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An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. Further extensions use time reversal and time translation operations on magnetic and electrodynamic systems. An alternative description of magnetic structures to that with symmetry groups, the Shubnikov groups, is using representations of space groups. The same can be done for aperiodic crystals. A discussion of the relation between the two approaches is given. Representations of space groups and superspace groups play a role in the study of physical properties. These, and generalizations of them, are discussed for aperiodic crystals. They are used, in particular, for the characterization of phase transitions between aperiodic crystal phases.

Symmetry plays an important role in the study of physical systems. A symmetry group is a group of transformations leaving both the structure of the physical system and the physical equations invariant. If new systems show new degrees of freedom, the symmetry groups usually have to be adapted. For example, macroscopic properties of crystals are described using point groups. When the internal microscopic structure is taken into account, space groups take over the role of point groups. Mathematically speaking, these space groups are extensions of the point group by the translations: the translation subgroup is an invariant subgroup, and the quotient of the space group by the translation subgroup is isomorphic to the point group. (Sometimes, one says that the translation group is extended by the point group.) Considering magnetic structures, the symmetry groups become magnetic space groups, which, in turn, become generalized magnetic space-time groups when time-dependent electromagnetic fields are considered. These groups are discrete subgroups of the four-dimensional Galilean group, have a four-dimensional subgroup of translations and a four-dimensional point group, the elements of which are combinations of 3D orthogonal transformations with time reversal. They are the prototype of the superspace groups used for the description of incommensurate modulated phases. The full list of four-dimensional magnetic space-time groups was published in [

Section 2 gives a short review of the theory of superspace. In Section 3, it is shown that the superspace approach may also be used for the study of physical properties of a periodic or aperiodic crystal in an external electromagnetic field. Section 4 discusses the additional symmetry operators for systems with magnetic moments. Here, magnetic groups and magnetic space-time groups are discussed. In Section 5, two alternative ways of describing aperiodic crystals, by superspace groups or using representations, are compared. A general procedure to go from one formulation to the other is discussed. In Section 6, a more general type of representation, the so-called projective unitary-antiunitary (PUA) representations, is discussed. Earlier results are completed to cover all aperiodic crystals up to rank six. In Section 7, it is shown that in aperiodic crystals, there is a larger variety of phase transitions than for 3D lattice periodic systems. Several transitions are discussed where the superspace group changes. Finally, in Section 8, yet another type of symmetry operations is discussed: scale operations. These are important for quasi-periodic tilings, quasicrystals and in the context of relativistic space-time symmetries. The present paper is not meant as a review. For a historic overview, see [

When incommensurate modulated phases were discovered, the role of space groups was taken over by superspace groups. The diffraction pattern of incommensurate (IC) modulated phases has sharp diffraction spots at positions:

One then defines a density function in _{I}_{E}_{I}_{I}_{E}_{I}_{I}_{E}_{I}_{I}

The action of an element, _{s}_{I}_{E}_{E}_{I}_{I}

This description in Fourier space is an alternative to crystallography in direct space. This has already been discussed in [

A further extension of the symmetry groups are the generalized magnetic space-time groups mentioned before (Section 2). These are discrete subgroups of the 4D Poincaré group and group extensions of a magnetic point group with a 4D lattice group. They occur as symmetry groups of time-dependent electromagnetic fields. If the 4D space were Euclidean, the space groups would simply be the 4D space groups. However, in the 4D Minkowski space-time, one has to make a distinction between space-like, time-like and isotropic vectors (with positive, negative and zero norm, resp., for the metric −1,1,1,1 in the space,

The space-time symmetry of an electromagnetic field is a subgroup of the Poincaré group. For example, the symmetry of a transverse monochromatic plane wave is a (continuous) Lie group with a component of the identity generated by [

For a constant homogeneous magnetic field along the

The situation, however, is different when one considers the problem of a quantum mechanical charged particle in such a combination of fields, because in the Hamilton operator, the potential, _{t}(_{x}^{2}

The formulation of the problem of an electron in an aperiodic crystal in a homogeneous magnetic field is now straight-forward, if one uses the superspace formulation. However, one should be aware of the difficulty that the existence of a superspace group does not reduce the problem of electrons or phonons as it does for 3D lattice periodic crystals. For the latter, the phonon problem is reduced from an infinite to a finite problem, with the number of degrees of freedom three times the number of particles in the unit cell. Because the number of points on the atomic surfaces (the infinite collection of points in the

If a crystal structure has magnetic moments, an additional symmetry operation is the time reversal,

In the same way as for 3D space groups, one may proceed for superspace groups. Combing superspace group elements with _{I}_{I}_{I}_{I}_{I}_{I}

If an incommensurate magnetic structure has magnetic moments given by _{I}_{I}_{I}_{ij}S_{i}S_{j}_{d}_{d}

If the magnetic structure is given by spinors _{1})_{2}) = _{1}_{2}) for elements _{1} and _{2} from H, but

An example of a rather complicated incommensurate magnetic structure is ErFe4Ge2 [_{2}/mnm [_{2}, of the little group of _{2} + Γ_{3}. According to what will be discussed in Section 5, this leads to two different superspace groups, Pn2_{1}m(0_{1}m(0_{d} (

The first symmetry considerations of magnetic structures used magnetic space groups (Shubnikov groups) [

A similar difference of the description method existed in the early years of aperiodic crystals. Incommensurate modulated phases often have their origin in the instability of a phonon mode. The latter are usually described using representations of the basic structure.

Incommensurate modulated phases originate from lattice periodic structures by an incommensurate displacive or occupational modulation. These are vector valued (for displacive modulation) or scalar (for occupational modulation) functions on the set of atomic positions in a basic structure. This leads to an alternative way of describing these aperiodic structures, in terms of representations of the symmetry group of the basic structure. This is a 3D space group [_{q}, the subgroup of the point group that leaves

The symmetry group of the lower symmetry structure is the subgroup for which the restriction of the representation to this group is the identity representation. If the high-symmetry group is a space group or superspace group, and the modulation is incommensurate, then there is no space group symmetry for the lower symmetry phase. Then, one still can characterize the structure using the representation (which is irreducible in simple Landau theory). This is an alternative to the use of superspace groups.

The relation between the representation of the space group and the superspace group is easily demonstrated if the star of the modulation vector, _{q} is one-dimensional.

_{m}_{q}_{q}^{m}

The relation between representations of the space group and the resulting superspace group is more complicated when the star of _{q} is not one-dimensional [_{q} is more-dimensional, then the vectors in the representation space may determine different symmetries for the low symmetry phase. As an example, consider the case of high-symmetry group P4mm and _{q} = 4 mm. When the displacement is in the _{x}_{y}_{q}, the two corresponding superspace groups are different. They are P4(00

What is said here for the transition from a lattice periodic 3D structure to an incommensurate modulated structure holds also for transitions from an aperiodic structure with one superspace group to one with another superspace group. Furthermore, in that case, the transition may be described either with representations of the first group or with a transition from one superspace group to another.

Not only the usual (vector) representations play a role. Besides them, one has to consider projective representations,

A PUA-representation of Group G with Subgroup H of Index 1 or 2 is a mapping of the Group G into a group of unitary operators or a group of unitary and antiunitary operators, D(g), such that H is presented by unitary operators and the following relation holds:
_{1}, _{2}) is of absolute value one. For a conventional representation, the value of ^{g}^{g}

The PUA-representations of Group G are conventional (co)-representations of a group, which is an extension of G with the so-called (co-)multiplicator, M(G,H), but (in this case, at least) also a group extension of G by a group, _{2}, because all multiplicators are products of Groups _{2}. These groups are found by enlarging the number of generators and the number of defining relations. For example, the icosahedral group has generators ^{5} = ^{3} = (^{2} = ^{5} = ^{3} = ^{2} = ^{2} = ^{−1}^{−1} = ^{−1}^{−1} = _{6}) corresponds to the so-called double group, which plays a role for systems with spin. The matrices of this representation are SU(2)-matrices.

The projective representations of the point groups are needed for some of the irreducible representations of non-symmorphic superspace groups, those with a wave vector on the border of the Brillouin Zone. This is well known for three-dimensional space groups and holds also in arbitrary dimensions. An irreducible representation of the little group of the vector, _{1}32(5^{2}32), which has generators:
_{1}/5, gives a non-trivial factor system for the irreducible representation. Therefore, states with this wave vector are necessarily doubly degenerate.

Incommensurate modulated phases usually originate from a lattice periodic structure. According to the Landau theory of phase transitions, the transitions can be described in terms of irreducible representations of the “high symmetry” phase. In this case, the transitions are treated using irreducible representations of the 3D space group of the lattice periodic structure. However, phase transitions happen to occur also from one aperiodic structure to another. Here, one has to distinguish two types. For the first, the rank of the Fourier module (the dimension of the superspace) does not change. In the second, this rank is different for the two phases. Moreover, Landau theory can only be used for the situation in which the two phases have a group-subgroup relation between their symmetry groups. Therefore, one has to distinguish four types: yes or no for the group-subgroup relation and yes or no for the change in the dimension. The phase transitions occur in all three classes of aperiodic crystals and also in systems with magnetic moments.

When the two phases have a group-subgroup relation and the rank of the crystal does not change, the transition may be described using irreps of the high symmetry space group, resp. the superspace group. The theory of these irreps is well known. These representations are also sufficient for systems with magnetic moments, if the magnetic moments are real. If the spins are complex, one has to consider the co-representations.

When the rank of the crystal changes at the phase transition, _{2}-dimensional group, G_{2}, and that of the lower-dimensional structure is the _{1}-dimensional group, G_{1}, then G_{2} may be a subgroup of the direct product of G_{1} with the Euclidean group in _{2}_{1} dimensions, and one can again use irreps of the latter to describe the structure of the former. The irreps of E(

Some examples of different types of phase transitions in aperiodic crystals are the following.

K_{2}SeO_{4} [_{y}, m_{z}

A compound with an incommensurate modulation, that goes down in temperature as low as one has measured, is biphenyl [_{1}/a. In this phase, a soft mode develops with wave vectors _{1}/a(_{1}/

A complicated magnetic structure has been found in ErFe_{4}Ge_{2} [

A phase transition with a change of rank has also been found in composites. An example is nonadecane-urea [_{1}(00

The phase transition from the four-dimensional structure with superspace group P6_{1}22(00_{1}(00_{q} is six. Its irreps are one-dimensional, and for each, there exists a compensating phase shift (compensating gauge transformation). The resulting superspace group depends on these phases, which are unknown until a precise structure determination will have been made.

An example of a phase transition in quasicrystals is the transition from an icosahedral quasicrystal to a tetrahedral (a subgroup of m3̄) or a rhombohedral (subgroup of 3̄m) approximant. Such a transition is usually first-order. This is a transition from a lattice periodic structure in six dimensions to a three-dimensional lattice periodic structure [^{2}3m)), to the tetrahedral group, m3̄, or a rhombohedral group. For some P-type icosahedral quasicrystals (with primitive icosahedral superspace group;, approximants are known [

Interesting cases of phase transitions are those where two phase transitions are coupled; e.g., this is the case of multiferroics. Examples are TbMnO3 and DyMnO3 [_{N}_{d}_{c}_{d}_{1}cn(00_{3}, similar behavior is found, but for the lock-in transition under an external magnetic field. The basic structure has space group Pmcn [

Only for a number of phase transitions (semi-)microscopic models exist [

Symmetries that are non-crystallographic in three dimensions, like octagonal or decagonal groups, have matrix representations, which are irreducible if one considers integers, but reducible for real numbers. For real representations, this means that there may be invariant subspaces of the superspace and lattice transformations, leaving these subspaces also invariant. An example is an aperiodic crystal, or tiling, with octagonal symmetry of 8 mm. The superspace has dimension four in that case. The symmetry group has generators _{E}_{I}_{e}_{I}_{E}_{I}

Other examples of scale transformations that leave the points of the Fourier module invariant combine scale factors with a rotation. An example of such a roto-scale is the following. The Fourier module of the Penrose tiling has a symmetry group with two generators. One is a rotation over 2_{E}_{I}_{E}

In superspace, the scale transformations are non-physical, because the laws of nature are, generally, not invariant under scale transformations. This is different for crystallographic systems with relativistic symmetry [

We have argued that new phenomena in solid state materials may lead to a generalization of symmetry considerations. This applies to the generalizations from point groups, to space groups, space-time groups and superspace groups, but also to magnetic space groups and magnetic superspace groups. Moreover, this procedure can be extended to lattice periodic and aperiodic crystals in external electromagnetic fields, both static and time-dependent. For the study of states in aperiodic crystals, we have seen a completion of the list of ordinary and projective unitary-antiunitary representations (multiplier co-representations). Finally, a short overview is given of the various types of phase transitions that may occur in aperiodic crystals.

The author thanks Tohoku University, Japan, and especially An Pang Tsai for the hospitality during a stay when part of this paper was written.

The author declares no conflict of interest.

_{4}Ge

_{2}

_{4}Ge

_{2}

_{2}SeO

_{4}

_{65}Cu

_{20}Fe

_{15}quasicrystal

_{3}

_{3}(R = Gd,Dy,Tb)

_{3}

_{2}O

_{5}

The embedding into superspace of a magnetic wave (left, one unit cell) and a displacive wave (right, two unit cells) for a case where the functions are sinusoidal and both patterns are invariant under

Projective unitary-antiunitary (PUA) representations for groups relevant for aperiodic crystals. The groups are characterized by the values of the elements, _{i}_{i}_{i}

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | ||||
---|---|---|---|---|---|---|---|---|---|

^{l} |
^{m} |
(^{n} |
^{2} |
^{−1}^{−1} |
^{−1}^{−1} | ||||

_{2n+1} |
_{2n+1} |
e | _{1} |
^{2n+1} |
|||||

_{2n} |
_{2n} |
e | _{1} |
^{2n} |
|||||

_{2n} |
_{n} |
^{2} |
_{2} |
±1 | |||||

_{2n+1} |
_{2n+1} |
e | _{1} |
^{2n+1} |
^{2} |
^{2}^{2} |
|||

_{2n+1} |
_{2n+1} |
_{2} |
^{2n+1} |
||||||

_{2n} |
_{2n} |
e | _{2} |
±^{2n} |
^{2} |
^{2}^{2} |
|||

_{2n} |
_{2n} |
_{2} × _{2} |
^{2n} |
±1 | ±1 | ||||

_{2n} |
_{n} |
_{2} × _{2} |
±1 | ^{2} |
±1 | ||||

e | _{2} |
^{5} |
^{3} |
±^{2}^{2} |
|||||

_{2} |
_{2} |
e | _{2} × _{2} |
^{5} |
^{3} |
±^{2}^{2} |
^{2} |
±^{2} |
^{2} |

_{2} |
_{2} × _{2} × _{2} |
^{5} |
±^{3} |
±^{2}^{2} |
^{2} |
^{2} |
^{2} | ||

| |||||||||

_{n}_{N} |

Character table for the covering group of the icosahedral point group,

Γ | − |
− |
^{2} |
−^{2} |
− |
||||
---|---|---|---|---|---|---|---|---|---|

| |||||||||

order: | 1 | 1 | 12 | 12 | 12 | 12 | 20 | 20 | 30 |

Γ_{1} |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Γ_{2} |
3 | 3 | 1 + |
1 + |
− |
− |
0 | 0 | −1 |

Γ_{3} |
3 | 3 | − |
− |
1 + |
1 + |
0 | 0 | −1 |

Γ_{4} |
4 | 4 | −1 | −1 | −1 | −1 | 1 | 1 | 0 |

Γ_{5} |
5 | 5 | 0 | 0 | 0 | 0 | −1 | −1 | 1 |

| |||||||||

Γ_{6} |
2 | −2 | 1 + |
−1 − |
− |
1 | −1 | 0 | |

Γ_{7} |
2 | −2 | − |
−1− |
1 + |
1 | −1 | 0 | |

Γ_{8} |
4 | −4 | 1 | −1 | −1 | 1 | −1 | 1 | 0 |

Γ_{9} |
6 | −6 | −1 | 1 | 1 | −1 | 0 | 0 | 0 |