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The Majorana lattice gauge theory purely composed of Majorana fermions onsquare lattice is studied throughly. The ground state is obtained exactly andexhibits the coexistence of symmetry breaking and topological order. The $Z_2$symmetry breaking of matter fields leads to the intertwined antiferromagneticspin order and $\eta$-pairing order. The topological order is reflected in the$Z_2$ quantum spin liquid ground state of gauge fields. The Majorana latticegauge theory, alternatively can be viewed as interacting Majorana fermionmodel, is possibly realized on a Majorana-zero-mode lattice.

Majorana lattice gauge theory: symmetry breaking, topological order and intertwined orders all in one