Paper Info
Title
Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit
Abstract
Using the Lieb-Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites with the carrier space of irreducible representations of U(N) described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M) x U(N - M). We embed this N-component fermion mixture in Fock space through a Schwinger-Jordan (boson and fermion) representation of U(N)-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). U(N)-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 (N - M) x M matrix points Z on the Grassmannian phase space. They adopt the form of a U(N) displaced/rotated highest-weight vector, or a multinomial Bose-Einstein condensate in the flux occupation number representation. Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength) U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.
Year
DOI
Venue
2022
10.3390/sym14050872
SYMMETRY-BASEL
Keywords
DocType
Volume
N-component fermion mixtures, quantum Hall ferromagnets, unitary group representations, boson Schwinger-Jordan realizations, Young tableaux, Lieb-Mattis theorem, Grassmannian sigma models
Journal
14
Issue
ISSN
Citations 
5
2073-8994
0
PageRank 
References 
Authors
0.34
0
3
Name
Order
Citations
PageRank
Manuel Calixto110.77
Alberto Mayorgas210.77
Julio Guerrero310.77