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Transition from topological to chaos in the nonlinear Su-Schrieffer-Heeger model (2403.03038v3)

Published 5 Mar 2024 in cond-mat.mes-hall, cond-mat.stat-mech, and nlin.CD

Abstract: Recent studies on topological materials are expanding into the nonlinear regime, while the central principle, namely the bulk-edge correspondence, is yet to be elucidated in the strongly nonlinear regime. Here, we reveal that nonlinear topological edge modes can exhibit the transition to spatial chaos by increasing nonlinearity, which can be a universal mechanism of the breakdown of the bulk-edge correspondence. Specifically, we unveil the underlying dynamical system describing the spatial distribution of zero modes and show the emergence of chaos. We also propose the correspondence between the absolute value of the topological invariant and the dimension of the stable manifold under sufficiently weak nonlinearity. Our results provide a general guiding principle to investigate the nonlinear bulk-edge correspondence that can potentially be extended to arbitrary dimensions.

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Summary

  • The paper demonstrates that increased nonlinearity transitions topological edge modes into chaotic zero modes, breaking bulk-edge correspondence.
  • The study links the spatial distribution of zero modes to a discrete dynamical system exhibiting period-doubling bifurcations.
  • The paper reveals a correlation between the topological invariant's absolute value and the stable manifold dimension, suggesting extensions to higher dimensions.

Unveiling the Transition from Topological States to Chaos in Nonlinear Su-Schrieffer-Heeger Models

Introduction

In the continuum of studies on topological insulators and their fascinating properties, a paper explores the nonlinear domain, specifically targeting the Su-Schrieffer-Heeger (SSH) model. The central thesis of this investigation is the transition of nonlinear topological edge modes into spatial chaos by increasing nonlinearity, suggesting an intriguing mechanism for the breakdown of bulk-edge correspondence in nonlinear systems.

Overview of Findings

The exploration reveals several key insights:

  • Transition to Chaos: It is articulated how in the domain of strong nonlinear effects, there exists a transition from topological edge modes to spatially chaotic zero modes. This is portrayed as a potentially universal mechanism leading to the breakdown of bulk-edge correspondence in nonlinear systems.
  • Dynamical System Description: The research deciphers that the spatial distribution of zero modes aligns with a discrete dynamical system. Through analysis of a one-dimensional model, it is shown how the system undergoes period-doubling bifurcation to chaos, correlating this bifurcation point with the collapse of bulk-edge correspondence.
  • Topological Invariant Implications: Under the regimen of sufficiently weak nonlinearity, the paper proposes a correlation between the absolute value of the topological invariant and the dimension of the stable manifold in the dynamical system describing zero modes. This introduces a new perspective on investigating nonlinear bulk-edge correspondence.
  • Extensions and Generalizations: Crucially, these findings are presented with the potential for extension to higher dimensions, laying a groundwork for further elucidation of nonlinear topological insulators.

Implications of the Research

The research has several theoretical and practical implications. Theoretically, it enriches the understanding of nonlinear topological physics by introducing chaos theory into the discussion of bulk-edge correspondence. It suggests that traditional ways of looking at topological phases may need reevaluation under nonlinear conditions. Practically, the insights could influence the design and understanding of materials and systems where nonlinearity plays a significant role, such as photonic topological insulators, quantum simulations, and nonlinear electronic systems.

Prospects for Future Developments

The paper opens numerous avenues for future exploration. A natural extension would be to investigate the occurrence and implications of this transition in higher-dimensional topological insulators. Another intriguing direction is the exploration of similar transitions in systems with different types of nonlinearity or in systems where nonlinearity and non-Hermiticity both play significant roles. Additionally, understanding the practical conditions under which these transitions could be observed and possibly leveraged for technological applications remains a crucial next step.

Conclusion

This research adds a significant layer of depth to the understanding of nonlinear effects in topological insulators, articulating the role of chaos in the breakdown of bulk-edge correspondence. It bridges the gap between traditional topological physics and nonlinear dynamics, offering a fresh perspective on the interplay between topological invariants and the dynamical systems describing the spatial distribution of edge modes. The implications of this paper span both the theoretical and practical realms, potentially guiding the development of new technologies based on nonlinear topological phases.