Crafting Complexity: Characterization and Tuning of Non-Hermitian Topological Phases

Abstract

Quantum mechanics relies on self-adjoint or Hermitian operators, ensuring real eigenenergies and preserving probability via a unitary time evolution [1]. However, this applies strictly to closed systems. In open systems, where energy or particles are exchanged among subsystems, there is an overall energy or probability norm change. One way to model these dissipative systems invokes complex energy eigenvalues and a non-Hermitian (NH) Hamiltonian. On the other hand, topological phases with their unique protected states, have been in the limelight in the past decade [2]. In recent years, the emergence of NH topological systems marks a remarkable fusion of two distinct fields, offering new insights into dissipation-controlled topological phases.

NH topological systems exhibit a unique class of spectral degeneracies called exceptional points (EPs), where both eigenvalues and eigenvectors merge, rendering the Hamiltonian non-diagonalizable [3,4]. Enclosing such EPs reveals quantized invariants, unveiling their underlying topological nature. This exceptional topology offers a range of novel properties, including skin effects and NH semimetals with exotic Fermi surfaces. Remarkable progress on the experimental front has also been evident, giving rise to new avenues including unidirectional transport and topological lasers.

In this thesis, I show that tropical geometry, an emerging field at the intersection of algebraic and polyhedral geometry, offers a novel framework for characterizing NH systems [5]. Through Newton's polygon method and the concept of amoebas, we develop a unified tropical geometric approach, demonstrating its versatility to study different facets of NH systems. We showcase its utility in selecting higher-order EPs, predicting skin effects, and uncovering universal properties in disordered systems.

Next, we propose and show that application of light leads to an intriguing platform for controlling exceptional topology in NH systems [6]. We use a combination of analytical and numerical calculations to illustrate the topological properties and map out the topological phase transitions arising from the application of light. We further extend the concept to enable tuning of van Hove singularities in NH interface systems [7].

Furthermore, to explore the connection between single-particle and many-body states, we investigate many-body phases in the NH Su-Schrieffer-Heeger model, unveiling distinct non-equilibrium phases and their transitions governed by exceptional topology [8]. Through a systematic analysis, we elucidate the rich interplay of non-equilibrium phases, quantum entanglement, and topology in NH finite-sized systems.

Finally, we turn our attention to characterizing topological states in complex multi-band NH systems. We propose a decimation framework, leveraging real space renormalization group techniques. This systematic approach enables a streamlined analysis of various phases, their transitions, and NH topological features, including disorder-induced effects and tunable flat bands [9].

Overall, this thesis offers a powerful framework for understanding and characterizing the intricate properties of NH topological phases, with potential to drive advancements in theoretical and experimental studies of these intriguing systems.

References: [1] P. A. M. Dirac, The principles of quantum mechanics, Oxford University Press (1981). [2] X.-G. Wen, Rev. Mod. Phys. 89, 041004 (2017). [3] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021). [4] A. Banerjee, R. Sarkar, S. Dey and A. Narayan, J. Phys.: Condens. Matter 35, 333001 (2023). [5] A. Banerjee, R. Jaiswal, M. Manjunath, and A. Narayan, PNAS 120 e2302572120 (2023). [6] A. Banerjee and A. Narayan, Phys. Rev. B 102, 205423 (2020). [7] A. Banerjee, D. Chowdhury, and A. Narayan, Phys. Rev. B 108, 235148, (2023). [8] A. Banerjee, S. Hegde, A. Agarwala, and A. Narayan, Phys. Rev. B 105, 205403 (2022). [9] A. Banerjee, A. Bandyopadhyay, R. Sarkar and A. Narayan, Physical Review B 110, 085431 (2024).