Session: 01-02-01: Topological Phononics

Paper Number: 149869

149869 - Hierachical Topological States in the Su-Schrieffer-Heeger Model 

As a special class of mechanical metamaterials and phononic crystals, topological mechanical metamaterials and phononic crystals endowed with anomalous wave manipulation capabilities have attracted significant attention over the past decade. Analogous to topological insulators in quantum physics, where a topological invariant classifies different quantum states of matter, mechanical systems can also derive a topological invariant from the spectral evolution of eigenvectors, or mode shapes, from a unit cell analysis. The topological invariant can then be used to determine the number and types of topologically protected surface, edge, and corner states that confine phonon modes both statically and dynamically, a concept usually referred to as bulk-edge correspondence.

In recent years, there has been a rise in higher-order topological insulators. Conventionally, the band topology of these systems is manifested in a hierarchy of dimensions, requiring the bulk unit cell to be at least two-dimensional (2D). For instance, by gapping a doubly degenerate Dirac point in a 2D lattice, a bulk topological transition allows the existence of two first-order 1D topologically protected edge states within the bulk bandgap. Breaking the crystalline symmetries that introduce pseudo-Kramers degeneracy at the boundaries results in the gapping of these 1D topological edge states, leading to a 1D Dirac bandgap. Crossing two different 1D topologically different edges forms a 0D corner with a localized state within the 1D Dirac bandgap. As a result, such a localized corner state is a second-order topological state due to the topological difference of the first-order topological states. With this conventional method, one can achieve up to third-order topological states in a 3D bulk lattice.

In our presentation, we will discuss the emergence of higher-order topological states in a simple 1D Su-Schrieffer-Heeger model without the need for dimension reduction as seen in previous studies. We will also introduce a new generalized topology characterization method for hierarchical topological states. Additionally, we demonstrated that the number of topological domain-wall states can be predicted by accounting for the difference in winding numbers of the two domains, regardless of their topological hierarchical orders. In other words, even with different topological hierarchical orders about the domain boundary, the number of topologically protected boundary states is only determined by the difference between the topological indices of the two domains. Our study proposes a novel mechanism to achieve multiple (and infinite) ranks of topological orders in lattices as low as 1D. However, the orders of these topological states do not alter the robust bulk-edge correspondence.

This research is supported by the Vermont Space Grant Consortium Graduate Research Fellowship.

Presenting Author: Jihong Ma University of Vermont

Presenting Author Biography: Jihong Ma is an assistant professor at the University of Vermont. She obtained her Ph.D. in Mechanical Engineering from the University of Minnesota-Twin Cities in 2017, and her B.Eng. in Engineering Mechanics from Xi'an Jiaotong University (China) in 2012. Her Ph.D. thesis focused on computational heat transport in nanomaterials. She then worked as a Postdoctoral Associate in the Department of Civil, Environmental, and Geo- Engineering at the University of Minnesota-Twin Cities, studying topological metamaterials from 2017 to 2019, and at Oak Ridge National Laboratory - Center for Nanophase Materials Sciences on soft matter simulations.

Dr. Ma is working on the structure-property relationship of materials at multiple scales (from nano- to macro-) via a combination of theoretical analysis, numerical simulations, and experimental characterizations. Her current active research projects include investigations of self-healing polymers, polymeric membranes for carbon capture, organic electronics, quantum dots, dynamics of phononic crystals, and nanoscale thermal and electrical transport.

Authors:

Joel Pyfrom University of Vermont
Jihong Ma University of Vermont