The symmetry of crystals and the topology of electrons

The concept of topology, having played a perpetual center stage role in the field of mathematics and physics for already a few centuries, heuristically pertains to the study of properties of geometrical objects that are preserved under smooth deformations. Recently, it was uncovered that such topological structures may in particular emerge in the quantum world of electron systems that are subjected to the presence of discrete symmetries such as time reversal symmetry, particle hole symmetry and chiral symmetry. Nonetheless, the role of the most prominent and necessarily present symmetry in this regard, being that of the underlying lattice, has remained elusive. In this work we signify the profound interplay between topological order in electronic band theory and the associated crystal symmetries by exploring the physics of defects, with the emphasis on dislocations. We find that in addition to a plethora of interesting mechanisms, which are related to the defects themselves and culminate, inter alia, in helical propagating fermionic modes bound to freely deformable channels and exotic isospinless graphene-like states that can be linked to illustrious field theoretical anomalies, these results in turn expose an extension of the classification of topological electrons systems.Theoretical Physic

Переяславська рада у сучасній російській навчальній літературі

Topological band-insulators (TBIs) represent a new class of quantum materials that in the presence of time-reversal symmetry (TRS) feature an insulating bulk bandgap together with metallic edge or surface states protected by a Z 2 topological invariant [1,2,3,4]. Recently, an extra layer in this Z 2 classification of TBIs has been uncovered by considering the crystal symmetries [5]. Dislocation lines being the unique topological defects related to the lattice translations play a fundamental role in this endeavor. We here elucidate the general rule governing their response in three-dimensional TBIs and uncover their role in this classification. According to that K-b-t rule, the lattice topology, represented by dislocation lines oriented in the direction t with the Burgers vector b , conspires with the electronic-band topology, characterized by the band-inversion momentum K inv , to produce gapless propagating modes along these line defects, which were discovered in Ref. [6]. For sufficiently symmetric crystals, this conspiracy leads to the topologically-protected metallic states inside the dislocation loops, which could also be important for applications. Finally, these findings are experimentally consequential as dislocation defects are ubiquitous in the real crystals

Topological classification of crystalline insulators through band structure combinatorics

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented match the mathematical structure underlying the topological classification of these crystals in terms of K-theory and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify all allowed topological phases of spinless particles in crystals in class A. Employing this classification, we study transitions between topological phases within class A that are driven by band inversions at high-symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure and to identify whether or not they give rise to intermediate Weyl semimetallic phases

Classification of nematic order in 2+1D: dislocation melting and O(2)/Z_N lattice gauge theory

Theoretical Physic

Topological classification of crystalline insulators through band structure combinatorics

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented match the mathematical structure underlying the topological classification of these crystals in terms of K-theory and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify all allowed topological phases of spinless particles in crystals in class A. Employing this classification, we study transitions between topological phases within class A that are driven by band inversions at high-symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure and to identify whether or not they give rise to intermediate Weyl semimetallic phases

The conspiracy of electronic topology and crystal symmetry: Dislocation-line modes in topological band-insulators

Topological band-insulators (TBIs) represent a new class of quantum materials that in the presence of time-reversal symmetry (TRS) feature an insulating bulk bandgap together with metallic edge or surface states protected by a Z 2 topological invariant [1,2,3,4]. Recently, an extra layer in this Z 2 classification of TBIs has been uncovered by considering the crystal symmetries [5]. Dislocation lines being the unique topological defects related to the lattice translations play a fundamental role in this endeavor. We here elucidate the general rule governing their response in three-dimensional TBIs and uncover their role in this classification. According to that K-b-t rule, the lattice topology, represented by dislocation lines oriented in the direction t with the Burgers vector b , conspires with the electronic-band topology, characterized by the band-inversion momentum K inv , to produce gapless propagating modes along these line defects, which were discovered in Ref. [6]. For sufficiently symmetric crystals, this conspiracy leads to the topologically-protected metallic states inside the dislocation loops, which could also be important for applications. Finally, these findings are experimentally consequential as dislocation defects are ubiquitous in the real crystals