  Recent works have established universal entanglement properties and
demonstrated validity of single-particle eigenstate thermalization in
quantum-chaotic quadratic Hamiltonians. However, a common property of all
quantum-chaotic quadratic Hamiltonians studied in this context so far is the
presence of random terms that act as a source of disorder. Here we introduce
tight-binding billiards in two dimensions, which are described by
non-interacting spinless fermions on a disorder-free square lattice subject to
curved open (hard-wall) boundaries. We show that many properties of
tight-binding billiards match those of quantum-chaotic quadratic Hamiltonians:
the average entanglement entropy of many-body eigenstates approaches the random
matrix theory predictions and one-body observables in single-particle
eigenstates obey the single-particle eigenstate thermalization hypothesis. On
the other hand, a degenerate subset of single-particle eigenstates at zero
energy (i.e., the zero modes) can be described as chiral particles whose
wavefunctions are confined to one of the sublattices.
