Exploring Novel Quantum Phases in both Periodic and Fractal Topological Systems 公开

Abstract

The collective electronic behavior in materials, fundamental to condensed matter physics, often arises from the periodic potential landscape created by an ordered atomic lattice, which allows the application of Bloch’s theorem and traditional band theory. These frameworks reveal characteristic energy bands and gaps in periodic systems, with features such as Van Hove singularities (VHS) enhancing interaction-driven quantum phases like superconductivity, magnetism, and charge density waves. Interestingly, higher-order VHS (HOVHS), arising from higher-order saddle points in band dispersion, exhibit an even stronger density of states divergence, amplifying correlation effects and providing fertile ground for novel electronic phases. In this thesis, we focus on the interplay of HOVHS and broken time-reversal symmetry (TRS) in topological systems.

We investigate this phenomenon in the context of two distinct systems. First, we examine the interplay between HOVHS and broken TRS on the surface of 3D topological insulators (TIs). Specifically, we explore the emergence of topological Chern bands on the surface of TIs, which host higher-order Van Hove singularities characterized by a power-law divergence in the density of states. These singularities arise from the interaction between a time-reversal symmetry-breaking Zeeman field, induced by proximity to a ferromagnetic insulator, and a time-reversal invariant moir\'e potential influencing the surface electrons. The singularities are modulated by the relative energy scales of the Zeeman field and moir\'e potential, providing a novel platform for realizing exotic Lifshitz transitions in topological bands. 

We next turn our attention to the kagome lattice, a geometrically frustrated system with the potential for hosting exotic quantum phases. By introducing time-reversal-breaking next-nearest-neighbor (NNN) hopping, we induce non-trivial topological band structures characterized by higher Chern numbers. This intricate interplay between geometry and topology gives rise to a rich landscape of HOVHS in the electronic band structure, controlled by the phase and magnitude of the NNN hopping. This classification of HOVHS in kagome systems

provides a platform to explore unconventional electronic orders induced by electronic correlations. Furthermore, the NNN hopping induces topological bands with higher Chern numbers obeying a sublattice interference, where electronic states in specific bands became maximally localized on particular sublattices at high-symmetry points in the Brillouin zone.

The second focus of this thesis extends beyond conventional band theory by exploring electronic systems with fractal symmetry, thereby challenging traditional frameworks that rely on translational invariance. By employing fractal geometries, we construct potential landscapes that lack periodicity, aiming to understand how self-similarity influences the behavior of Dirac fermions on the surfaces of 3D topological insulators (TIs). We investigate a novel class of states that arise from the coupling of surface Dirac fermions to a time-reversal symmetric fractal potential, which breaks translational symmetry while preserving self-similarity. Using large-scale exact diagonalization, scaling analysis of the inverse participation ratio, and the box-counting method, we identify the emergence of self-similar Dirac fermions with fractal dimension for a surface potential modeled after the Sierpinski carpet fractal. These states, characterized by their fractal dimension, open promising avenues for exploring exotic transport phenomena and the potential for quantum information storage in topological systems with fractal dimensionality.

In summary, this thesis systematically studies novel quantum phases in both periodic and non-periodic topological systems. By investigating systems both within and beyond the constraints of translational invariance, this research highlights how unique electronic behaviors can emerge when established paradigms of band theory are pushed into new regimes.

Table of Contents

Introduction Chern Bands with Higher-Order Van Hove Singularities on Topological Moir\'e Surface States Classification of Higher-Order Van Hove Singularities in Kagome Topological Quantum Fractality on the Surface of 3D Topological Insulators Conclusion and Outlook

About this Dissertation

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School	Laney Graduate School
Department	Physics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Physics, Condensed Matter
关键词	Fractals Kagome systems Moire surface states higher-order van hove singularities Topological Insulators
Committee Chair / Thesis Advisor	Luiz H Santos, Emory University
Committee Members	Stefan Boettcher, Emory University Wladimir Benalcazar, Emory University Ajit Srivastava, Emory University Raphael Ribeiro, Emory University

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