AG Mathematische Physik, Alexander Cerjan: An Operator-Based Approach to Topological Physics:,Band Structures and Bloch Eigenstates not Required
Alexander Cerjan
An Operator-Based Approach to Topological Physics:
Band Structures and Bloch Eigenstates not Required
Abstract:
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, and show recent experimental observations of bulk-boundary correspondence in 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).
This work is part-funded by Sandia National Laboratories (SNL). SNL is managed and operated by NTESS
under DOE NNSA contract DE-NA0003525.
References:
[1] W. Cheng*, A. Cerjan*, S.-Y. Chen, E. Prodan, T. A. Loring, and C. Prodan, Revealing topology in
metals using experimental protocols inspired by K-theory, Nature Communications 14, 3071 (2023).
[2] A. Cerjan and T. A. Loring, An operator-based approach to topological photonics, Nanophotonics
11, 4765 (2022).
[3] S. Wong, T. A. Loring, and A. Cerjan, Probing topology in nonlinear topological materials using
numerical K-theory, arXiv:2307.08374.