Hyperspectral unmixing is one of the crucial steps for many hyperspectral applications. The problem of hyperspectral unmixing has proved to be a difficult task in unsupervised work settings where the endmembers and abundances are both unknown. In addition, this task becomes more challenging in the case that the spectral bands are degraded by noise. This paper presents a robust model for unsupervised hyperspectral unmixing. Specifically, our model is developed with the correntropy-based metric where the nonnegative constraints on both endmembers and abundances are imposed to keep physical significance. Besides, a sparsity prior is explicitly formulated to constrain the distribution of the abundances of each endmember. To solve our model, a half-quadratic optimization technique is developed to convert the original complex optimization problem into an iteratively reweighted nonnegative matrix factorization with sparsity constraints. As a result, the optimization of our model can adaptively assign small weights to noisy bands and put more emphasis on noise-free bands. In addition, with sparsity constraints, our model can naturally generate sparse abundances. Experiments on synthetic and real data demonstrate the effectiveness of our model in comparison to the related state-of-the-art unmixing models.