## Topology and broken symmetries:

## from driven quantum matter to active metamaterials

##### DIEP Workshop | 1st - 3rd July 2019 | @ University Library Uithof | Booth Hall room |Utrecht, Netherlands

## Andrea Alù

## Vincenzo Vitelli

## Sebastian Huber

## Alexander Khanikaev

## Stefano Gonella

## Carl Bender

## Aashish Clerk

## Massimo Ruzzene

## Zhong Wang

## Vladimir Gritsev

## Netanel Lindner

## Ewold Verhagen

## Cristiane de Morais

## Ornella Mattei

## Organizers: Corentin Coulais, Romain Fleury, Jasper Van Wezel and Jácome Armas

## Ronny Thomale

## Miguel A. Bandres

## An invitation to bridge the quantum and classical worlds

###### Over the last few years, there has been an explosion of activity in the fields of topological matter, active matter and designer matter. Concepts of symmetries and topology have been found as essential guiding principles to capture the emergence of new phases of matter such as frustration-free disorder, protected edge modes or non-reciprocal transport. All of these remain elusive, however, for far-from-equilibrium systems. This workshop aims at bringing together a broad group of scientists from the quantum and the classical worlds to identify the unifying concepts for the understanding of emergent electronic, mechanical, acoustic and electromagnetic properties of driven/active materials and metamaterials.

## Invited speakers (confirmed)

## Andrea Alù

## (CUNY)

## Aashish Clerk

## (University of Chicago)

## Stefano Gonella

## (University of Minnesota)

## Ewold Verhagen

## (AMOLF)

## Vincenzo Vitelli

## (University of Chicago)

## Vladimir Gritsev

## (University of Amsterdam)

## Massimo Ruzzene

## (Georgia Tech)

## Miguel A. Bandres

## (University of Central Florida)

## Cristiane de Morais Smith

## (Utrecht University)

## Netanel Lindner

## (Technion)

## Ronny Thomale

## (Universität Würzburg)

## Carl Bender

## (Washington University)

## Alexander Khanikaev

## (CUNY)

## Zhong Wang

## (Tsinghua University)

## Ornella Mattei

## (University of Utah)

## Organisers

## Corentin Coulais

## (University of Amsterdam)

## Romain Fleury

## (EPFL)

## Jasper Van Wezel

## (University of Amsterdam)

## Jácome Armas

## (University of Amsterdam)

###### Programme:

###### â€‹

###### Monday, 1st of July

###### 09:30-10:00 Registration & Coffee

###### 10:00-10:15 Opening Remarks

###### 10:15-11:00 Vincenzo Vitelli (abstract)

###### 11:00-11:45 Andrea Alù (abstract)

###### 11:45-12:00 Coffee & Tea

###### 12:00-12:15 Marc Serra-Garcia (abstract)

###### 12:15-12:30 Harry Pirie (abstract)

###### 12:30-14:00 Lunch Break (at the foyer)

###### 14:00-14:45 Cristiane de Morais Smith (abstract)

###### 14:45-15:15 Poster pitches

###### 15:15-16:15 Posters & Coffee

###### 16:15-16:45 Massimo Ruzzene (abstract)

###### 16:45-17:15 Ronny Thomale (abstract)

###### 17:15-17:45 Alexander Khanikaev (abstract)

###### 17:45-18:45 Drinks

###### â€‹

###### Tuesday, 2nd of July

###### 09:30-10:00 Coffee

###### 10:00-10:45 Aashish Clerk (abstract)

###### 10:45-11:30 Netanel Lindner (abstract)

###### 11:30-11:45 Coffee & Tea

###### 11:45-12:00 Radoslaw Kolkowski (abstract)

###### 12:00-12:15 Anton Souslov (abstract)

###### 12:15-12:30 Martin van Hecke (abstract)

###### 12:30-14:00 Posters & Lunch Break (at the foyer)

###### 14:00-14:30 Ewold Verhagen (abstract)

###### 14:30-15:00 Ornella Mattei (abstract)

###### 15:00-15:30 Coffee & Tea

###### 15:30-16:00 Vladimir Gritsev (abstract)

###### 16:00-16:15 Martin Brandenbourger (abstract)

###### 19:00-22:00 Dinner (at Oudaen)

###### â€‹

###### Wednesday, 3rd of July

###### 09:30-10:00 Coffee

###### 10:00-10:45 Carl Bender (abstract)

###### 10:45-11:30 Miguel Bandres (abstract)

###### 11:30-11:45 Coffee & Tea

###### 11:45-12:00 Nicolas Bachelard (abstract)

###### 12:00-12:15 Simon Lieu (abstract)

###### 12:15-14:00 Posters & Lunch Break (at the foyer)

###### 14:00-14:15 Mehul Makwana (abstract)

###### 14:15-14:30 Andres Arrieta (abstract)

###### 14:30-15:00 Coffee & Tea

###### 15:00-15:30 Zhong Wang (abstract)

###### 15:30-16:00 Stefano Gonella (abstract)

###### 16:00-16:15 Closing Remarks

###### Abstract (Andrea Alù): Topological and non-reciprocal metamaterials

###### In this talk, I discuss our recent research activity in electromagnetics, nano-optics, quantum optics, acoustics and mechanics, showing how suitably tailored meta-atoms and suitable arrangements of them open exciting venues to realize non-reciprocal devices for light, radio-waves and sound, largely breaking Lorentz reciprocity and realize isolation without the need of magnetic bias. Our approaches are based on using suitably tailored mechanical motion, spatio-temporal modulation, and large nonlinearities in coupled resonator systems, and have enabled magnetic-free circulators and isolators for sound, microwaves, THz and optical frequencies, non-reciprocal antennas, emitters and absorbers breaking Kirchhoff’s law, self-induced isolation for high-intensities triggered by nonlinearities, broadband delays overcoming the delay-bandwidth limit, and a new generation of topological insulators for light, sound, and static systems in mechanics. In the talk, I will also discuss the impact of these concepts from basic science to practical technology from classical light to quantum optics and computing.

###### Abstract (Alexander Khanikaev): Higher-Order Topological Photonic States Induced by Long Range Interactions

The discovery of topological phases has led to a paradigm shift in understanding of states of condensed matter and facilitated breakthrough in designing novel topological states within engineered photonics and acoustic materials. While originally limited by the dimensionality of the protected states that is one dimension lower than the host TI material, the recent discovery of higher-order topological insulators (HOTIs) provides the potential to overcome this dimensionality limitations by offering topological protection over an extended range of dimensionalities. Here we demonstrate 2D photonic HOTI with topological states of two dimensions lower than that of the host system. We consider a photonic metacrystal of distorted Kagome lattice geometry which exhibits topological bulk polarization, leading to the emergence of 1D topological edge states and of higher order 0D states confined to corners of the structure. Interestingly, in addition to the corner states known to exist in systems described within tight binding model with nearest neighbor approximation, we discover a new class of corner states unique to systems with long-range interactions, including photonic systems. The new type of corner states us shown to emerge from topological edge states which localize at the corners due to interactions beyond nearest neighbor, which is evidenced by both analytical and ab-initio eigenvalue studies.

Our finding demonstrates that photonic HOTIs may possess richer physics compared to their condensed matter counterparts, thus offering opportunities for engineering novel designer electromagnetic states with topological robustness.

###### Abstract (Nicolas Bachelard): Collective Mechanisms for the Self-organization of Dynamic Photonic and Phononic Crystals out of Thermodynamic Equilibrium

In photonics the propagation of waves is typically shaped through a spatial modulation of the optical environment. For instance, in devices achieved by repeatedly stacking layers with different properties, like a photonic crystal, the propagation of waves can be forbidden over continuous energy bandwidths, known as band-gaps. Such systems are routinely manufactured either through top-down techniques—such as e-beam lithography—or bottom-up processes—such as chemical self-assembly. However, these structures are intrinsically static and arise in thermodynamic equilibrium. They form inherently rigid devices, thus difficult to reconfigure and very sensitive to inhomogeneity and imperfections.

The observation of Nature teaches that complex materials and systems can be obtained through alternative mechanisms. Flocks of birds or schools of fish are examples of systems of high complexity that spontaneously self-organize from “simple” elements. Through constant dissipation of energy and entropy production, individual elements are continuously attracted toward a collective steady state that resides far from thermodynamic equilibrium. This attraction provides the ensemble with the ability to adapt to environmental perturbations (e.g. to avoid predators), and to collectively create coherent motions despite important heterogeneities in its populations.

In this talk, we reproduce collective mechanisms to force the self-organization of wave devices far from thermodynamic equilibrium. We explain how an ensemble of dissimilar mobile particles can be driven to spontaneously form a coherent band-gap structure, which is able to self-heal or continuously adapt to its environment. Then, we demonstrate that similar mechanisms can be enforced to drive the emergence of coherent responses in solid-state platforms that are bestowed with resilience properties and complex nonlinear memory effects.

###### Abstract (Ronny Thomale): Topolectrical circuits

First developed by Alessandro Volta and Felix Savary in the early 19th century, circuits consisting of resistor, inductor and capacitor (RLC) components are now omnipresent in modern technology. The behavior of an RLC circuit is governed by its circuit Laplacian, which is analogous to the Hamiltonian describing the energetics of a physical system. We show that “topolectrical” boundary resonances (TBRs) appear in the impedance read-out of a circuit whenever its Laplacian bandstructure resembles that of topological semimetals - materials with extensive degenerate edge modes known as Fermi arcs that also harbor enigmatic transport properties. Such TBRs not only provide unambiguous and highly robust signals for the presence of a topological phase, but also promise diverse applicability within high density electronic mode processing. Due to the versatility of electronic circuits, our topological semimetal construction can be generalized to topolectrical phases with any desired lattice symmetry, spatial dimension, and even quasiperiodicity. Topolectrical circuits establish a bridge between electrical engineering and topological states of matter, where the accessibility, scalability, and operability of electronics promises to synergize with the intricate boundary properties of topological phases.

###### Abstract (Vincenzo Vitelli): Topology and activity in mechanical metamaterials

TBA

###### Abstract (Stefano Gonella): Edge Modes and Asymmetric Wave Transport in Topological Lattices

In this work, we investigate the phononic and topological characteristics of Maxwell lattices. While this class of metamaterials has been the object of extensive theoretical investigation, the classical treatment has been limited to ideal configurations and confined to the static limit. Here, we explicitly explore the dynamics of structural lattices in which the ideal hinges that appear in the theoretical models are replaced by ligaments capable of supporting bending deformation. This is the scenario practically observed in structures realized using cutting or printing fabrication techniques. This change in configuration results in a systematic migration of all the response features, including the availability of edge modes, to finite frequencies. Through a laser-enabled experimental characterization, we reveal how the zero-energy floppy edge modes predicted for topological configurations morph into finite-frequency wave modes that localize at the edges, thus coexisting and overlapping with the low-frequency bulk phonon modes. Finally, the experiments reveal that the polarization endows topological lattices with pronounced asymmetric wave transport capabilities available over broad low-frequency ranges, which can be exposed through transmissibility analysis.

###### Abstract (Ornella Mattei): Field patterns: a new type of wave in space-time microstructures

Field patterns are a new type of wave propagating in one-dimensional linear media with moduli that vary both in space and time. Specifically, the geometry of these space-time materials is commensurate with the slope of the characteristic lines so that a disturbance does not generate a complicate cascade of subsequent disturbances, but rather concentrates on a periodic space-time pattern, that we call field pattern. Field patterns present spectacularly novel features. One of the most interesting ones is the appearance of a wave generated from an instantaneous source, whose amplitude, unlike a conventional wake, does not tend to zero away from the wave front. Furthermore, very interestingly, the band structure associated with these special space-time geometries is infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions, a basis for which are generalized functions each concentrated on a field pattern.

###### Abstract (Radoslaw Kolkowski): Topology of exceptional points in resonant metasurfaces with gain and loss

Exceptional points (EPs) are square root singularities featured by a variety of non-Hermitian systems. EPs have been observed in the dispersion maps of photonic crystals subjected to radiation damping (Zhen et al., Nature 2015) and are expected to appear frequently at the photonic band degeneracies in active metamaterials, where the

interplay between gain and loss gives rise to transition points between broken and unbroken parity-time symmetry. Recently, topological properties of EPs have been addressed (Zhou et al., Science 2018), which was enabled by identification of a topological invariant for non-Hermitian systems (Shen et al., PRL 2018). We adapt this extended topological band theory in our semi-analytical Green function methods to investigate the topology of EPs in arrays of resonant scatterers with gain and loss. Our results are compared which predictions of a simple tight-binding model. We show that the frequency-momentum map of effective polarizability is capable of probing the MoÌˆbius strip topology of photonic modes around the isolated EPs.

###### Abstract (Anton Souslov): Topological waves and odd viscosity in chiral active fluids

Active liquids are composed of self-driven microbots that endow the liquid with a unique set of mechanical characteristics. I will present two designs for materials exhibiting topological states: one using periodic confinement of an active liquid and another using a bulk fluid without periodic order. In a periodic lattice, geometry of confinement controls the structure of topological waves. Without periodic order, topological edge waves can arise in a rotating fluid as a result of the Coriolis force that breaks Galilean invariance and opens a gap at low frequency. I will explore how the number and spatial profile of topological edge states depends on an anomalous response coefficient called odd (or Hall) viscosity. For large odd viscosity, this transverse response can be measured via the profile shape of topologically robust edge waves. As the sign of odd viscosity changes, a topological phase transition occurs without closing the bulk band gap.

###### Abstract (Martin Brandenbourger): Non-reciprocal robotic metamaterials

Most materials transmit physical signals symmetrically by virtue of reciprocity. Recently, there has been a lot of excitement to break this fundamental symmetry to achieve one- way transport of energy. To date, large levels of non-reciprocity have been realized using broken spatial or temporal symmetries, yet mostly in the vicinity of resonances bandgaps or using nonlinearities, thereby non-reciprocal transmission remains limited to narrow ranges of frequencies or input magnitudes and sensitive to attenuation. Here, we introduce a novel robotic mechanical metamaterial that overcomes these limits. Each building block is a minimal robot that is able to sense the behavior of its neighbor and act consequently. We leverage that platform to inject work into the mechanical degrees of freedom and break reciprocity at the level of the interactions between the unit cells. We show theoretically that first-of-their-kind asymmetric standing waves at all frequencies and unidirectionally amplified propagating waves emerge. We demonstrate experimentally and numerically that this property leads to tunable, giant, broadband and attenuation-free non-reciprocal performances, namely a level of 50dB linear non-reciprocal isolation over 3.5 decades in frequency, as well as one-way amplification of pulses. These findings significantly advance the field of active metamaterials for non-reciprocity and beyond, provide new vistas to harness recent progress in robotics and opens avenues to channel mechanical energy in unprecedented ways.

###### Abstract (Martin van Hecke): Non-Abelian Metamaterials

Under most circumstances, materials respond in a nearly linear fashion, and the sequence of external actuations is relevant. However, non-Abelian behavior, where the

sequence of actuations matters, opens new avenues towards programmable material behavior, but also may be a general feature of frustrated systems. Here we introduce a class of mechanical metamaterials that are non-Abelian and present a general strategy to characterize and exploit this behavior.

###### Abstract (Simon Lieu): Non-Hermitian topological band theory for open quantum systems

Topological band theory was developed to predict and explain robust features in the electronic structure of insulators and superconductors. Are these ideas applicable to

photonic bands or the steady-state of open quantum systems? In this talk, I will discuss the topology of "non-Hermitian Hamiltonians." While non-Hermitian Hamiltonians have been experimentally realized in classical optics with gain/loss, this talk will focus on applications to dissipative atomic systems using a Lindblad approach. Non-Hermitian symmetries of a Lindbladian can be used to constrain topological effects in quantum matter. From a mathematical point of view, non-Hermiticity of the Hamiltonian results in a richer set of topological symmetry classes. I highlight the importance of the "Bernard- LeClair" classes which generalize the Altland-Zirnbauer (AZ) classes in the absence of Hermiticity. These classes represent a new direction in symmetry-protected topological phases which go beyond the AZ and crystalline symmetries.

###### Abstract (Mehul Makwana): Geometrically engineering topological networks

Predictive theory to geometrically engineer materials in continuum systems to have desired symmetry-induced effects is developed here by bridging the gap between quantum and continuum descriptions. We emphasise a predictive approach, the strength of which is demonstrated by the ability to design well-defined broadband edge states and valley-Hall networks. We solely use semi-analytical models; this means we can concentrate cleanly upon issues such as group theory, and its influence upon the effects we see, without numerical distractions. A specific example shown will be that of the first topological three-way energy-splitter that solely exists for square structures. Strategically combining four structured domains creates the first ever three-way topological energy-splitter; remarkably, this is only possible using a square, or rectangular, lattice, and not the graphene-like structures more commonly used in valleytronics. To achieve this effect, the two mirror symmetries, present within all fully-symmetric square structures, are broken; this leads to two nondistinct interfaces upon which valley-Hall states reside. These interfaces are related to each other via the time-reversal operator and it is this subtlety that allows us to ignite the third outgoing lead; this is unlike graphene-like structures that yield two distinct interfaces on which two modes of opposite parity reside. The geometrical construction of our structured medium allows for the three-way splitter to be adiabatically converted into a wave steerer around sharp bends. Due to the tunability of the energies directionality by geometry, our results have far-reaching implications for applications such as beam-splitters, switches and filters across wave physics. Experiments are currently underway in a myriad of regimes. Preliminary experimental results for a structured elastic plate will also be shown.

###### Abstract (Andres Arrieta): Broandband Response of Bistable Lattices from Transition Waves

The last two decades has seen an explosion in the interest on metamaterials. In terms of mechanical metamaterials, unconventional properties for both static and dynamic— such as negative Poisson’s ratio and negative density, etc—have been demonstrated. In terms of dynamics, most efforts have focused on utilizing linear effects such as formation of bandgaps from scattering or local resonance. This fundamental reliance band gaps to restricts the applicability of such metamaterials to finite frequency bands and high frequencies (i.e., in the acoustic band). This problem is related to the dimensional limit constraining resonant and scattering impeding metamaterials for low structural frequencies in view of the inverse relationship between size and natural frequency. We show that strongly nonlinear waves can allow for relaxing the strong connection of unit cell size to unconventional dynamical properties. Concretely, we show that in lattices supporting topological solitary waves or transition waves, their strong and localized nature creates long-range interactions that indicating low frequency unconventional properties can be attained. This opens new possibilities for designing metamaterials exhibiting unconventional properties at low frequencies—structural range.

###### Abstract (Carl Bender): PT symmetry in quantum mechanics and quantum field theory

PT-symmetric quantum theory began with an analysis of the strange-looking non-Hermitian Hamiltonian $H=p^2+x^2(ix)^\epsilon$. This Hamiltonian is PT symmetric and the eigenvalues of this Hamiltonian are discrete, real, and positive} when $\epsilon\geq0$. In this talk we describe the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\epsilon$ in D-dimensional spacetime, where $\phi$ is a pseudoscalar field. We show how to calculate all of the Green's functions as series in powers of $\epsilon$ directly from the Euclidean partition function. We derive exact finite expressions for the vacuum energy density, the renormalized mass, and the connected n-point Green's functions for all n and $0\leq D<2$. For $D\geq2$ the one-point Green's function and the renormalized mass become infinite, but perturbative renormalization can be performed. The beautiful spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.

###### Abstract (Cristiane de Morais Smith): Atom-by-atom engineering of topological states of matter

Feynman’s original idea of using one quantum system that can be controlled and manipulated at will to simulate the behavior of another more complex one has flourished during the last decades in the field of cold atoms. More recently, this concept started to be developed in nanophotonics and in condensed matter. In this talk, I will discuss a few recent experiments, in which 2D electron lattices were engineered on the nanoscale. The first is the Lieb lattice, and the second is a Sierpinski gasket , which has dimension D = 1.58. The realization of fractal lattices opens up the path to electronics in fractional dimensions. Finally, I will show how to realize topological states of matter using the same procedure. We investigate the robustness of the zero modes in a breathing Kagome lattice, which is the first experimental realization of a designed electronic higher-order topological insulator. Then, we investigate the importance of the sample termination in determining the existence of topological edge modes in crystalline topological insulators. We focus on the breathing Kekule lattice, with two different types of termination. In all cases, we observe an excellent agreement between the theoretical predictions and the experimental results. This technique is proving itself to be a very versatile framework for the engineering and control of electronic systems.

###### Abstract (Vladimir Gritsev): Integrable dynamics

I will discuss exactly-solvable models of dynamics for a class of isolated and dissipative Floquet-driven quantum models.

###### Abstract (Harry Pirie): Topological phononic logic

Topological metamaterials have robust properties engineered from their macroscopic arrangement, rather than their microscopic constituency. They can be designed by starting from Dirac metamaterials with either symmetry-enforced or accidental degeneracy. The latter case provides greater flexibility because a large number of tuning parameters can break the degeneracy to induce a topological phase. However, the design of a topological logic element—a switch that can be controlled by the output of a separate switch—remains elusive. Here we numerically demonstrate a topological logic gate for ultrasound by exploiting the large phase space of accidental degeneracies in a honeycomb lattice. We find that a degeneracy can be broken by six physical parameters, and we show how to tune these parameters to create a phononic switch that transitions between a topological waveguide and a trivial insulator by ultrasonic heating. Our design scheme is directly applicable to photonic crystals or nano-structured quantum materials, providing a new direction for developing a field-effect topological transistor.

###### Abstract (Miguel Bandres): Topological Insulator Laser

We propose and experimentally demonstrate a fundamentally new approach in exploiting topological effects in a unique way: the topological insulator laser – a laser whose lasing mode is topologically protected. In contrast to Hermitian condensed matter systems, the design of such a laser is highly non-standard – a laser is an open, non-Hermitian (due to gain/loss) and nonlinear (due to gain saturation) system. We demonstrate experimentally an all-dielectric magnetic-free topological insulator laser, and show that the underlying topological properties lead to a highly efficient laser, robust to defects and disorder, with single mode lasing even at very high gain values. The proposed concept alters the current understanding of the interplay between disorder and lasing, and at the same time opens exciting possibilities in topological physics.

###### Abstract (Ewold Verhagen): Topological photons and phonons with silicon nanophotonics

We demonstrate nonreciprocal and topologically protected transport of both light and sound in nanophotonic systems. We show how nano-optomechanical interactions and suitable symmetry breaking induces one-way propagation on silicon chips mimicking that of electrons in topological insulators. On the one hand, we directly observe photonic topological edge states in silicon photonic crystals at telecom wavelengths. These exploit spatial symmetry breaking to make photons mimick the quantum spin Hall effect, in a system that obeys time-reversal symmetry. Far-field imaging and spectroscopy allows to map propagation and extract the canonical linear edge state dispersion, and reveal that photonic pseudospin is encoded in the far-field polarization. On the other hand, we create optomechanical systems in which time-reversal symmetry is broken through photon-phonon interactions. We demonstrate how the radiation pressure of a drive laser can be used to create synthetic magnetic fields for photonic transport, resulting in nonreciprocal function including isolation and circulation. We show how such principles can be extended to nanomechanical transport, and demonstrate the basic building block of an on-chip acoustic quantum Hall system.

###### Abstract (Marc Serra-Garcia): Advanced mechanical systems with discrete models: From HOTI’s to mechanical computers

TBA

###### Abstract (Aashish Clerk): Non-reciprocal and effective non-Hermitian quantum dynamics

TBA

###### Abstract (Zhong Wang): Non-Hermitian skin effect and generalized bulk-boundary correspondence

Non-Hermitian Hamiltonians can exhibit the unusual behavior that all the eigenstates are localized at the boundary, which is dubbed the non-Hermitian skin effect. In the first part, we show that the non-Hermitian skin effect establishes a generalized bulk-boundary correspondence, in which the bulk topological invariants are defined in a generalized Brillouin zone. These non-Bloch topological invariants faithfully predict the number of topological edge modes. In the second part, we show that the non-Hermitian skin effect has dramatic consequences in the dynamics of open quantum systems governed by the Lindblad master equation. Specifically, we show that the long-time damping behavior of an open quantum system can be drastically affected by the non-Hermitian skin effect.

###### Abstract (Massimo Ruzzene): Modulated Metamaterials: Mode localization and Topological Pumping

We investigate spatial stiffness modulations defined by the sampling of a two-dimensional surface, which provide 1D elastic lattices with topological properties that are usually attributed to 2D crystals. Quasi-periodic lattices are first presented as metamaterials that result from commensurate or non-commensurate spatial modulations. These modulations lead to unique topological properties that translate into mode localization in one-dimensional and two-dimensional systems. These properties can be as interesting and perhaps more useful than the interface modes that are found in periodic structures, as the quasi-periodicity framework provides a consistent methodology that leads to vibration confinement in systems that are not ordered, but are described by deterministic property distributions. Beam structures with quasiperiodic arrangements of grounding springs and resonators are presented as structural components which support a variety of localized modes and that are suitable for the experimental characterization of the dynamic behavior of these configurations. Next, we show that cyclic variation of the phase of the stiffness modulation accumulates a phase in the Bloch eigenmodes, which is characterized by integer valued Chern numbers for the associated bands. The resulting non-trivial gaps are spanned by modes whose edge location is determined by the stiffness phase, which can be modulated to drive an edge-to-edge transition. Based on these principles, a topological pump is implemented in the form of a plate whose stiffness varies periodically in one direction, and is adiabatically modulated along a second direction, along which the edge-to-edge transition occurs. The plate effectively implements the stacking and coupling of a family of modulated 1D lattices along a second spatial dimension, along which the adiabatic variation of the stiffness phase drives the transition from a localized state at one boundary, to a bulk state and, finally, to another localized state at the opposite boundary.

###### Abstract (Netanel Lindner): Local Inverse Problems in Condensed Matter

Recovering an unknown Hamiltonian from a set of local measurements is an important task underlying many applications, including certification of noisy quantum devices and investigation of novel phases of matter. We show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling polynomially with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian, allowing the method to recover a large family of time-dependent Hamiltonians. The method can also be adapted to open quantum systems, for which it reconstructs the Lindblad evolution operator. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.