Date of Degree
Condensed Matter Physics | Engineering Physics | Optics | Quantum Physics
One of the best tools we have for the edification of physics is the analogy. When we take our classical set of states and dynamical variables in phase space and treat them as vectors and Hermitian operators respectively in Hilbert space through the canonical quantization, we lose out on a lot of the intuition developed with the previous classical physics. With classical physics, through our own experiences and understanding of how systems should behave, we create easy-to-understand analogies: we compare the Bohr model of the atom to the motion of the planets, we compare electrical circuits to the flow of water through pipes of varying cross-sectional radii, and we compare spacetime itself to a rubber sheet. In this dissertation, we will continue this time-honored tradition by mapping condensed matter systems to classical wave systems, i.e. by comparing the behavior of electrons in crystalline solids to the behavior of acoustic or electromagnetic waves in their respective media. We will pay particular attention to what are called topological condensed matter systems.
Topology in condensed matter has a history spanning back a few decades ever since Thouless, Kohmoto, Nightingale, and den Nijs discovered the relationship between the quantized Hall conductivity and the first Chern number in the 1980s, thus establishing a link between physical observable properties and the topology hidden within the band structure. Afterward, a whole host of topological phenomena have been discovered in condensed matter systems, which have been termed topological insulators. The general scheme of a topological insulator is that it behaves as a normal insulator within the bulk but at the boundary there exist conductive states that propagate robustly (i.e. insensitive to disorder or perturbations) or states that exponentially localize in zero-dimensions. Both of these boundary properties are guaranteed by the underlying bulk topology of the Brillouin zone manifold for the infinite lattice structure.
These extraordinary properties generated much interest with physicists who sought to explore and understand these systems, including through the use of analogies. It is well known that we can map classical wave systems onto quantum systems (such as the correspondence between the electromagnetic wave equation in the paraxial limit with the Schrodinger equation), so it became of substantial interest of physicists to derive classical analogues of these novel systems full of rich physics. It was soon discovered that by finely tuning the geometric and material parameters of classical wave lattices (such as in acoustics and photonics), we are able to engineer and introduce the same topology that exists in the topological insulator for those respective systems.
Chapter 1 is an introduction to the theory of topology in band structure, including an introduction to topology in general, the Berry phase, different types of topological invariants, the bulk-boundary correspondence, the modern theory of polarization, and selected types of topological insulators such as the bulk-polarized insulator, the Chern insulator, the insulator, and the Floquet topological insulator. We outline many different types of symmetries of the lattice structure and how they might be used to allow for easier calculation of the topological invariant or how they must be enforced on the bulk Hamiltonian in order to provide topological protection of the conductive or exponentially localized boundary states.
In chapter 2 we take the selected types of topological insulators and discuss both the theory of the mapping onto classical wave systems and then the realization of those systems through experiment. We discuss for both cases of acoustics and photonics.
In chapter 3, whereas the previous examples only had topological states localized exactly one dimension below the bulk (the boundary), we discuss a new class of topological states localized in more than one dimension of a D-dimensional system, referred to as Wannier-type higher-order topological (HOT) states. We will discuss the theory behind both the two-dimensional Kagome lattice Wannier-type HOT insulator and the three-dimensional pyrochlore Wannier-type HOT insulator using the methods learned in chapter 2, plus evidence from first-principles finite element method (FEM) wave simulations. Then, we will design and demonstrate the results from the experiment of the three-dimensional topological acoustic metamaterial supporting third-order (0D) topological corner states along with second-order (1D) edge states and first-order (2D) surface states within the same topological bandgap.
In chapter 4, we extend the modern theory of polarization from chapter 1 and introduce the quantized multipole topological insulator (QMTI), described by higher-order electric multipole moments in the bulk. These QMTIs reveal new types of gapped boundaries, which themselves represent lower-dimensional topological phases and host topologically protected zero-dimensional zero-dimensional corner states. We will discuss the difference between the Wannier-type from chapter 3, then move onto the theory of the QMTI, and then design and demonstrate the results from the experiment of the acoustic analogue of the quantized octupole topological insulator (QOTI) with spectroscopic evidence of a topological hierarchy of states in our metamaterial, observing 3rd order corner states, 2nd order hinge states and 1st order surface states.
In chapter 5, we propose, fabricate and experimentally test a reconfigurable one-dimensional acoustic array which emulates an electron moving through a magnetic field and allows for the emergence of a fractal energy spectrum described by the Hofstadter butterfly. We map experimentally the full Hofstadter butterfly spectrum from spectroscopy of the acoustic system. 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.
In chapter 6, we move away from topology and we demonstrate experimentally the control of sound waves by using two types of engineered acoustic systems, where synthetic pseudo-spin emerges either as a consequence of the evanescent nature of the field or due to the lattice symmetry. In the first type of system, we show that the evanescent sound waves bound to the perforated films possess transverse angular momentum locked to their propagation direction which facilitates their highly directional excitation in such acoustic metasurfaces. As the second example, we demonstrate that the lattice symmetries of acoustic Kagome array also offer transverse synthetic pseudo-spin which is locked to the modes linear momentum, thus enabling control of sound propagation in the bulk of the array. We experimentally confirm show that by spinning the source field we can excite directionally bulk states of both types of systems.
Weiner, Matthew, "Emulating Condensed Matter Systems in Classical Wave Metamaterials" (2020). CUNY Academic Works.