Search for localized Wannier functions of topological band structures via compressed sensing
Search for localized Wannier functions of topological band structures via compressed sensing
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DOI：
10.1103/physrevb.90.115110
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发表时间：
2014-05
影响因子：
3.7
通讯作者：
J. C. Budich;J. Eisert;E. Bergholtz;S. Diehl;P. Zoller
中科院分区：
文献类型：
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作者：
J. C. Budich;J. Eisert;E. Bergholtz;S. Diehl;P. Zoller
我们研究了Wannier函数的能带结构、拓扑和局域化性质之间的相互作用。为此，我们扩展了最近提出的基于压缩感知的范例，用于搜索最大局部化的Wannier函数[Ozolins等人，PNAS110,18368(2013年)]。我们开发了一个实用的工具箱，它可以搜索完全符合所研究的平移不变量子晶格系统的基本物理对称性的最大局域Wannier函数。最显著的是，这使我们能够系统地识别带结构的拓扑等价类的最局部化代表，即绝热地连接到一般初始代表的最局部化的Wannier函数集。我们还详细介绍了压缩感知方案，并找到了一种特别简单和高效的实现方法，其中迭代的每一步都是格点数目$N$中的$O(N\logN)$算法。我们给出了一维拓扑超导体的基准结果，展示了这些工具的威力。此外，我们还利用我们的方法解决了二维拓扑绝缘体等对称保护拓扑态是否存在紧Wannier函数的问题。这种函数的存在意味着具有严格有限范围跳跃的精确平带模型。在这里，我们找到了这些函数不存在的数字证据。我们简要讨论了张量网络方法在耗散态制备和状态变分集设计中的应用。
We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., PNAS 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an $O(N \log N)$ algorithm in the number of lattice sites $N$. We present benchmark results on one-dimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question whether compact Wannier functions can exist for symmetry protected topological states like topological insulators in two dimensions. The existence of such functions would imply exact flat band models with strictly finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipative state preparation and in devising variational sets of states for tensor network methods.