Hilbert space and ground state structure of bilayer quantum Hall systems at $\nu=2/\lambda$

We analyze the Hilbert space and ground state structure of bilayer quantum Hall (BLQH) systems at fractional filling factors $\nu=2/\lambda$ ($\lambda$ odd) and we also study the large $SU(4)$ isospin-$\lambda$ limit. The model Hamiltonian is an adaptation of the $\nu=2$ case [Z.F. Ezawa {\it et al.}, Phys. Rev. {B71} (2005) 125318] to the many-body situation (arbitrary $\lambda$ flux quanta per electron). The semiclassical regime and quantum phase diagram (in terms of layer distance, Zeeeman, tunneling, etc, control parameters) is obtained by using previously introduced Grassmannian $\mathbb{G}^4_{2}=U(4)/[U(2)\times U(2)]$ coherent states as variational states. The existence of three quantum phases (spin, canted and ppin) is common to any $\lambda$, but the phase transition points depend on $\lambda$, and the instance $\lambda=1$ is recovered as a particular case. We also analyze the quantum case through a numerical diagonalization of the Hamiltonian and compare with the mean-field results, which give a good approximation in the spin and ppin phases but not in the canted phase, where we detect exactly $\lambda$ energy level crossings between the ground and first excited state for given values of the tunneling gap. An energy band structure at low and high interlayer tunneling (spin and ppin phases, respectively) also appears depending on angular momentum and layer population imbalance quantum numbers.