Topology-change surgery for crystals

Abstract

We performed the topology-change surgery for ring-shaped crystals of tantalum triselenide (TaSe 3 ) to investigate the interplay between the closed-ring topology and elasticity/plasticity of the crystals. We cut the TaSe 3 rings using a focused ion beam and observed that the curvatures of the open rings increased from their initial curvatures. We found that a change in the radius is proportional to inverse square of the thickness of the crystals, which corresponds to an inhomogeneous distribution of edge dislocations. From the distribution, we suggest the existence of cylindrical domain walls in the ring-shaped crystals as a result of the crystal topology. The surgery method in Mathematics, which changes the topology of a space by cut and glue, is a powerful tool to extract topological properties, as Perelman solved the Poincaré conjecture.1,2 Even in nature, the topology-change surgery is an important approach since the interplay between global topology and local symmetry would play a fundamental role in cosmology, condensed matter physics, and biology.3-5 By the discovery of rings, Möbius rings and Hopf links of crystals of transition metal-trichalcogenide compound systems (NbSe 3 , NbS 3 , TaSe 3 , and TaS 3 ), 6-9 we obtained experimental systems for the interplay between the closed-ring topology and local order parameters. Using the topological crystals, effects of the closed-ring topology on charge density waves 10-17 and superconductivity 18 have been investigated. As well as the ordered phases in electron systems, the crystal orders of the topological crystals have rich problems of the interplay between the global topology. The crystal structures of the topological crystals must be different from normal crystals due to a topological charge, namely, a global disclination. 19 As shown in Fig. 1(a) (left), the global disclination is obtained as the total rotation angle of a crystal axis (b axis) of a ring-shaped crystal along a closed curve on the crystal, where it is 2π for the ring-shaped crystals, and this value is topologically robust against continuous deformations. Due to the global disclination of power 2π , the crystal lattice cannot be mapped onto that of perfect crystals globally. Moreover, the topological crystals must be frustrated geometrically by the deformation energy from bending or twist, and the deformation energy would induce topological defects into a topological crystal for relaxation. Hayashi et al. pointed out theoretically that the geometrical frustrations (curvature) of the crystals induce topological defects (edge dislocations), as analogous with vortices in type II superconductors.20 The x-ray diffraction measurements for ring-shaped crystals of NbSe 3 confirmed that the averaged bending strain in a ring-shaped crystal is relaxed by defects. 21 However, the distributions of strain and defects have never investigated experimentally since the x-ray diffraction measurement has less spatial resolution for the small crystals due to the spot size of x ray. In principle, the conventional...

Polyhedral Topological Crystals

“…Topological crystals with a closed curve do not have an edge, and hence, slipping dislocations cannot exit from the crystal. When many dislocations are generated, cylinder-shaped walls of dislocations are expected to be formed as a result of the arrangement of the dislocations on parallel lines. , Here, we considered two dislocations moving on different crystal lines or planes. When the two dislocations are sufficiently far apart, a repulsive force is exerted; however, when the distance is short, an attractive force is exerted.…”

“…When many dislocations are generated, cylinder-shaped walls of dislocations are expected to be formed as a result of arrangement on parallel line. 16,17 Here, we considered two dislocations moving on different crystal lines or planes.…”

“…On the other hand, a topological crystal does not display discrete translational or rotational symmetries, because the orientation of the crystal axes is smoothly changing. This also indicates that the distribution of dislocations/disclinations introduced into a topological crystal [as shown in Figure (right)] must be smooth. , When a topological crystal needs to be bent, geometrical frustration is relaxed through the inclusion of edge dislocations, and other types of dislocations, such as screw and mixed dislocations, do not need to be considered. − However, the interplay between the global topology of the crystals and the distribution of dislocations/disclinations is still unclear.…”

“…It also indicates that the distribution of the dislocations/disclinations introduced in the topological crystals [as shown in Figure 1 (RIGHT)] must be smooth. 16,17 When the topological crystal needs to be bent, geometrical frustration is relaxed by inclusion of the edge dislocation, and other kinds of dislocation, such as screw and mixed dislocation are not necessary to be considered. [18][19][20] However, the interplay between the global topology of the crystals and the distribution of dislocations/disclinations has been unclear yet.…”

Manipulation of the Magnetoresistance by Strain in Topological TaSe3

“…[16][17][18][19] Bending creates nonuniform strain along the b-axis of TaSe 3 and induces the deformation energy, thus generating defects/disorder in a crystal for relaxation. [20] Cylindrical domain walls from the edge dislocations and vortex have been suggested in ring-shaped crystals. [17] Given the observed nontrivial topological band and extremely large magnetoresistance (XMR) in as-grown ribbon-shaped samples [21][22][23] and a uniform strain-induced topological phase transition, [15] we are inspired to investigate the magnetotransport properties in ring-shaped TaSe 3 single crystals by applying magnetic field up to 60 T. Compared to as-grown unbended samples, the MR in ring-shaped samples reduces three orders of magnitude.…”