Quantum Theory of Landau and Peierls Electrons From the Central Extensions of Their Symmetry Groups

Abstract

By Wigner’s theorem on symmetries, the total state space of a quantum system whose symmetries form the group G is the collection of all projective unitary representations of G; these are, in turn, realised as certain unitary representations of the set of all central extensions of G by U(1). Exploiting this relationship, we present in this paper a new approach to the quantum mechanics of an electron in a uniform magnetic field B, in the plane (the Landau electron) and on the 2-torus in the presence of a periodic potential V whose periodicity is that of an N×N lattice (the Peierls electron). For the Landau electron, G is the Euclidean group E(2) whose central extensions arise from the Heisenberg Lie group central extensions, determined by B, of the translation subgroup. The state space is a unitary representation of the direct product of two such groups corresponding to B and -B and the Hamiltonian is a unique element of the universal enveloping algebra of the centrally-extended E(2). The complete quantum theory of the Landau electron follows directly. For the Peierls electron, lattice translation-invariance is possible only if the flux per unit cell Φ takes rational values with denominator N. The state space is a unitary representation of the direct product of a finite Heisenberg group, which is a central extension of the translation group, and a Heisenberg Lie group, both characterised by Φ. The following new results are rigorous consequences. In the empty lattice limit V=0, the energy spectrum is the Landau spectrum with degeneracy equal to the total flux through the sample. As V moves away from zero, every Landau level splits into NΦ discrete sublevels, each of degeneracy N. More generally, for V≠0 of any strength and (periodic) form, and B such that Φ is nonintegral, every point in the spectrum has multiplicity N. The degeneracy is thus proportional to the linear size rather than the area of the sample. Throughout the paper, vector potentials and gauges are dispensed with and many misconceptions thereby removed.

Product of Projective Representations in Description of Multi-Electron States in An External Magnetic Field

“…as a central extension of the translation group. Similar approach was also proposed by Divakaran and Rajagopal [10]. These investigations gave a basis to construct and take into account all representations, including those considered previously by Zak [4] as "non-physical" [11].…”

The Landau electron problem on a cylinder

“…( 1), we can write a cocycle corresponding to c as γ(g, h) = e iβ(g,h) . But β is not uniquely defined by (5). In many (but not all [4]) cases, we can choose β itself to be antisymmetric: β = 1 2 α, which is a canonical choice.…”

“…We expect the symmetry group of a charged particle moving on the plane with a uniform magnetic field B, perpendicular to it to be the Euclidean group, E 2 . To apply the general procedure outlined above to this case, we note that central extensions of E 2 are completely determined by those of its translation subgroup T [5]. T is the group of two dimensional vectors, x, y, • • •, under addition.…”

“…The belief that the fundamental physical quantity is the field itself -which is translationally invariantrather than the vector potential -which is not invariant -is thereby vindicated. Gauges and gauge transformations turn out to be artifacts of the many equivalent ways of choosing the 2-cocycle and of constructing the unique irreducible URs of Tλ [5] and can be very naturally accommodated.…”

The role of a form of vector potential — normalization of the antisymmetric gauge

“…Nevertheless, it seems that the representations introduced by Brown (10) could be used in Zak's approach to construct (vector) representations of T ′ (finite or not) according to (1). A comparison of (10) and (13) shows that for odd N Brown and Zak used different representations. However, for even N (b = 2) we have…”