In this thesis we aim to show that single-body fermionic systems on a lattice can exhibit quantum chaotic behaviour. Up until now, quantum chaos on a lattice was usually studied only for interacting many-body systems and single-body systems with disorder. We investigate whether irregular boundaries of a two-dimensional lattice break the integrability of the translationally invariant free fermion model on a square lattice. In the first part of the thesis we describe the measures of quantum chaos: the agreement of eigenstate statistics with random matrix predictions, the form of the average von Neumann and 2. Rényi entanglement entropy of many-body Hamiltonian eigenstates, and the agreement of the form of the matrix elements of local observables with the eigenstate thermalization hypothesis ansatz. We define discrete billiard systems - single-body fermionic systems on a two-dimensional lattice with a boundary - and describe their generic properties. I introduce three discrete billiards, which are classically chaotic as well as quantum chaotic in their continuous limit, and test the measures of quantum chaos for them.
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