We extend non-Hermitian topological quantum walks on a Su-Schrieffer-Heeger (SSH) lattice [Phys. Rev. Lett. 102, 065703 (2009)] to the case of non-Markovian evolution. This non-Markovian model is established by coupling each unit cell in the SSH lattice to a reservoir formed by a quasicontinuum of levels. We find a topological transition in this model even in the case of non-Markovian evolution where the walker may visit the reservoir and return to the SSH lattice at a later time. The existence of a topological transition does, however, depend on the low-frequency properties of the reservoir, characterized by a spectral density $J\left(\epsilon \right)\propto {\left|\epsilon \right|}^{\alpha }$. In particular, we find a robust topological transition for a sub-Ohmic ($\alpha <1$) and Ohmic ($\alpha =1$) reservoir, but no topological transition for a super-Ohmic ($\alpha >1$) reservoir. This behavior is directly related to the well-known localization transition for the spin-boson model. We confirm the presence of non-Markovian dynamics by explicitly evaluating a measure of Markovianity for this model.

中文翻译：

---

非马尔可夫耗散量子行走的拓扑转变

我们在Su-Schrieffer-Heeger（SSH）晶格上扩展了非Hermitian拓扑量子步[ Phys。牧师 102，065703（2009）]，以非马尔科夫演进的情况下。通过将SSH晶格中的每个晶胞耦合到由准连续谱构成的储层，可以建立这种非马尔可夫模型。即使在非马尔可夫演化的情况下，我们也能在此模型中找到拓扑过渡，在这种情况下，步行者可能会访问水库并在稍后时间返回SSH晶格。但是，拓扑过渡的存在确实取决于储层的低频特性，其特征在于频谱密度$Ĵ（\epsilon ）\propto {\left|\epsilon \right|}^{\alpha }$。特别是，我们发现了亚欧姆（$\alpha <\mathrm{1个}$）和Ohmic（$\alpha =\mathrm{1个}$）储层，但对于超欧姆（$\alpha >\mathrm{1个}$）水库。此行为与自旋玻色子模型的众所周知的定位转换直接相关。通过明确评估该模型的马尔可夫性度量，我们确认了非马尔可夫动力学的存在。