now that goal has been achieved — twice. Two groups working independently produced superlattices by placing ultraclean graphene (Ponomarenko et al.) or bilayer graphene (Kim et al.) on a hexagonal boron nitride substrate and crystallographically aligning the films at a precise angle to produce moire pattern superstructures. Electronic transport measurements on the moire superlattices provide clear evidence for Hofstadter's spectrum. The demonstrated experimental access to a fractal spectrum offers opportunities for the study of complex chaotic effects in a tunable quantum system. Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties1,2,3,4,5,6,7,8. In previous studies (see, for example, refs 1, 2, 3, 4, 5, 6, 7, 8), it proved difficult to realize superlattices with short periodicities and weak disorder, and most of their observed features could be explained in terms of cyclotron orbits commensurate with the superlattice1,2,3,4. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter’s butterfly9) has been limited to the observation of new low-field oscillations5 and an internal structure within Landau levels6,7,8. Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate’s moire potential10,11,12 acts as a superlattice and leads to profound changes in the graphene’s electronic spectrum. Second-generation Dirac points13,14,15,16,17,18,19,20,21,22 appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. The latter indicates that the effective sign of the charge carriers changes within graphene’s conduction and valence bands. Strong magnetic fields lead to Zak-type cloning23 of the third generation of Dirac points, which are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems7,8,9,22,23,24 and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures25.
1,135 citations