# Hilbert space structure of the low energy sector of U(N) quantum Hall ferromagnets and their classical limit

## TL;DR

This paper explores the mathematical structure of low energy states in U(N) quantum Hall ferromagnets, linking Hilbert space representations with Grassmannian phase spaces and providing a semi-classical description of excitations.

## Contribution

It introduces a detailed representation of the low energy sector using Young tableaux, Gelfand-Tsetlin patterns, and coherent states, connecting quantum states with Grassmannian sigma models.

## Key findings

- Hilbert space of low energy states identified with U(N) irreducible representations.
- Explicit matrix elements of U(N)-spin operators in Gelfand-Tsetlin basis provided.
- Semi-classical description of skyrmion excitations via Grassmannian coherent states.

## 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.