  Matrix product states (MPS) have proven to be a very successful tool to study
lattice systems with local degrees of freedom such as spins or bosons.
Topologically ordered systems can support anyonic particles which are labeled
by conserved topological charges and collectively carry non-local degrees of
freedom. In this paper we extend the formalism of MPS to lattice systems of
anyons. The anyonic MPS is constructed from tensors that explicitly conserve
topological charge. We describe how to adapt the time-evolving block decimation
(TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local
and charge-conserving Hamiltonian. To demonstrate the effectiveness of anyonic
TEBD algorithm, we used it to simulate (i) the ground state (using imaginary
time evolution) of an infinite 1D critical system of (a) Ising anyons and (b)
Fibonacci anyons both of which are well studied, and (ii) the real time
dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a
ladder geometry with an anyonic flux threading each island of the ladder. Our
results pertaining to (ii) give insight into the transport properties of
anyons. The anyonic MPS formalism can be readily adapted to study systems with
conserved symmetry charges, as this is equivalent to a specialization of the
more general anyonic case.
