  We study the modular Hamiltonians of an interval for the massless Dirac
fermion on the half-line. The most general boundary conditions ensuring the
global energy conservation lead to consider two phases, where either the vector
or the axial symmetry is preserved. In these two phases we derive the
corresponding modular Hamiltonian in explicit form. Its density involves a
bi-local term localised in two points of the interval, one conjugate to the
other. The associated modular flows are also established. Depending on the
phase, they mix fields with different chirality or charge that follow different
modular trajectories. Accordingly, the modular flow preserves either the vector
or the axial symmetry. We compute the two-point correlation functions along the
modular flow and show that they satisfy the Kubo-Martin-Schwinger condition in
both phases. The entanglement entropies are also derived.
