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

Entropy growth of shift-invariant states on a quantum spin chain
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量子自旋链上平移不变态的熵增长

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发表时间：
2003
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通讯作者：
M. Mosonyi
M. Mosonyi
中科院分区：
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作者：
M. Fannes;B. Haegeman;M. Mosonyi

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我们研究量子自旋链上纯平移不变态的熵。与经典情况不同，对长度 N 的区间的局部限制通常是混合的，因此具有非零熵 SN，此外，它在 N 中单调递增。我们对总熵的渐近性感兴趣。我们详细研究了从 CAR 代数上的准自由态导出的一类态。它们的特征是单位间隔的可测量子集。由于已知熵密度会消失，因此 SN 在 N 中是次线性的。对于对应于有限多个间隔的并集的状态，SN 的增长速度慢于 log2 N。数值计算表明存在 log N 行为。对于具有无限多个间隔的情况，我们提出一类状态，其中熵 SN 随着 Nα 增加，其中 α 可以取 (0,1) 中的任何值。
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 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 characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, SN is sublinear in N. For states corresponding to unions of finitely many intervals, SN is shown to grow slower than log2 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 SN increases as Nα where α can take any value in (0,1).