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Mutual Composite Fermion and composite Boson approaches to balanced and imbalanced bilayer quantum Hall system: an electronic analogy of the Helium 4 system

We use both Mutual Composite Fermion (MCF) and Composite Boson (CB) approach to study balanced and im-balanced Bi-Layer Quantum Hall systems (BLQH) and make critical comparisons between the two approaches. We find the CB approach is superior to the MCF approach in studying ground states with different kinds of broken symmetries. In the phase representation of the CB theory, we first study the Excitonic superfluid state (ESF). The theory puts spin and charge degree freedoms in the same footing, explicitly bring out the spin-charge connection and classify all the possible excitations in a systematic way. Then in the dual density representation of the CB theory, we study possible intermediate phases as the distance increases. We propose there are two critical distances $ d_{c1} < d_{c2} $ and three phases as the distance increases. When $ 0 < d < d_{c1} $, the system is in the ESF state which breaks the internal $ U(1) $ symmetry, when $ d_{c1} < d < d_{c2} $, the system is in an Pseudo-spin density wave (PSDW) state which breaks the translational symmetry, there is a first order transition at $ d_{c1} $ driven by the collapsing of magneto-roton minimum at a finite wavevector in the pseudo-spin channel. When $ d_{c2} < d < \infty $, the system becomes two weakly coupled $ ν=1/2 $ Composite Fermion Fermi Liquid (FL) state. There is also a first order transition at $ d= d_{c2} $. We construct a quantum Ginzburg Landau action to describe the transition from ESF to PSDW which break the two completely different symmetries. By using the QGL action, we explicitly show that the PSDW takes a square lattice and analyze in detail the properties of the PSDW at zero and finite temperature.