Broken Symmetry of Two-Component $\mathit{\nu }=1/2$ Quantum Hall States

We show that the $\nu =1/2\mathrm{}$ quantum Hall states in bilayer systems are triplet $p$-wave pairing states of composite fermions exactly the same as ${}^3$He superfluids. The observed persistence (though weakening) of the $1/2$ state in the two- to one-component crossover region is identical to the well known $A$ to $A_1$ transition in ${}^3$He. This illustrates the remarkable phenomenon of “incompressible deformation” of quantum Hall states. The broken symmetry of the $\nu =1/2\mathrm{}$ state is a “pairing” vector $\mathbf{d}$, which implies a “pseudo” magnetization $\propto \mathit{i}\mathbf{d}\times {\mathbf{d}}^{*}$. In the presence of layer tunneling, the (331) state ( $\mathbf{d}$ real) is unstable against other states with “magnetization” ( $\mathbf{d}$ complex).