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.
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