Intrinsically Polar Elastic Metamaterials

In many applications, one needs to combine materials with varying properties to achieve certain functionalities. For example, the inner layer of a helmet should be soft for cushioning while the outer shell should be rigid to provide protection. Over time, these combined materials either separate or wear and tear, risking the exposure of an undesired material property. This work presents a design principle for a material that gains unique properties from its 3D microstructure, consisting of repeating basic building blocks, rather than its material composition. The 3D printed specimens show, at two of its opposing faces along the same axis, different stiffness (i.e., soft on one face and hard on the other). The realized material is protected by design (i.e., topology) against cuts and tears: No matter how material is removed, either layer by layer, or in arbitrary cuts through the repeating building blocks, two opposing faces remain largely different in their mechanical response

Observation of a phononic quadrupole topological insulator

The modern theory of charge polarization in solids is based on a generalization of Berry’s phase. The possibility of the quantization of this phase arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures. Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic or phononic crystals. The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states. However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides in higher dimensions, and thereby open up a new path for the design of metamaterials

Observation of a phononic quadrupole topological insulator

The modern theory of charge polarization in solids is based on a generalization of Berry’s phase. The possibility of the quantization of this phase arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures. Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic or phononic crystals. The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states. However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides in higher dimensions, and thereby open up a new path for the design of metamaterials

Topology in Linear Mechanical Metamaterials

This thesis deals with topological effects in linear mechanical metamaterials governed by Newton's equation of motion. Mechanical metamaterials are artificially designed structures whose behavior arises from a scale much larger than its constituting units. The topological effects taken into account are the ones known from single-particle condensed matter physics. While first introduced in the context of quantum mechanics, they are not bound to it and appear in other contexts as well. As such, the field of topological mechanical metamaterials is the youngest offspring implementing ideas from topological band theory and beyond. This thesis is part of this development and its contributions are fourfold. The first contribution is a systematic approach to import topological effects from single-particle condensed matter physics to mechanical metamaterials. We show how to bring Newton's equation of motion into a form akin the Schrödinger equation. This then allows for a direct import of the desired physics. Besides, through this transformation we combine previously unconnected approaches to carry over topological effects, and set out how further ones transfer. A central feature of topological materials is the presence of robust surface modes that are of interest in sight of applications. However, some types of topological effects can only be implemented if classical time-reversal symmetry is broken, which in turn can be hard to achieve for versatile and affordable mechanical metamaterial. We theoretically and experimentally demonstrate how a passive, time-reversal-invariant topological material can be built, by implementing the topological aspects of the quantum spin Hall effect in a mechanical metamaterial. We experimentally characterize the edge (surface) modes and discuss their topological protection. The quantum spin Hall effect is protected by quantum time-reversal symmetry, which translates into a local symmetry in its mechanical version. By deliberately breaking this symmetry, we show how the topological edge channels can be switched on and off. Furthermore, we experimentally show that it is sufficient to break the symmetry in a very small spatial region of the system to obtain an almost perfect switching behavior. Taking into account additional energy terms renders topological invariants often ill-defined. Nevertheless, they offer a structured approach to engineer peculiar surface physics. We create such a metamaterial with a polar elastic response. When straining the material with a point-like object on a certain surface, it provides a very soft response, whereas when poked on the opposite surface the material is hard. The effect is robust, and in particular not prone to wearing

Free fermions on a line: Asymptotics of the entanglement entropy and entanglement spectrum from full counting statistics

We consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length L, the leading asymptotic behavior of this entropy is known to be logarithmic in L. We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers

Switchable topological phonon channels

Guiding energy deliberately is one of the central elements in engineering and information processing. It is often achieved by designing specific transport channels in a suitable material. Topological metamaterials offer a way to construct stable and efficient channels of unprecedented versatility. However, due to their stability it can be tricky to terminate them or to temporarily shut them off without changing the material properties massively. While a lot of effort was put into realizing mechanical topological metamaterials, almost no works deal with manipulating their edge channels in sight of applications. Here, we take a step in this direction, by taking advantage of local symmetry breaking potentials to build a switchable topological phonon channel.ISSN:1367-263