  We investigate the low-lying excitation spectrum and ground-state properties
of narrow graphene nanoribbons with zigzag edge configurations. Nanoribbons of
comparable widths have been synthesized very recently [P. Ruffieux, \emph{et
al.} Nature \textbf{531}, 489 (2016)], and their descriptions require more
sophisticated methods since in this regime conventional methods, like
mean-field or density-functional theory with local density approximation, fail
to capture the enhanced quantum fluctuations. Using the unbiased density-matrix
renormalization group algorithm we calculate the charge gaps with high accuracy
for different widths and interaction strengths and compare them with mean-field
results. It turns out that the gaps are much smaller in the former case due to
the proper treatment of quantum fluctuations. Applying the elements of quantum
information theory we also reveal the entanglement structure inside a ribbon
and examine the spectrum of subsystem density matrices to understand the origin
of entanglement. We examine the possibility of magnetic ordering and the effect
of magnetic field. Our findings are relevant for understanding the gap values
in different recent experiments and the deviations between them.
