Abstract
The Majorana lattice gauge theory purely composed of Majorana fermions on square lattice is studied throughly. The ground state is obtained exactly and exhibits the coexistence of symmetry breaking and topological order. The Z_2Z2 symmetry breaking of matter fields leads to the intertwined antiferromagnetic spin order and \etaη-pairing order. The topological order is reflected in the Z_2Z2 quantum spin liquid ground state of gauge fields. The Majorana lattice gauge theory, alternatively can be viewed as interacting Majorana fermion model, is possibly realized on a Majorana-zero-mode lattice.
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