Some properties of Grassmannian $U(4)/U(2)^2$ coherent states and an entropic conjecture

TL;DR

This paper studies the properties of $U(4)$ coherent states, extending spin coherent states with an additional degree of freedom, and explores their applications in quantum Hall systems and entropy bounds.

Contribution

It introduces a matrix version of spin $U(2)$ coherent states with a pseudospin degree of freedom and extends Wehrl's entropy to Grassmannian coherent states, proposing a new entropy conjecture.

Findings

Analysis of $U(4)$ coherent states properties

Application to bilayer quantum Hall systems

Proposed entropy lower bound conjecture

Abstract

We analyze mathematical and physical properties of a previously introduced [J. Phys. A47, 115302 (2014)] family of $U(4)$ coherent states (CS). They constitute a matrix version of standard spin $U(2)$ CS when we add an extra (pseudospin) dichotomous degree of freedom: layer, sublattice, two-well, nucleon, etc. Applications to bilayer quantum Hall systems at fractions of filling factor $\nu=2$ are discussed, where Haldane's sphere picture is generalized to a Grassmannian picture. We also extend Wehrl's definition of entropy from Glauber to Grassmannian CS and state a conjecture on the entropy lower bound.