  We consider the problem of two coupled Luttinger liquids both at half filling
and at low doping levels, to investigate the problem of competing orders in
quasi-one-dimensional strongly correlated systems. We use bosonization and
renormalization group equations to investigate the phase diagrams, to determine
the allowed phases and to establish approximate boundaries among them. Because
of the chiral translation and reflection symmetry in the charge mode away from
half filling, orders of charge density wave (CDW) and spin-Peierls (SP)
diagonal current (DC) and $d$-density wave (DDW) form two doublets and thus can
be at most quasi-long range ordered. At half-filling, umklapp terms break this
symmetry down to a discrete group and thus Ising-type ordered phases appear as
a result of spontaneous breaking of the residual symmetries. Quantum disordered
Haldane phases are also found, with finite amplitudes of pairing orders and
triplet counterparts of CDW, SP, DC and DDW. Relations with recent numerical
results and implications to similar problems in two dimensions are discussed.
