We measured the magnetic torque of the ring-shaped crystals of TaSe 3 by using piezoresistive cantilevers to investigate the superconducting topological properties. We measured three ring samples and we observed that the magnetic torque of the ring crystals oscillates with increasing external magnetic field. The magnetic periods of Sample A ͑radius: 37.9 m͒, Sample B ͑24.5 m͒, and Sample C ͑15.6 m͒ were 2.16Ϯ 6.6ϫ 10 −2 G, 4.69Ϯ 1.8ϫ 10 −1 G, and 5.44Ϯ 3.8ϫ 10 −1 G, respectively. We found that hundreds of vortices collectively and simultaneously penetrated the ring crystal. When such a phenomenon occurs, it is natural that the vortices in the ring crystal would be positioned along the circumference. From these results, we suggest that vortices exist as cylinder vortices in the rings, and this phenomenon provides unambiguous experimental evidence of the topological effect in a superconductor.

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...

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Abstract. Topology has many applications in modern condensed matter physics. We report that superconducting properties are changed by topology. We have measured the magnetic torque of the ring-shaped crystals of TaSe3 by using piezoresistive cantilevers in order to investigate the superconducting topological properties. We measured two ring samples. The outer radius of sample A was 37.9 µm and that of sample B was 24.5 µm. We found that the magnetic torque of the ring crystals changes periodically by increasing of the external magnetic field. The periodicity of sample A was 1.85 Gauss and that of sample B was 4.75 Gauss, respectively. We found that the period is proportional to the circumference of the ring crystal rather than the area enclosed by the outer circumference. In that case, it is natural that vortices in the ring crystal were placed along the circumference. From these results, we suggest that vortices exist as cylinder vortices in the rings at the results of topological effects of superconductive ring-shaped crystals, and this matter give a new experimental evidence of topological effect in superconductor. IntroductionWhat kind of effects on the superconducting state, such as a mixed state of type II superconductors, are caused by topology?At the mixed state vortices penetrate the superconductor when the applied magnetic field increases beyond H c1 . Typically, Abrikosov lattice is the most stable configuration of voltices with repulsive force interaction to each other. However, when the system size is small, shape of superconductors, such as boundary condition, affects penetration and arrangement of vortices [1]. Similarly, topology of systems has a potential of affecting the superconducting properties. For example, closed systems like ring crystals [2], vortices may not relocate to become stable like bulk superconductors, because they are trapped globally in the hole of the crystals. Furthermore, systems like Möbius ring crystals [3], there is the possibility that vortices cannot penetrate crystals. These speculations suggest that topology of systems plays an important role in physical properties.The purpose of this research is to investigate the effects of topology on superconducting properties. The ring-shaped crystals of quasi-one-dimensional superconductor TaSe 3 are suitable to investigate these topological effects, because it is expected that TaSe 3 causes topological effects for its filamentary structure. Besides that, a measurement without electrode is appropriate to investigate those effects, because this method conserves topology of systems. Therefore, we adopted magnetic torque measurement without electrode by using micro cantilevers [4]. As a result, we found a phenomenon that the magnetic torque of the ring crystals oscillates

geometries. We also studied the CDW properties in ring-shaped crystals. In this case, the "mixed state" of CDW analogous to that in type-II superconductors is expected which may show quite different electric response from ordinary CDW state. We also comment on some crystal properties of these "topological crystals".