Spacegroup ========== Group theory helps reduce the complexity of the on-lattice model. Elements of group theory ------------------------ A group :math:`G` is defined as a set of elements together with an operation. It must satisfy 4 group axioms: #. **Closure**: .. math:: a \cdot b \in G \quad \forall a,\; b \in G #. **Associativity**: .. math:: (a \cdot b) \cdot c = a \cdot (b \cdot c) \quad \forall a,\; b,\; c \in G #. **Identity element**: .. math:: \exists e \in G \quad \text{such that} \quad a \cdot e = e \cdot a = a \quad \forall a \in G #. **Inverse element**: .. math:: \exists g^{-1} \in G \quad \text{such that} \quad g \cdot g^{-1} = g^{-1} \cdot g = e \quad \forall g \in G A subgroup :math:`H` of a group :math:`G` is a subset of elements of :math:`G` satisfying the group axioms above. Given a group :math:`G` of size :math:`|G|=N` and a subgroup :math:`H` of size :math:`M`, one can partition :math:`G` into :math:`N/M` cosets. The partition is performed by left-multiplying (left-cosets) or right-multiplying (right-cosets) :math:`H` by the elements of :math:`G`. A subgroup :math:`N` is called normal if the left- and the right-cosets are equal: .. math:: N \; \text{normal if} \; g \cdot N = N \cdot g \quad \forall g \in G Crystal structure ----------------- Any crystal structure is composed of a ``lattice`` and a ``motif``. * The ``lattice`` is an infinite set of points (lattice points) regularly spaced. It is defined by 3 lattice vectors :math:`(\vec{a}_1, \, \vec{a}_2, \, \vec{a}_3)` that span the entire 3-dimensional space. The lattice exibits the translational symmetry character of the crystal structure. Any lattice point :math:`\vec{T}_\vec{n}` is defined as: .. math:: \vec{T}_\vec{n} = n_1 \times \vec{a}_1 + n_2 \times \vec{a}_2 + n_3 \times \vec{a}_3 n_1, n_2, n_3 \in \mathbb{Z} * The ``motif`` is the smallest pattern of atoms that is repeated at every lattice points. The motif is represented by a set of basis vectors :math:`\{\vec{b}_i\}` that points to the positions of each atom. Any basis vector :math:`\vec{b}_i` is defined as: .. math:: \vec{b}_i = x_{i1} \times \vec{a}_1 + x_{i2} \times \vec{a}_2 + x_{i3} \times \vec{a}_3 x_{i1}, x_{i2}, x_{i3} \in \mathbb{R} The crystal structure can be fully described by the ``unitcell``, which is the smallest repeating building block of a crystal. The unitcell is composed of the motif and its boundaries are defined by the lattice parameters. The motif is chosen such that all atoms lie in the unitcell, i.e. :math:`x_{i1}, x_{i2}, x_{i3} \in [0,1[`. It is convenient to write any atomic site :math:`\vec{s}` in the crystal structure in the so-called unitcell coordinates :math:`(n_1, n_2, n_3, m)` defined as: .. math:: \vec{s} = n_1 \times \vec{a}_1 + n_2 \times \vec{a}_2 + n_3 \times \vec{a}_3 + \vec{b}_m = \vec{T}_\vec{n} + \vec{b}_m n_1, n_2, n_3 \in \mathbb{Z} \, , \; m=1,2,\dots, |\{\vec{b}_i\}| .. figure:: figures/crystal.png :height: 400 :align: center Illustration of a 2D crystal structure containing 3 atoms in the unitcell. There are 2 symmetrically unique atomic sites, one containing atomic site 1 and the other containing atomic sites 2 and 3. An atomic site vector :math:`\vec{s}` can be splitted into a lattice translation :math:`\vec{T}_{\vec{n}}` and a basis vector :math:`\vec{b}_i`. Symmetry -------- The group of all symmetries that leave the crystal unchanged is called the ``spacegroup`` :math:`G`. It always contains a normal subgroup :math:`N` composed of all translations :math:`\vec{T}`. We can choose a set of coset representatives :math:`R=\{ r \}` such that the origin lies within the unitcell. The spacegroup can thus be partitioned as: .. math:: g = t_g \cdot r_g = r_g' \cdot t_g' \quad \forall g \in G \quad \text{with} \quad r_g, r_g' \in R, \; t_g, t_g' \in N The group action :math:`\alpha` of the spacegroup `G` on the set of atomic sites :math:`S` is defined as: .. math:: \alpha: \quad G \times S &\longrightarrow S \\ (g , \vec{s}) &\longmapsto g \cdot \vec{s} with: .. math:: g \cdot \vec{s} = (t_g \cdot r_g) \cdot (\vec{T}_\vec{n} + \vec{b}_m) = \vec{T}_{(t_g \cdot r_g) \cdot \vec{n}} + r_g \cdot \vec{b}_m = \vec{T}_\vec{n'} + \vec{b}_{m'} As a result, applying a symmetry operation on an atomic site might change translate to another unitcell (:math:`\vec{T}_\vec{n'}`) and change the basis vector (:math:`\vec{b}_{m'}`). Two atomic sites related by such a symmetry are called symmetrically equivalent. In the illustrative 2D crystal example above, applying a mirror reflection at the :math:`\vec{b}_2`-axis sends basis vectors 2 onto basis vectors 3 and vice versa, but leaves basis vectors 1 on basis vectors 1. It results that atomic sites 2 and 3 are symmetrically equivalent. Sites symmetrically equivalent to atomic site 1 belong to equivalency 1, while sites symmetrically equivalent to atomic site 2 (or equivalently to atomic site 3) belong to equivalency 2. Orbits ------ A ``cluster`` :math:`C` is a finite set :math:`\{\vec{s}_i\}` of distinct atomic sites in the crystal structure. We call a point cluster a cluster composed of only only 1 atomic site, a pair cluster a cluster composed of 2 atomic sites, ... The body order of a cluster is the number of atomic sites that belong to the cluster (1 for point clusters, 2 for pair clusters, ...). The group action :math:`\alpha` defined above can be applied to all sites belonging to the cluster :math:`C`. The ``stabilizer`` and the ``orbit`` are two important sets that result from the application of the group action on a cluster: * The ``stabilizer`` :math:`G_C` is the subgroup of the spacegroup that leaves the cluster :math:`C` unchanged (up to a change in order (= permutation of sites) of the atomic sites): .. math:: G_C = \{ g \in G \; : \; g \cdot C = C \} * The ``orbit`` :math:`G \cdot C` is the set of all clusters related to the cluster :math:`C` by a symmetry operation. In order to avoid repetitions, the :math:`G \cdot C` can be equivalently defined as the set of representatives of the cosets :math:`G / G_C` (`Orbit-Stabilizer Theorem `_). .. math:: G \cdot C = \{ g \cdot C , \; \forall g \in G \} = \big\{ g \in G \; : \; \bigcup_g g \cdot G_C = G \big\} Additionally, from :math:`G_C`, one can obtain the set :math:`\{ \pi \}` of invariant ``permutations`` since all possible orders the atomic sites can arrange are contained in :math:`G_C`. The permutations of the cluster :math:`C` characterize the symmetry of the cluster. As an example, any pair cluster between symmetrically equivalent sites shows inversion symmetry. As a result, one can permute the atomic sites (i.e., :math:`C = [s_1, s_2] = [s_2, s_1]`). .. math:: \{ \pi \} = \{ \pi \; : \; g \cdot [s_{i_1}, s_{i_2}, \dots , s_{i_{|C|}}] = [s_{i_{\pi(1)}}, s_{i_{\pi(2)}}, \dots , s_{i_{\pi(|C|)}}] \; ;& \\ \; [s_{i_1}, s_{i_2}, \dots , s_{i_{|C|}}] \in C , g \in G_C \}& ``Site-centric orbits`` :math:`G^i \cdot C` to atomic site :math:`\vec{s}_i` is defined as a subset of :math:`G_C` such that: .. math:: G^i \cdot C = \{ g \cdot C \; : \; \vec{s}_i \in g \cdot C , g \in G\} In the example illustrated below, the orbit of a triplet cluster ABC is shown. The spacegroup is composed of the identity, the horizontal mirrir, the vertical mirror, and the 180°-degree rotation operations in addition to all lattice translations. The stabilizer is composed of the identity and the horizontal mirror operations. The invariant permutations correponds to the order the sites follow after performing one of these two operations. It results that there are two possible invariant permutations that are (123) and (132), corresponding to the identity and the horizontal mirroroperations, respectively. The smallest orbit is composed of the clusters in the left columns, with representatives being the clusters resulting after the identity and the vertical mirror operations. It can be seen that this orbit is invariant to the spacegroup once all lattice translations are considered. The orbit is said **globally invariant**. It happens this orbit is also site-centric to atomic site A, but not to the other atomic sites. It is possible to define another orbit that considers all clusters that have at least one atom in the unitcell. This orbit corresponds to all clusters represented in the figure below (all three columns). The orbit is said **locally invariant** and ensures that the site-centric orbits of all three atoms composing the cluster are contained. .. figure:: figures/orbits.png :width: 1000 :align: center Illustration of an orbit of a cluster containing 3 atomic sites (sites A, B, and C). The identity and the horizontal mirror operations leave the cluster unchanged, up to a permutation of the atomic site B and C. These operations forms the stabilizer of the cluster. The orbit is composed of 2 variants of the cluster, one where the atomic site A points to the left and one where it points to the right. These two variants corresponds to the 2 cosets of the spacegroup with respect to the stabilizer subgroup (4 spacegroup operations (without acounting for the lattice translations) / 2 operations in the stabilizer = 2). The additional lattice translation are supplementary operations that accounts for all possible clusters that have at least one atom within the unitcell in order to have site-centric orbits. The first column represents the site-centric orbit for atomic site A, while the two other colomns constitutes the site-centric orbit for atomic site B and C. The fact that permuting atomic sites B and C do not change the cluster indicates that these two sites are symmetrically equivent within the cluster.