  We reveal that local interactions in graphene allow novel spin liquids
between the semi-metal and antiferromagnetic Mott insulating phases, identified
with algebraic spin liquid and Z$_{2}$ spin liquid, respectively. We argue that
the algebraic spin liquid can be regarded as the two dimensional realization of
one dimensional spin dynamics, where antiferromagnetic correlations show
exactly the same power-law dependence as valence bond correlations. Nature of
the Z$_{2}$ spin liquid turns out to be $d + i d'$ singlet pairing, but time
reversal symmetry is preserved, taking $d + i d'$ in one valley and $d - i d'$
in the other valley. We propose the quantized thermal valley Hall effect as an
essential feature of this gapped spin liquid state. Quantum phase transitions
among the semi-metal, algebraic spin liquid, and Z$_{2}$ spin liquid are shown
to be continuous while the transition from the Z$_{2}$ spin liquid to the
antiferromagnetic Mott insulator turns out to be the first order. We emphasize
that both algebraic spin liquid and $d \pm id'$ Z$_{2}$ spin liquid can be
verified by the quantum Monte Carlo simulation, showing the enhanced symmetry
in the algebraic spin liquid and the quantized thermal valley Hall effect in
the Z$_{2}$ spin liquid.
