Rényi entropy of highly entangled spin chains

Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an "area law" which means that the entanglement entropy is proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. In this paper, we first analytically compute the Rényi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Rényi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Rényi entropy is known as a function of its parameter. We find nonanalytic behavior of the Rényi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Rényi entropy of Rokhsar-Kivelson states in two dimensions.