Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
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DOI：
10.22331/q-2018-09-24-95
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发表时间：
2018-09-24
期刊：
影响因子：
6.4
通讯作者：
Werner, R. F.
中科院分区：
文献类型：
--
作者：
Cedzich, C.;Geib, T.;Werner, R. F.
我们提供了翻译不变的一维量子步行的分类，相对于保留单位性，局部性，翻译不变性，间隙条件和对十倍方式的某些对称性的连续变形。该分类在很大程度上匹配了最近获得的（Arxiv：1611.04439）的类似设置的分类。但是，翻译不变的情况有一些更好的区别，因为某些步行只能通过沿途破坏翻译不变性来连接，仅保留偶数偶数的站点。同样，如果只有通过添加一个琐碎的步行而被认为是等效的，即，即不允许在细胞之间跳跃的步行，则分类也崩溃了。一般分类的索引可以在实践中仅用于与某些翻译不变的步行相关的步行。我们证明了涵盖所有对称类型的缠绕数量的简单公式集合。此外，我们确定了当地条件的强度，并表明频带结构的连续性是对绕线数的拓扑分类的最小要求，这意味着步行的换向器的紧凑性与一半 - 空间投影，这种条件也是一般理论的基础。为了将理论应用于大型但有限的大块碎片的连接，需要确定固定的Schrodinger方程的渐近行为。我们显示指数的行为，并提供一种计算衰减常数的实用方法。
We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv: 1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schrodinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.