论文标题
一维非自动量子步行的Witten指数与无间隙的时间进化
The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution
论文作者
论文摘要
离散时间量子步行索引理论的最新发展使我们能够为一对单一的时间进步$ u $和$ \ mathbb {z} _2 $ - grading operator $ \\ vargamma $满足这些文章的$ u^u^* f vargmma u^** f vargmma u^** f vargmma u^* ugmma u^** ugma，\ vargmma in contex $ u^u^* vargmma in vargmma conthery $ u^* vargmma in vargmma in v vargmma contex $ u^* f affimage u^*** f vargamma扩展到包含非独立$ U $。现有的单一$ u $文献利用了必不可少的假设，即$ u $本质上是陷入困境的；也就是说，我们要求$ u $的基本频谱既包含$ -1 $也不包含$+1 $来定义相关索引。事实证明，如果给定的时间进步$ u $是独立的，则不再需要这个假设。作为一个具体的例子，我们将考虑由Mochizuki-kim-Obuse引入的一维整数晶格上的众所周知的非单身量子步行模型。
Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a pair of a unitary time-evolution $U$ and a $\mathbb{Z}_2$-grading operator $\varGamma$ satisfying the chiral symmetry condition $U^* = \varGamma U \varGamma.$ In this paper, this index theory will be extended to encompass non-unitary $U$. The existing literature for unitary $U$ makes use of the indispensable assumption that $U$ is essentially gapped; that is, we require that the essential spectrum of $U$ contains neither $-1$ nor $+1$ to define the associated index. It turns out that this assumption is no longer necessary, if the given time-evolution $U$ is non-unitary. As a concrete example, we shall consider a well-known non-unitary quantum walk model on the one-dimensional integer lattice, introduced by Mochizuki-Kim-Obuse.
