Date of Degree
Alexander B. Khanikaev
Vinod M. Menon
Condensed Matter Physics | Engineering Physics | Optics | Other Physics
Topology, Higher Order topological phase, NonHermitian
Topological phenomena in condensed matter physics have been investigated intensively in the past decades since the discovery of the integer quantum Hall effect (IQHE). For the IQHE, the energy band can be characterized by its topological invariants (Chern number or TKNN invariant), which relates to the quantized Hall conductance directly. Later, this expression was recognized as the first Chern class of a U(1) principal fiber bundle on a torus, where the fibers and torus correspond respectively to the magnetic Bloch waves and the magnetic Brillouin zone. And then, the discoveries of time-reversal symmetric topological insulators in two and three dimensions opened a new field in modern physics, the so-called symmetry-protected topological (SPT) phases. Non-interacting and disorder free fermionic SPT phases, i.e., topological band insulators and topological superconductors, are reasonably well understood based on the Time-Reversal symmetry, Particle-Hole symmetry, Chiral symmetry, and the symmetry of crystals. The studies of topological phases also inspired an exotic scheme to realize quantum computation, i.e., using Majorana fermions to construct q-bits. For example, the Kitaev chain, which is a toy model of topological superconductors, can host two unpaired Majorana fermions at its boundaries. The Majorana fermions can be used to realize a robust quantum computer based on the non-locality and the non-Abelian statistics of the Majorana modes. Recently, the interplay between non-Hermicity and topology yields rich and novel physics, e.g., non-Hermitian skin effect, non-Hermitian exceptional points, non-Hermitian bulk-boundary correspondence breaking. And a new kind of topological phases, known as higher order topological insulators (HOTIs), have drawn attention of researchers. HOTIs hold higher order topological boundary states, including 0D corner states in 2D structures, as well as 0D and 1D states in 3D systems, which are induced by non-trivial bulk polarization.
In recent years, the topological concepts and methods also motivated many exotic results on classical systems, such as electromagnetic waves, acoustic waves, mechanical waves and electric circuits with and without non-Hermitian terms in the effective Hamiltonian of the system (gain-loss, nonreciprocal coupling). Differing from band insulators in condensed matter physics, there is no Fermi level and classical waves can be excited in very broad band of frequencies.
This dissertation is structured as follows. In chapter 0, I present a brief overview of the role of topology in physics, and I will take several toy models as concrete examples to describe the power of the topological concepts in the discovery of novel phases in modern physics.
In chapter 1, we demonstrate that, indeed, for a topological system in higher dimensions one can construct a 1D deterministic aperiodic counterpart which retains its spectrum and topological characteristics. We consider a 4D quantized Hexadecapole higher-order topological insulator (HOTI) which supports topological corner modes. We apply the Lanczos transformation and map it onto an equivalent deterministic aperiodic 1D array (DAA) emulating 4D HOTI in 1D. We observe topological zero-energy 0D states of the DAA -- the direct counterparts of corner states in 4D HOTI and the hallmark of multipole topological phase, which is meaningless in lower dimensions. To explain this paradox, we show that higher-dimension invariant, the multipole polarization, retains its quantization in the DAA, yet changes its meaning by becoming a nonlocal correlator in the 1D system. By introducing nonlocal topological phases of DAAs our discovery opens a new direction in topological physics. It also unveils the possibility to engineer multiple localized resonances via dimensional reduction and their unique features, such as precise spectral properties stemming from their topological nature, opens remarkable opportunities for practical applications, from robust resonators to sensors and aperiodic topological lasers.
In chapter 2, we propose a non-Hermitian topological system protected by the generalized rotational symmetry which invokes rotation in space and Hermitian conjugation. The system, described by the tight-binding model with reciprocal imaginary next-nearest-neighbor hopping, is found to host two pairs of in-gap edge modes in the gapped topological phase, and is characterized by the non-Hermitian (NH) Chern number . The quantization of the non-Hermitian Chern number is shown to be protected by the generalized rotational symmetry (g) of the system. Our finding paves the way towards novel non-Hermitian topological systems characterized by large values of topological invariants and hosting multiple in-gap edge states, which can be used for topologically resilient multiplexing.
In chapter 3, we demonstrate that the exotic physics of relativistic Dirac atoms can be emulated in photonic structures with engineered pseudo-spins and synthetic gauge fields acting on them. We designed and experimentally realized silicon photonic crystals with doubly degenerate (spin-full) Dirac dispersion and a spatially variable mass term aimed at generating trapping potentials. Such analogues of Dirac atoms, their energy levels and corresponding orbitals are then characterized through optical imaging in real and momentum space. We fabricated artificial Dirac atoms operating in the mid-infrared region to realize a hierarchy of photonic modes with distinct radiation profiles directly analogous to various atomic orbitals endowed with some unique characteristics. Our work demonstrates that structured photonic materials – a well-established versatile platform to control light – can serve as a powerful table-top playground to explore relativistic phenomena that have so far evaded experimental observation. In addition to the fundamental interest in the structure of such orbitals, the proposed system offers a route for designing new types of nanophotonic devices, spin-full resonators, and topological light sources compatible with integrated photonics platforms.
Chen, Kai, "Topological Shadow of Higher-Order Topological Phases and Non-Hermitian Phases Protected by Generalized Rotational Symmetry" (2022). CUNY Academic Works.
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