Entropy growth of shift-invariant states on a quantum spin chain

We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length N are typically mixed and have therefore a non-zero entropy SN which is, moreover, monotonically increasing in N . We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterised by a measurable subset of the unit interval. As the entropy density is known to vanishes, SN is sublinear in N . For states corresponding to unions of finitely many intervals, SN is shown to grow slower than (logN)2. Numerical calculations suggest a logN behaviour. For the case with infinitely many intervals, we present a class of states for which the entropy SN increases as N α where α can take any value in (0, 1).