Enrique Daltabuit^{4} and Harold V. McIntosh
Instituto Politécnico Nacional
Centro de Investigación y de Estudios Avanzados
Para estudiar los grupos de simetría de una red cristalográfica en presencia de un campo magnético deben considerarse las representaciones proyectivas de dichos grupos, debido al cambio de fase que sufren las funciones de onda y que proviene de las transformaciones de norma que acompañan a las simetrías geométricas. Estas representaciones proyectivas son representaciones ordinarias de un grupo que es isomórfico a los grupos generalizados de Dirac. Las representaciones irreducibles y las funciones adaptadas a la simetría de dichos grupos se puede obtener mediante técnicas que resultan del método de representaciones inducidas para los grupos que se pueden exhibir como productos semi-directos.
In studying the symmetry groups of a crystal lattice in the presence of a magnetic field, the projective representations of these groups must be considered, because of the phase change in the wave functions which is due to the gauge transformations which accompany the geometric symmetries. These projective representations are ordinary representations of a group which is isomorphic to the generalized Dirac groups. The irreducible representations of the symmetry adapted functions of these groups can be obtained by means of techniques arising from the method of induced representations for those groups which can be exhibited as semi-direct products.
If we transform the vector potential of a magnetic field in such a way that
Thus, while such a transformation alters the Hamiltonian H = (p- e/c A)^{2} + V(x,y,z) and thus is not a symmetry of the problem, it nevertheless does not affect any observable quantities which depend upon the absolute value of the wave function. A symmetry operation for such a Hamiltonian thus consists of the gauge transformation (1) together with the corresponding phase change (2).
Also we know that [2] coordinate transformations such as translations, rotations or reflections, or combinations of these, induce changes in the vector potentials that in some cases can be gauge transformations of the form (1), so that these transformations are not strictly symmetry operations even in the presence of a symmetric potential, although it is possible to compensate their effect with an appropriate gauge transformation.
Such transformations change the wave function in such a way that
The most general symmetry in the case of a uniform magnetic field consists of a general translation together with an arbitrary rotation about the field direction and a possible reflection perpendicular to the field axis.
To prove this let us remember that Harper [3] proved that any translation is a symmetry and that the corresponding gauge transformation is
Now if R is the matrix that represents a transformation of coordinates formed by a rotation together with a reflection, the vector potential that appears after such a transformation is
Now, the integrability conditions for ,
that is the conditions which must be satisfied so that we can write (8) in the form of (1), are
Now breaking M into its symmetric and antisymmetric parts, M=M^{s}+M^{a},
(9') reduces to
So we see that the integrability condition is that the antisymmetric part of the vector potential matrix commutes with the transformation matrix, which means that they have a common set of eigenvectors. The real eigenvector of M^{a},
Now, introducing the integrability condition (10) in (8), we see that
(15) |
D(t_{1})D(t_{2}) | = | ||
(17) | |||
D(t_{1}+t_{2}) | = |
Now let us turn our attention to the group of transformations whose matrices obey the integrability condition (10). This group is formed by rotations about the axis of the field and reflections perpendicular to the field axis.
We see that the matrices
Thus it is seen that this representation is also projective. We moreover see that
Let us now turn our attention toward the structure of the general symmetry group. If we apply a rotation or reflection to a vector r, followed by a translation, the resulting vector is
r'' = R_{2}r'+t_{2} = R_{2}R_{1}r+R_{2}t_{1}+t_{2} | (24) |
In view of this, we notice that the group is a semidirect product of the group of translations and the group of rotations-reflections, considering the latter group as an automorphism group for the first.
Returning to (15), we see that
which shows that the representation of the entire group is also projective.
Since the semidirect product group as a whole is non-commutative one cannot immediately deduce a commutation rule for the elements of the type which defines a ``Dirac group'' (ref. [4]), but bearing in mind that we have a projective representation,
a_{1} | = | ||
(25) | |||
a_{2} | = |
In summary, we have laid the groundwork to show that when the change of phase due to gauge transformation is taken into account, one obtains a projective representation of an appropriate two-dimensional lattice group as the symmetry group of a particle moving in a periodic potential in the presence of a uniform magnetic field.
Given such a projective representation, one can proceed to determine the possible irreducible representations to which it might correspond by deducing appropriate exchange relations and applying the theory of Dirac groups.