If the translation vector of a glide plane operation is itself an element of the translation group, then the corresponding glide plane symmetry reduces to a combination of reflection symmetry and translational symmetry. For any symmetry group containing glide plane symmetry, the translation vector of any glide plane operation is one half of an element of the translation group. In the present paper, we derive a formula for the Z 2 topological invariant protected by glide symmetry in a nonprimitive lattice from that in a primitive lattice. In the case of glide plane symmetry, the symmetry group of an object contains a glide plane operation, and hence the group generated by it. We study the topological crystalline insulator phase protected by the glide symmetry, which is characterized by the Z 2 topological number. Combining two equal glide plane operations gives a pure translation with a translation vector that is twice that of the glide plane operation, so the even powers of the glide plane operation form a translation group. The isometry group generated by just a glide plane operation is an infinite cyclic group. The glide plane operation in the strict sense and the pure reflection are two of the four kinds of indirect isometries in 3D. Thus the effect of a reflection combined with any translation is a glide plane operation in the general sense, with as special case just a reflection. However, a glide plane operation with a nonzero translation vector in the plane cannot be reduced like that. In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. Symmetry elements are imaginary objects that perform the symmetry operation. The combination of a reflection in a plane and a translation in a perpendicular direction is a reflection in a parallel plane. The 7 symmetry operators are : translation, rotation, reflection (mirror), inversion (center of symmetry), roto-inversion (inversion axis), glide (translation + reflection), and screw (translation + rotation). Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector. Reversing the order of combining gives the same result. In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane. The latter is often called the diamond glide plane as it features in the diamond structure. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a fourth of either a face or space diagonal of the unit cell. Beyond advances made in intricate topological quantum systems, the latter, theoretical finding suggests that single-particle wave functions, and therefore, by the same token, sound, light or vibrations could bind to an analogous classical vortex.Glide planes are noted by a, b or c, depending on which axis the glide is along. The translational component can be half of the translation unit. Similarly, the distortion of the hexagonal lattice of graphene by a so-called Kekulé distortion field induces an equivalent zero mode, that is, a topological mid-gap state in the Dirac spectrum 4. Glide planes: A combination of a reflection and a translation parallel to the reflection plane. Here we show that the nodal point can be moved to the point (k 0) considering. This property was first demonstrated for a square-lattice phononic crystal at the X point of the first Brillouin zone (wavenumber k /a with a the lattice constant), for a half-lattice-constant glide. Interestingly, the binding mechanism of such localized defects is identical to Majorana bound states in p-wave superconductors 3. Nodal points can be artificially synthesized using glide-reflection symmetries at crystal interfaces. Originally, the Jackiw–Rossi model 2 described topological defects in the form of vortices in quantum field theory showing that localized zero-energy fermionic states bind at the vortex core. Topological bound states can form around these defects much in the same way edge and surface states bind to one- and two-dimensional (1D and 2D) interfaces, respectively. Nodal points can be artificially synthesized using glide-reflection symmetries at crystal interfaces. Topological defects are local kinks or obstructions in an order parameter field where domain walls, superconductor vortices or dislocations are few of many prominent examples 1.

Glide reflection in lattice