In recent years, there has been a surge of activities in proposing "exactly solvable" quantum spin chains with surprising high amount of ground state entanglement--exponentially more than critical systems that have $\log(n)$ von Neumann entropy. We discuss these models from first principles. For a spin chain of length $n$, we prove that the ground state entanglement entropy is $\sqrt(n)$ and in some cases even extensive (i.e., extensive $n$) despite the underlying Hamiltonian being: (1) Local (2) Having a unique ground state and (3) Translationally invariant in the bulk. These models have rich connections with combinatorics, random walks, Markov chains, and universality of Brownian excursions. Lastly, we develop techniques for proving the gap. As a consequence, the gap of Motzkin and Fredkin spin chains are proved to vanish as 1/n^c with c>2; this rules out the possibility of these models to be relativistic conformal field theories in the continuum limit. Time permitting we will discuss more recent developments in this direction and 'generic' aspects of local spin chains.
Movassagh, Farhi, Goldstone, Nagaj, Osborne, Shor, PRA (2010) . Bravyi, Caha, Movassagh, Nagaj and Shor, PRL (2012) . Movassagh and Shor, PNAS (2016) . Levine, Movassagh, Journal of Physics A: Mathematical and Theoretical (2017) . Movassagh, Ann. of Math Sci and App, (2018), arXiv 1609.09160 . Movassagh, PRL (2017)