  We write down a class of two-dimensional quantum spin-1/2 Hamiltonians whose
eigenspectra are exactly solvable via the Jordan-Wigner transformation. The
general structure corresponds to a suitable grid composed of XY or XX-Ising
spin chains and ZZ-Ising spin chains and is generalizable to higher dimensions.
They can host stacks of one-dimensional spinless fermion liquids with gapless
excitations and power-law correlations coexisting with ordered spin moments
(localized spinless fermions). Bond-dependent couplings thus can be an
alternate mechanism than geometric frustration of SU(2)-symmetric couplings to
obtain spinless fermionic excitations. Put in a different way, bond-dependent
couplings allow for an embedding of one-dimensional spinless fermion
(Tomonoga-Luttinger) liquids and solids and also Majorana excitations in higher
dimensions. They can accommodate a simpler set of site-local conserved
quantities apart from the more intricate, interlocked set of plaquette-local or
bond-local conserved quantities in Kitaev's honeycomb model with Majorana
excitations. The proposed grid structure may provide an architecture for
quantum engineering with controllable qubits.
