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($M$)$\times$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 $(N-M)\times M$ 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.

24 pages, no figures. Version to appear in the Special Issue "Topological Spin Textures: From Fundamentals to Applications" of Symmetry

References in corpus (21)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Correlated Insulator Behaviour at Half-Filling in Magic Angle Graphene Superlattices
• Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions
• Identifying Topological Order by Entanglement Entropy
• Ultracold Fermi Gases with Emergent SU(N) Symmetry
• Ultracold fermions and the SU(N) Hubbard model
• Ultracold Gases of Ytterbium: Ferromagnetism and Mott States in an SU(6) Fermi System
• Collective Modes and Skyrmion Excitations in Graphene SU(4) Quantum Hall Ferromagnets
• Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
• Inverse participation ratio and localization in topological insulator phase transitions
• Identifying topological-band insulator transitions in silicene and other 2D gapped Dirac materials by means of Rényi-Wehrl entropy
• Band inversion at critical magnetic fields in a silicene quantum dot
• Application of path-integral quantization to indistinguishable particle systems topologically confined by a magnetic field
• Magnetotransport Study of the Canted Antiferromagnetic Phase in Bilayer $\nu =2$ Quantum Hall State
• Role of mixed permutation symmetry sectors in the thermodynamic limit of critical three-level Lipkin-Meshkov-Glick atom models
• Entropic uncertainty relations and topological-band insulator transitions in 2D gapped Dirac materials
• Hilbert space and ground state structure of bilayer quantum Hall systems at $\nu =2/\lambda$
• Husimi function and phase-space analysis of bilayer quantum Hall systems at $\nu =2/\lambda$
• Quantum mechanism of extremely high energy processes at neutron star collapse and of quasar luminosity
• Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin-Meshkov-Glick D-level atom models

Cited by in corpus (1)

• Lieb-Mattis ordering theorem of electronic energy levels in the thermodynamic limit