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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 $\mathbb G_2^4=U(4)/U(2)^2$, orthonormal basis, $U(4)$ generators and their matrix elements in the reproducing kernel Hilbert space $\mathcal H_λ(\mathbb G_2^4)$ of analytic square-integrable holomorphic functions on $\mathbb G_2^4$, which carries a unitary irreducible representation of $U(4)$ with index $λ\in\mathbb N$. 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 $\mathbb G_2^4$-CS can be regarded as a generalized (matrix) version.