  Few-body physics for anyons has been intensively studied within the
anyon-Hubbard model, including the quantum walk and Bloch oscillations of
two-anyon states. However, the known theoretical proposal and experimental
simulations of two-anyon states in one-dimensional lattice have been carried
out by expanding the wavefunction in terms of non-orthogonal basis vectors,
which introduces extra non-physical degrees of freedom. In the present work, we
deduce the finite difference equations for the two-anyon state in the
one-dimensional lattice by solving the Schr\"odinger equation with orthogonal
basis vectors. Such an orthogonal scheme gives all the orthogonal physical
eigenstates for the time-independent two-anyon Schr\"odinger equation, while
the conventional (non-orthogonal) method produces a lot of non-physical
redundant eigen-solutions whose components violate the anyonic relations. The
dynamical property of the two-anyon states in a sufficiently large lattice has
been investigated and compared in both the orthogonal and conventional schemes,
which proves to depend crucially on the initial states. When the initial states
with two anyons on the same site or (next-)neighboring sites are suitably
chosen to be in accordance with the anyonic coefficient relation, we observe
exactly the same dynamical behavior in the two schemes, including the revival
probability, the probability density function, and the two-body correlation,
otherwise, the conventional scheme will produce erroneous results which not any
more describe anyons. The period of the Bloch oscillation in the
pseudo-fermionic limit is found to be twice that in the bosonic limit, while
the oscillations disappear for statistical parameters in between. Our findings
are vital for quantum simulations of few-body physics with anyons in the
lattice.
