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It is shown theoretically that a one-dimensional crystal with time reversal symmetry is characterized by a Z_{2} topological invariant that predicts the existence or otherwise of edge states. This is confirmed experimentally through the…

Mesoscale and Nanoscale Physics · Physics 2016-03-09 Timothy J. Atherton , Celia A. M. Butler , Melita C. Taylor , Ian R. Hooper , Alastair P. Hibbins , J. Roy Sambles , Harsh Mathur

Band topology is both constrained and enriched by the presence of symmetry. The importance of anti-unitary symmetries such as time reversal was recognized early on leading to the classification of topological band structures based on the…

Strongly Correlated Electrons · Physics 2022-03-14 A. Corticelli , R. Moessner , P. A. McClarty

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical MHD, it is important to verify that such simulations are in agreement with…

Data Analysis, Statistics and Probability · Physics 2018-09-12 Irina Makarenko , Paul Bushby , Andrew Fletcher , Robin Henderson , Nikolay Makarenko , Anvar Shukurov

Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, such as geometric phases, topological…

Mesoscale and Nanoscale Physics · Physics 2024-05-07 ZhaoXiang Fang , Ming Gong , Guang-Can Guo , Yongxu Fu , Long Xiong

Crystalline symmetries play an important role in the classification of band structures, and the rich variety of spatial symmetries in solids leads to various topological crystalline phases (TCPs). However, compared with topological…

Mesoscale and Nanoscale Physics · Physics 2019-03-12 Feng Tang , Hoi Chun Po , Ashvin Vishwanath , Xiangang Wan

We construct a general theory of $Z_2$ topological phase transitions in two-dimensional systems with time-reversal symmetry. We investigate the possibilities of $Z_2$ topological phase transitions at band inversions at all high-symmetry…

Mesoscale and Nanoscale Physics · Physics 2023-03-13 Ren Sasaki , Yutaro Tanaka , Shuichi Murakami

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the…

Strongly Correlated Electrons · Physics 2021-12-01 Jans Henke , Mert Kurttutan , Jorrit Kruthoff , Jasper van Wezel

Topology is a fundamental aspect of quantum physics, and it has led to key breakthroughs and results in various fields of quantum materials. In condensed matters, this has culminated in the recent discovery of symmetry-protected topological…

Materials Science · Physics 2023-10-24 Baokai Wang , Yi-Chun Hung , Xiaoting Zhou , Tzen Ong , Hsin Lin

Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent…

Strongly Correlated Electrons · Physics 2026-04-23 Naren Manjunath , Maissam Barkeshli

The topological classification of gapped band structures depends on the particular definition of topological equivalence. For translation-invariant systems, stable equivalence is defined by a lack of restrictions on the numbers of occupied…

Mesoscale and Nanoscale Physics · Physics 2024-01-29 Piet W. Brouwer , Vatsal Dwivedi

There are two prominent applications of the mathematical concept of topology to the physics of materials: band topology, which classifies different topological insulators and semimetals, and topological defects that represent immutable…

Materials Science · Physics 2022-08-11 Zhi-Kang Lin , Qiang Wang , Yang Liu , Haoran Xue , Baile Zhang , Yidong Chong , Jian-Hua Jiang

Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and…

Strongly Correlated Electrons · Physics 2026-01-16 Weiguang Cao , Masahito Yamazaki , Linhao Li

Topological materials (TMs) showcase intriguing physical properties defying expectations based on conventional materials, and hold promise for the development of devices with new functionalities. While several theoretically proposed TMs…

Mesoscale and Nanoscale Physics · Physics 2019-04-19 Feng Tang , Hoi Chun Po , Ashvin Vishwanath , Xiangang Wan

Topological properties of solid states have sparked considerable recent interest due to their importance in the physics of lattices with a non-trivial basis and their potential in the design of novel materials. Here we describe an…

Applied Physics · Physics 2020-02-19 Henning U. Voss , Douglas J. Ballon

We show that compositions of time-reversal and spatial symmetries, also known as the magnetic-space-group symmetries, protect topological invariants as well as surface states that are distinct from those of all preceding topological states.…

Mesoscale and Nanoscale Physics · Physics 2022-07-13 Bingrui Peng , Yi Jiang , Zhong Fang , Hongming Weng , Chen Fang

Topological materials discovery has evolved at a rapid pace over the past 15 years following the identification of the first nonmagnetic topological insulators (TIs), topological crystalline insulators (TCIs), and 3D topological semimetals…

Topological flat bands in two-dimensional (2D) moir\'e materials have emerged as promising platforms for exploring the interplay between topology and correlation effects. However, realistic calculations of moir\'e band topology using…

Mesoscale and Nanoscale Physics · Physics 2025-11-25 Yunzhe Liu , Ethan Angerhofer , Kaijie Yang , Chao-Xing Liu , Jiabin Yu

The topological mechanics is a perfect tool that can bridge the gap between the quantum and Newtonian physics and mechanics of materials. It requires discrete models of the material with analogies with the topological characteristics of…

Applied Physics · Physics 2020-06-02 Mohamed Shaat

We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells), two-band models with any complex couplings and open…

Mesoscale and Nanoscale Physics · Physics 2025-05-28 Janet Zhong , Heming Wang , Alexander N Poddubny , Shanhui Fan

Macroscopic two-dimensional sonic crystals with inversion symmetry are studied to reveal higher-order topological physics in classical wave systems. By tuning a single geometry parameter, the band topology of the bulk and the edges can be…

Mesoscale and Nanoscale Physics · Physics 2020-09-30 Zhan Xiong , Zhi-Kang Lin , Hai-Xiao Wang , Xiujuan Zhang , Ming-Hui Lu , Yan-Feng Chen , Jian-Hua Jiang