Abstract
It is argued that there are a set of orthonormal basis states, which appear as highly degenerate ground states arising from spontaneous symmetry breaking with a type-B Goldstone mode, and they are scale-invariant, with a salient feature that the entanglement entropy S(n) scales logarithmically with the block size n in the thermodynamic limit. As it turns out, the prefactor is half the number of type-B Goldstone modes NB. This is achieved by performing an exact Schmidt decomposition of the orthonormal basis states, thus unveiling their self-similarities in the real space-the essence of a fractal. Combining with a field-theoretic prediction (Castro-Alvaredo and Doyon 2012 Phys. Rev. Lett. 108 120401), we are led to the identification of the fractal dimension df with the number of type-B Goldstone modes NB for the orthonormal basis states in quantum many-body systems undergoing spontaneous symmetry breaking.