量子自旋链上平移不变态的熵增长
Entropy growth of shift-invariant states on a quantum spin chain
摘要：
研究了量子自旋链上纯移不变态的熵。与经典情况不同，对长度为$N$的区间的局部限制通常是混合的，并且因此具有非零熵$S_N$，此外，该非零熵在$N$中单调增加。我们感兴趣的是总熵的渐近性。我们详细研究了一类由CAR代数上的准自由态导出的态。它们的特征在于单位间隔的可测量子集。由于已知熵密度为零，因此$S_N$在$N$中是次线性的。对于对应于许多间隔的工会的状态，$S_N$的增长速度慢于$(\log N)^2$。数值计算表明，一个$\log N$的行为。对于无穷多个区间的情形，我们给出了一类熵$S_N$随$N^α$增加的状态，其中$α$可以取$(0,1)$中的任意值.
英文摘要：
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 $S_N$ 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 characterised by a measurable subset of the unit interval. As the entropy density is known to vanishes, $S_N$ is sublinear in $N$. For states corresponding to unions of finitely many intervals, $S_N$ is shown to grow slower than $(\log N)^2$. Numerical calculations suggest a $\log N$ behaviour. For the case with infinitely many intervals, we present a class of states for which the entropy $S_N$ increases as $N^α$ where $α$ can take any value in $(0,1)$.
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