  We present a study of the scaling behavior of the R\'{e}nyi entanglement
entropy (REE) in SU($N$) spin chain Hamiltonians, in which all the spins
transform under the fundamental representation. These SU($N$) spin chains are
known to be quantum critical and described by a well known Wess-Zumino-Witten
(WZW) non-linear sigma model in the continuum limit. Numerical results from our
lattice Hamiltonian are obtained using stochastic series expansion (SSE)
quantum Monte Carlo for both closed and open boundary conditions. As expected
for this 1D critical system, the REE shows a logarithmic dependence on the
subsystem size with a prefector given by the central charge of the SU($N$) WZW
model. We study in detail the sub-leading oscillatory terms in the REE under
both periodic and open boundaries. Each oscillatory term is associated with a
WZW field and decays as a power law with an exponent proportional to the
scaling dimension of the corresponding field. We find that the use of periodic
boundaries (where oscillations are less prominent) allows for a better estimate
of the central charge, while using open boundaries allows for a better estimate
of the scaling dimensions. For completeness we also present numerical data on
the thermal R\'{e}nyi entropy which equally allows for extraction of the
central charge.
