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

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 47; no. 11; pp. 115302 - 115321
• Main Authors: Calixto, M, Pérez-Romero, E
• Format: Journal Article
• Language: English
• Published: IOP Publishing 21.03.2014
• Subjects: bilayer fractional quantum Hall effect; coherent states; composite fermion; Grassmannian coset; oscillator realization
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/47/11/115302

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 , orthonormal basis, U(4) generators and their matrix elements in the reproducing kernel Hilbert space of analytic square-integrable holomorphic functions on , which carries a unitary irreducible representation of U(4) with index . 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 to make 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 is also treated in parallel, of which Grassmannian -CS can be regarded as a generalized (matrix) version.