Common use of Some examples Clause in Contracts

Some examples. Before going to six dimensions, let us illustrate the above definitions with some easy examples of two-dimensional Z2 orbifolds, taken from Appendix 5.B. Space groups in the same Z-class Consider the affine class Z2-II–1–1, as defined in Appendix 5.B. As there are no roto-translations, the orbifolding group is equal to the point group and is generated by ϑ, a reflection at the horizontal axis. Now, let this reflection act on a lattice, first spanned by the basis vectors e = {e1, e2} and second spanned by f = {f 1, f 2}, see Figure 5.3. The two corresponding space groups read 0 −1 ⟩ f Se = ⟨(ϑ, 0), (1, e1), (1, e2)⟩ with ϑe = 1 0 , (5.21) Sf = ⟨(ϑ, 0), (1, f 1) (1, f 2 0 −1 , (5.22) where ϑe = ϑf because they are given in their corresponding lattice bases. How- ever, it is easy to see that they are related by the GL(2, Z) transformation 0 1 U = 1 1 with U−1 ϑe U = ϑf , (5.23) cf. Equation (5.19). Therefore, they belong to the same Z-class. Hence, as we actually knew from the start, they act on the same lattice and the matrix U just defines the associated change of basis precisely as in Equation (5.4). Space groups in the same Q-class, but different Z-classes Next, consider the space groups, 0 −1 S1–1 = ⟨(ϑ1–1, 0), (1, e1), (1, e2)⟩ with ϑ1–1,e = 1 0 , (5.24) S2–1 = ⟨(ϑ2–1, 0), (1, f 1), (1, f )⟩ with ϑ2–1,f = 0 1 , (5.25) 1 0 2 with lattices spanned by e1 = (1, 0), e2 = (0, 1) and f 1 = (1/2, 1/2), f 2 = (1/2, 1/2), respectively. The first space group belongs to the affine class Z2- II–1–1 and the second one to Z2-II–2–1, see Appendix 5.B. If we try to find the transformation V from Equation (5.20) that fulfills V −1 ϑ1–1,e V = ϑ2–1,f we see that y −y with x, y ∈ Q . (5.26) But for all values of x and y for which V −1 exists, either V or V −1 has non-integer entries. Therefore, the space groups Z2-II–1–1 and Z2-II–2–1 belong to the same Q-class, but to different Z-classes. In other words, these space groups are defined with inequivalent lattices. Indeed, the first space group possesses a primitive rectangular lattice, while the second one has a centered rectangular lattice, as we will see in detail in the following. There is an alternative way of seeing the relationship between the two space groups of the last example: one can amend one of the space groups by an additional translation. In general, this gives rise to a new lattice, and consequently to a different Z-class. In our case, let us take the Z2-II–1–1 affine class and add the non-lattice translation 1 τ = (e1 + e2) (5.27) to its space group. If we incorporate this translation into the lattice, we notice that this element changes the original primitive rectangular lattice to a centered rectangular lattice, with a fundamental cell of half area. The new lattice (see Figure 5.4) can be spanned by the basis vectors τ and e1 τ . We can interpret the inclusion of this additional translation as a “change of basis”, see Equation (5.4), but now generated by a matrix M GL(2, Q) instead of one from GL(2, Z). The transformation looks like Be M = Bτ with M = 1/2 1/2 1/2 −1/2 , (5.28) e2 e1 − τ e1 Figure 5.4: Change of a lattice by an additional translation: the basis of the orig- inal lattice is solid, the basis of the new one dashed. The additional lattice points are light gray. The action of ϑ is a reflection at the horizontal axis. Therefore, it maps e1 to itself, e2 to its negative and interchanges τ and e1 − τ . where Be and Bτ are matrices whose columns are (e1, e2) and (τ , e1 τ ), re- spectively. M is precisely the matrix in Equation (5.26) with values x = y = 1/2. Performing this basis change, the twist has to be transformed accordingly. Hence, the two Z-classes are related by a GL(2, Q) transformation M and the new space group with lattice Bτ is Z2-II–2–1. The geometrical action of the twist, however, is the same in both cases: it is a reflection at the horizontal axis (see Figure 5.4). That is the reason for the name geometrical crystal classes for Q-classes. A general method for including additional translations can be found in Appendix 5.A.2. The method of using additional translations has been used in [21] and [17] in order to classify six-dimensional space groups with point groups ZN ZN for N = 2, 3, 4, 6 (the classification of [17] is not fully exhaustive, see Section 5.5.1). In these works, the authors start with factorized lattices, i.e. lattices which are the orthogonal sum of three two-dimensional sublattices, on which the twists act diagonally. Then, in a second step additional translations are introduced. As we have shown here, adding such translations is equivalent to switching between Z- classes in the same Q-class. Hence, if one considers all possible lattices (Z-classes) additional translations do not give rise to new orbifolds. Space groups in different Q-classes Finally, consider the affine classes Z2-I–1–1 and Z2-II–1–1 defined in Appendix 5.B. If we try to find a transformation between both space groups generators, see Equa- tion (5.20), 0 −1 0 −1 V = V 1 0 , we obtain V = 0 x ∈/ GL(2, Q) ∀ x, y . (5.30) Therefore, the space groups Z2-I–1–1 and Z2-II–1–1 belong to different Q-classes (and also to different Z-classes). That is, the point groups are inequivalent: the twist of the first point group is a reflection at the origin and the twist of the second point group is a reflection at the horizontal axis.