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Adding interactions to many-body Hamiltonians of geometrically frustratedlattices often leads to diminished subspaces of localized states. In thispaper, we show how to construct interacting many-body Hamiltonians, startingfrom the non-interacting tight-binding Hamiltonians, that preserve or evenexpand these subspaces. The methods presented involve modifications in theone-body network representation of the many-body Hamiltonians which generatenew interacting terms in these Hamiltonians. The subspace of many-particlelocalized states can be preserved in the interacting Hamiltonian, by projectingthe interacting terms onto the subspace of many-body extended states or byconstructing the interacting Hamiltonian applying origami rules to the network.Expanded subspaces of localized states are found if interacting terms that mixsubspaces with different number of particles are introduced. Furthermore, wepresent numerical methods for the determination of many-body localized statesthat allows one to address larger clusters and larger number of particles thanthose accessible by full diagonalization of the interacting Hamiltonian. Thesemethods rely on the generalization of the concept of compact localized state inthe network. Finally, we suggest a method to determine localized states thatuse a considerable fraction of the network.

Methods for the construction of interacting many-body Hamiltonians with compact localized states in geometrically frustrated clusters