We outline the basic ideas involved in a recently proposed [17] derivation of a gauge theory for underdoped cuprates in the “spin gap phase”, performed essentially step by step starting from the t — J model, considered as a model Hamiltonian for the CuO2 layers. The basic tool is the U(1) x SU(2) Chern-Simons bosonization, to which it is dedicated a somewhat detailed discussion. The basic output is a “spingap” not vanishing in any direction and an antiferromagnetic correlation length proportional to the inverse square root of doping concentration, in agreement with data deduced from the neutron experiments. The model also exhibits a small half-pocket Fermi surface around (±π/2, ±π/2) and a linear in temperature dependence of in-plane resistivity in certain temperature range.
We consider the large-
N Chern-Simons theory for attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian collective-field approach based on the
1/N expansion. We show that the dynamics of low-lying density excitations around the ground-state vortex configuration is equivalent to that of the Sutherland model. The correspondence between the Chern-Simons coupling constant $\ensuremath{\lambda}$ and the Calogero-Sutherland statistical parameter ${\ensuremath{\lambda}}_{s}$ signals some sort of statistical transmutation accompanying the dimensional reduction of the initial problem.
We study $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory with integral coupling constants
(k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus
g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that
Z2 vortices in the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of
n pairs of
Z2 vortices on a sphere. These results allow us to show that
l=3 $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory is the low-energy effective theory for the
Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction $\ensuremath{\nu}=\frac{2}{2k\ensuremath{-}3}$. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ theory is more useful than an alternative $SU{(2)}_{4}\ifmmode\times\else\texttimes\fi{}U(1)∕U(1)$ Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the $\ensuremath{\nu}=2∕3$ phase diagram.