Coherent states on the Grassmannian U(4)/U(2)²: Oscillator realization and bilayer fractional quantum Hall systems

Bilayer quantum Hall (BLQH) systems, which underlie a U(4) symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian mathbb G₂⁴=U(4)/U(2)², orthonormal basis, U(4) generators and their matrix elements in the reproducing kernel Hilbert space mathcal H_λ\(mathbb G₂⁴\) of analytic square-integrable holomorphic functions on mathbb G₂⁴, which carries a unitary irreducible representation of U(4) with index λinmathbb N. A many-body representation of the previous construction is introduced through an oscillator realization of the U(4) Lie algebra generators in terms of eight boson operators. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index λ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor ν=2. The simpler, and better known, case of spin-s CS on the Riemann-Bloch sphere mathbb{S}²=U(2)/U(1)² is also treated in parallel, of which Grassmannian mathbb G₂⁴-CS can be regarded as a generalized (matrix) version.