Theoretical construction of weak topological crystalline insulators

2017 ◽  
Vol 31 (20) ◽  
pp. 1750136
Author(s):  
Qing-Li Zhu ◽  
Liang Hua ◽  
Ji-Mei Shen
Keyword(s):  
Point Group ◽  
Three Dimensional ◽  
Discrete Symmetry ◽  
Topological Phases ◽  
Symmetry Classes ◽  
New Type ◽  
Theoretical Construction ◽  
Topological Crystalline Insulators ◽  
Group Symmetries

Inspired by the discovery of topological crystalline insulators (TCIs) in three-dimensional materials such as Pb[Formula: see text]Sn[Formula: see text]Se(Te), the classification of topological insulators has been extended to other discrete symmetry classes such as crystal point group symmetries. In this paper, we construct and study a simple model of weak TCIs, which will serve as a more viable project in the experimental probe for such new type of topological phases.

scholarly journals Prediction of topological crystalline insulators and topological phase transitions in two-dimensional PbTe films

2017 ◽  
Vol 19 (43) ◽  
pp. 29647-29652 ◽  
Author(s):  
Yi-zhen Jia ◽  
Wei-xiao Ji ◽  
Chang-wen Zhang ◽  
Ping Li ◽  
Shu-feng Zhang ◽  
...  
Keyword(s):  
Phase Transitions ◽  
Three Dimensional ◽  
Topological Phases ◽  
Topological Phase ◽  
Two Dimensional ◽  
Topological Crystalline Insulators ◽  
Topological Phase Transitions

Topological phases, especially topological crystalline insulators (TCIs), have been intensively explored and observed experimentally in three-dimensional (3D) materials.

scholarly journals On three-dimensional misorientation spaces

2017 ◽  
Vol 473 (2206) ◽  
pp. 20170274 ◽  
Author(s):  
Robert Krakow ◽  
Robbie J. Bennett ◽  
Duncan N. Johnstone ◽  
Zoja Vukmanovic ◽  
Wilberth Solano-Alvarez ◽  
...  
Keyword(s):  
Materials Science ◽  
Point Group ◽  
Three Dimensional ◽  
Orientation Relationships ◽  
Active Deformation ◽  
Orientation Data ◽  
Analysis Methodology ◽  
Mapping Techniques ◽  
Group Symmetries ◽  
Directing Attention

Determining the local orientation of crystals in engineering and geological materials has become routine with the advent of modern crystallographic mapping techniques. These techniques enable many thousands of orientation measurements to be made, directing attention towards how such orientation data are best studied. Here, we provide a guide to the visualization of misorientation data in three-dimensional vector spaces, reduced by crystal symmetry, to reveal crystallographic orientation relationships. Domains for all point group symmetries are presented and an analysis methodology is developed and applied to identify crystallographic relationships, indicated by clusters in the misorientation space, in examples from materials science and geology. This analysis aids the determination of active deformation mechanisms and evaluation of cluster centres and spread enables more accurate description of transformation processes supporting arguments regarding provenance.

Opening, stellation and handle replacement of edges of the cube, tetrahedron and hexagonal prism: application to 3-connected three-dimensional polyhedra and (2, 3)-connected polyhedral units in three-dimensional nets

1994 ◽  
Vol 444 (1920) ◽  
pp. 217-238 ◽  
Keyword(s):  
Point Group ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Hexagonal Prism ◽  
High Symmetry ◽  
Regular Tetrahedron ◽  
Vertex Connectivity ◽  
3D Framework ◽  
Low Symmetry

Opening, stellation or handle replacement of edges of the regular cube, the regular tetrahedron and the semi-regular hexagonal prism (minimum point group symmetry 2/ m ) yields totals of 144, 11 and 205 geometrically distinct configurations respectively. Edge opening is restricted to a vertex connectivity of two or three. Sixty-five of the (2, 3)-connected polyhedral units were found in real three-dimensional (3D) framework structures. Low-symmetry configurations are as abundant as high-symmetry ones. With minor exceptions, those (2, 3)-connected patterns with the fewest converted edges are observed in each point group. The 113 polyhedra derived by double-stellation or double-handle replacement are uniformly three-connected. The ones observed in frameworks have a high symmetry. This study extends the general theory of 3D polyhedra and can be further expanded to other polyhedra and geometrical transformations. It may provide insight on the nature and growth of some seemingly complex framework structures that cannot otherwise be easily described topologically. The new polyhedral units are useful for classification of known framework structures in zeolites and related materials. New hypothetical nets generated from linkage of the units may solve unknown framework structures.

scholarly journals Acoustic Möbius insulators from projective symmetry

2021 ◽  
Author(s):  
Chunyin Qiu ◽  
Tianzi Li ◽  
Juan Du ◽  
Qicheng Zhang ◽  
Yitong Li ◽  
...  
Keyword(s):  
Critical Role ◽  
Three Dimensional ◽  
Real Space ◽  
Extensive Study ◽  
Topological Classification ◽  
Translation Symmetry ◽  
Topological Crystalline Insulators ◽  
First Time ◽  
Projective Translation

Abstract Symmetry plays a critical role in classifying phases of matter. This is exemplified by how crystalline symmetries enrich the topological classification of materials and enable unconventional phenomena in topologically nontrivial ones. After an extensive study over the past decade, the list of topological crystalline insulators and semimetals seems to be exhaustive and concluded. However, in the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topological physics. Here we demonstrate this novel idea by exploiting a projective translation symmetry and constructing a variety of Möbius-twisted topological phases. Experimentally, we realize two Möbius insulators in acoustic crystals for the first time: a two-dimensional one of first-order band topology and a three-dimensional one of higher-order band topology. We observe unambiguously the peculiar Möbius edge and hinge states via real-space visualization of their localiztions, momentum-space spectroscopy of their 4π periodicity, and phase-space winding of their projective translation eigenvalues. Not only does our work open a new avenue for artificial systems under the interplay between gauge and crystalline symmetries, but it also initializes a new framework for topological physics from projective symmetry.

scholarly journals Acoustic Möbius insulators from projective symmetry

2021 ◽  
Author(s):  
Chunyin Qiu ◽  
Tianzi Li ◽  
Juan Du ◽  
Qicheng Zhang ◽  
Yitong Li ◽  
...  
Keyword(s):  
Critical Role ◽  
Three Dimensional ◽  
Real Space ◽  
Extensive Study ◽  
Topological Classification ◽  
Translation Symmetry ◽  
Topological Crystalline Insulators ◽  
First Time ◽  
Projective Translation

Abstract Symmetry plays a critical role in classifying phases of matter. This is exemplified by how crystalline symmetries enrich the topological classification of materials and enable unconventional phenomena in topologically nontrivial ones. After an extensive study over the past decade, the list of topological crystalline insulators and semimetals seems to be exhaustive and concluded. However, in the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topological physics. Here we demonstrate this novel idea by exploiting a projective translation symmetry and constructing a variety of Möbius-twisted topological phases. Experimentally, we realize two Möbius insulators in acoustic crystals for the first time: a two-dimensional one of first-order band topology and a three-dimensional one of higher-order band topology. We observe unambiguously the peculiar Möbius edge and hinge states via real-space visualization of their localiztions, momentum-space spectroscopy of their 4π periodicity, and phase-space winding of their projective translation eigenvalues. Not only does our work open a new avenue for artificial systems under the interplay between gauge and crystalline symmetries, but it also initializes a new framework for topological physics from projective symmetry.

A Principal of Linear Covariance for Quantum Mechanics and Its Consequences Taking one Beyond Symmetry

1997 ◽  
Vol 52 (1-2) ◽  
pp. 46-48
Author(s):  
Oktay Sinanoğlu
Keyword(s):  
Quantum Mechanics ◽  
Superposition Principle ◽  
Point Group ◽  
Equivalence Classes ◽  
Quantum Mechanical ◽  
Quantum Numbers ◽  
Unitary Transformations ◽  
Linear Covariance ◽  
Group Symmetries

Abstract A Principle of Linear Covariance is stated which follows from the "superposition principle" of quantum mechanics. Accordingly, quantum mechanical equations should be written in linearly covariant form which makes them look the same under non-unitary as well as unitary transformations. The principle leads to a non-unitary classification of all molecules (and clusters and solids) into distinct equivalence classes giving hitherto unknown relations between isomeric molecules. One also gets kinetic and thermic selection rules for chemical reactions. All these are independent of, and far more general than any unitary or point group symmetries. The invariants found for each class of molecules or clusters allow qualitative electronic deductions and are more generally applicable than symmetry based quantum numbers.

Three-dimensional texture visualization approaches: theoretical analysis and examples

2017 ◽  
Vol 50 (2) ◽  
pp. 430-440 ◽  
Author(s):  
Patrick G. Callahan ◽  
McLean Echlin ◽  
Tresa M. Pollock ◽  
Saransh Singh ◽  
Marc De Graef
Keyword(s):  
Theoretical Analysis ◽  
Point Group ◽  
Three Dimensional ◽  
Euler Angle ◽  
Basic Theory ◽  
Euler Space ◽  
Contour Plots ◽  
The Creation ◽  
Group Symmetries

Crystallographic textures are commonly represented in terms of Euler angle triplets and contour plots of planar sections through Euler space. In this paper, the basic theory is provided for the creation of alternative orientation representations using three-dimensional visualizations. The use of homochoric, cubochoric, Rodrigues and stereographic orientation representations is discussed, and illustrations are provided of fundamental zones for all rotational point-group symmetries. A connection is made to the more traditional Euler space representations. An extensive set of three-dimensional visualizations in both standard and anaglyph movies is available.

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