Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of
quantum Hall/Heisenberg ferromagnets at filling factor
M for
L Landau/lattice sites with the carrier space of irreducible representations of
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Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of
quantum Hall/Heisenberg ferromagnets at filling factor
M for
L Landau/lattice sites with the carrier space of irreducible representations of
described by rectangular Young tableaux of
M rows and
L columns, and associated with Grassmannian phase spaces
. We embed this
N-component fermion mixture in Fock space through a Schwinger–Jordan (boson and fermion) representation of
-spin operators. We provide different realizations of basis vectors using Young diagrams, Gelfand–Tsetlin patterns and Fock states (for an electron/flux occupation number in the fermionic/bosonic representation).
-spin operator matrix elements in the Gelfand–Tsetlin basis are explicitly given. Coherent state excitations above the ground state are computed and labeled by complex
matrix points
Z on the Grassmannian phase space. They adopt the form of a
displaced/rotated highest-weight vector, or a multinomial Bose–Einstein condensate in the flux occupation number representation. Replacing
-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength)
-spin-wave coherent excitations (skyrmions) of
quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.
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