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 nonzero entropy S-N 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 characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, S-N is sublinear in N. For states corresponding to unions of finitely many intervals, S-N is shown to grow slower than log(2) N. Numerical calculations suggest a log N behavior. For the case with infinitely many intervals, we present a class of states for which the entropy S-N increases as N-alpha where alpha can take any value in (0,1). (C) 2003 American Institute of Physics.