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Motivated by the work of Kitaev, we construct an exactly solublespin-$\frac{1}2$ model on honeycomb lattice whose ground states are identicalto $\Delta_{1x}p_x+\Delta_{1y}p_y+i(\Delta_{2x}p_x+\Delta_{2y}p_y)$-wave pairedfermions on square lattice, with tunable paring order parameters. We derive auniversal phase diagram for this general p-wave theory which contains a gappedA phase and a topologically non-trivial B phase. We show that the gaplesscondition in the B phase is governed by a generalized inversion (G-inversion)symmetry under $p_x\leftrightarrow {\Delta_{1y}\over \Delta_{1x}} p_y$. TheG-inversion symmetric gapless B phase near the phase boundaries is described by1+1-dimensional gapless Majorana fermions in the asymptotic long wave lengthlimit, i.e. the $c=1/2$ conformal field theory. The gapped B phase hasG-inversion symmetry breaking and is the weak pairing phase described by theMoore-Read Pfaffian. We show that in the gapped B phase, vortex pairexcitations are separated from the ground state by a finite energy gap.

An exactly soluble model with tunable p-wave paired fermion ground states