An Operator-Based Approach to Topological Physics: Band Structures and Bloch Eigenstates not Required
Alexander Cerjan
Over the past two decades, the study of topological properties in physical systems has generated significant excitement, as such systems can realize robust boundary-localized states that have a wide range of applications. However, the theoretical frameworks that have been previously used to understand these phenomena are inextricably tied to band theory, usually requiring a system’s Bloch eigenstates or a projection onto the occupied subspace. Thus, the many successes of topological band theory also serve to highlight the current fundamental challenges facing the field, such as the difficulties in studying topology in aperiodic systems, non-linear and interacting systems, metallic systems, and quantitatively accounting for finite size effects.
In this talk, I will present an operator-based framework for topological physics that makes use of a system’s real-space description without the need to calculate its band structure or Bloch eigenstates. Instead, this framework is based on the system’s spectral localizer, and provides a set of local markers, protected by local gaps, for every symmetry class in every physical dimension. I will discuss how this operator-based framework can be used to identify topology in non-interacting metals and gapless heterostructures, as well as photonic systems with radiative environments, and show experimental observations of a topological acoustic metal metamaterial. Moreover, I will discuss how this framework can be directly applied to nonlinear systems and realistic photonic systems (i.e., Maxwell’s equations). Finally, I will demonstrate that the spectral localizer framework can be applied to 2D electron gasses, predicting quantized Hall conductance and demonstrating how Hofstadter’s butterfly emerges from a uniform system as a periodic potential is turned on.
This work is part-funded by Sandia National Laboratories (SNL). SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525.
This would be a remote talk on 11am eastern / 9am mountain, July 25 (Thursday)
The details of zoom meeting is as follows:
Meeting ID: 830 8475 5599
Passcode: 218047
ORGANIZER
Li Ge (CSI/GC-CUNY)