Quantum entanglement is the most surprising feature of quantum
mechanics, and plays a crucial role in quantum computation. Ground states of
quantum many-body systems typically exhibit the area law behavior in the
entanglement entropy, which measures the amount of entanglement between a
subsystem and the rest of the system. Recently, a class of solvable
one-dimensional spin models with local interactions has been constructed by
Mavassagh and Shor and by Salberger and Korepin, in which the ground state is
expressed as a superposition of random walks, and has much larger entanglement.
Its entanglement entropy is shown to be proportional to the square root of the
volume. In this talk, after a brief review of the models, we construct an
extension of these models based on the symmetric inverse semigroup, and discuss
properties of ground states with the entanglement entropy. If time permits, we
would like to discuss localization properties of excited states, which are new
features arising by the extension.