  The quantum spin-1/2 antiferromagnetic Heisenberg model on a two dimensional
triangular lattice geometry with spatial anisotropy is relevant to describe
materials like ${\rm Cs_2 Cu Cl_4}$ and organic compounds like
{$\kappa$-(ET)$_2$Cu$_2$(CN)$_3$}. The strength of the spatial anisotropy can
increase quantum fluctuations and can destabilize the magnetically ordered
state leading to non conventional spin liquid phases. In order to understand
these intriguing phenomena, quantum Monte Carlo methods are used to study this
model system as a function of the anisotropic strength, represented by the
ratio $J'/J$ between the intra-chain nearest neighbor coupling $J$ and the
inter-chain one $J'$. We have found evidence of two spin liquid regions. The
first one is stable for small values of the coupling $J'/J \alt 0.65$, and
appears gapless and fractionalized, whereas the second one is a more
conventional spin liquid with a small spin gap and is energetically favored in
the region $0.65\alt J'/J \alt 0.8$. We have also shown that in both spin
liquid phases there is no evidence of broken translation symmetry with dimer or
spin-Peirls order or any broken spatial reflection symmetry of the lattice. The
various phases are in good agreement with the experimental findings, thus
supporting the existence of spin liquid phases in two dimensional quantum
spin-1/2 systems.
