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Revealing the spatial nature of sublattice symmetry (2404.11398v3)

Published 17 Apr 2024 in cond-mat.mes-hall, cond-mat.other, and cond-mat.str-el

Abstract: The sublattice symmetry on a bipartite lattice is commonly regarded as the chiral symmetry in the AIII class of the tenfold Altland-Zirnbauer classification. Here, we reveal the spatial nature of sublattice symmetry, and show that this assertion holds only if the periodicity of primitive unit cells agrees with that of the sublattice labeling. In cases where the periodicity does not agree, sublattice symmetry is represented as a glide reflection in energy-momentum space, which inverts energy and simultaneously translates some $k$ by $π$, leading to substantially different physics. Particularly, it introduces novel constraints on zero modes in semimetals and completely alters the classification table of topological insulators compared to class AIII. Notably, the dimensions corresponding to trivial and nontrivial classifications are switched, and the nontrivial classification becomes $\mathbb{Z}_2$ instead of $\mathbb{Z}$. We have applied these results to several models, including the Hofstadter model both with and without dimerization.

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Summary

  • The paper demonstrates that sublattice symmetry splits into two classes: Class I matching conventional chiral symmetry and Class II characterized by glide reflection.
  • It reclassifies zero mode counts and introduces a Z₂ topological invariant in even dimensions, refining conventional topological paradigms.
  • The study’s insights suggest new design principles for metamaterials, with potential applications in electronic, photonic, and non-Hermitian quantum systems.

Revealing the Spatial Nature of Sublattice Symmetry

Introduction

The paper, titled "Revealing the Spatial Nature of Sublattice Symmetry," examines the spatial properties of sublattice symmetries in bipartite lattices. It challenges the established notion that sublattice symmetry is synonymous with the chiral symmetry of the AIII class from the tenfold Altland-Zirnbauer classification. The authors reveal that this symmetry's nature significantly influences the topological classification of materials, especially when the periodicity of primitive unit cells does not align with sublattice labeling.

Sublattice Symmetries Classification

The paper categorizes sublattice symmetries into two distinct types:

  1. Class I Sublattice Symmetry: This adheres to conventional chiral symmetry, observed when the periodicity of the unit cells matches the sublattice labeling, as exemplified by the Su-Schrieffer-Heeger (SSH) model (Figure 1a).
  2. Class II Sublattice Symmetry: Here, a discrepancy between unit cell periodicity and sublattice labeling leads to sublattice symmetry exhibiting glide reflection in energy-momentum space. This unique representation demands reevaluation of conventional topological classifications. Figure 1

    Figure 1: Illustrations of (a) Class I and (b) Class II sublattice symmetries.

Implications for Zero Modes

By examining semimetals, the paper explores how Class II sublattice symmetries impose unique constraints on zero modes. Notably, when the glide reflection symmetry is expressed in band structures, it leads to zero mode counts that differ significantly depending on the configuration of states per unit cell. Figure 2

Figure 2: Two scenarios demonstrating glide reflection's effect on zero modes.

Topological Classification

Central to the findings is a revised topological classification table for class-II sublattice symmetry. Unlike the traditional Z\mathbb{Z} classification of chiral symmetry, Class II exhibits a Z2\mathbb{Z}_2 classification in even dimensions, swapping the roles of trivial and nontrivial classifications between even and odd dimensions.

This reclassification has profound implications, altering the established paradigms within topological insulators and semimetals and providing a new framework for understanding materials where internal and spatial symmetries intertwine. Figure 3

Figure 3: Band structures for semimetal models showing variations in zero mode counts.

Topological Invariants and Edge States

The paper derives new topological invariants from the band topology in systems exhibiting Class II sublattice symmetry. These invariants elucidate the conditions for protected edge states in two-dimensional systems. The invariant's formulation demonstrates intricate relations between Berry phases and glide reflection symmetries.

The application to dimerized Hofstadter models highlights the realization of these edge states, underscoring the practical relevance of the revised topological analysis for artificial lattice systems. Figure 4

Figure 4: Topological invariant configurations and corresponding edge state spectra.

Conclusion

The elucidation of the spatial nature of sublattice symmetry challenges and extends the existing framework of topological classification in condensed matter physics. The expansion from traditional chiral symmetry to incorporate spatially significant glide reflections broadens the detection and categorization of topological phases in both theoretical and experimental domains.

Looking forward, the relevance of class-II sublattice symmetry can elevate the design of metamaterials, with potential applications in robust electronic, photonic, and acoustic devices. Furthermore, explorations into non-Hermitian systems may further amplify the breadth and depth of this symmetry's applications in future quantum materials research.

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Authors (2)

  1. Rong Xiao
  2. Y. X. Zhao
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