论文标题
材料科学中的循环旋转和量化配对
Cyclic cocycles and quantized pairings in materials science
论文作者
论文摘要
环状共同体和可分离的$ c^\ ast $ algebras的k理论之间的配对，通常与材料的物理响应系数有关。使用三个数值模拟，我们说明了其中一些不变的人如何在共生的整个Sobolev结构域中生存。这些有趣的现象可以通过源自Alain Connes的量化演算的索引定理来解释，现在在独立的电子图片中已被充分了解。在这里，我们回顾了有关与Bram Mesland合作获得的相关多效系统动态的最新发展。他们提供了相关派生的代数的完整表征，该代数在$ c^\ ast $ -Algebra上，由通用离散的Delone晶格索引的规范反信号关系。在这里认为，这些结果已经提供了在这个代数上产生有趣且相关的状态的手段，并确定与多个fermion系统的传输系数相对应的循环共生。对这些Cocycles配对的现有索引定理进行了审查和更新，重点是推动对Sobolev域的分析。给出了对多体设置的可能概括的评估。
The pairings between the cyclic cohomologies and the K-theories of separable $C^\ast$-algebras supply topological invariants that often relate to physical response coefficients of materials. Using three numerical simulations, we exemplify how some of these invariants survive throughout the full Sobolev domains of the cocycles. These interesting phenomena, which can be explained by index theorems derived from Alain Connes' quantized calculus, are now well understood in the independent electron picture. Here, we review recent developments addressing the dynamics of correlated many-fermions systems, obtained in collaboration with Bram Mesland. They supply a complete characterization of an algebra of relevant derivations over the $C^\ast$-algebra of canonical anti-commutation relations indexed by a generic discrete Delone lattice. It is argued here that these results already supply the means to generate interesting and relevant states over this algebra of derivations and to identify the cyclic cocycles corresponding to the transport coefficients of the many-fermion systems. The existing index theorems for the pairings of these cocycles, in the restrictive single fermion setting, are reviewed and updated with an emphasis on pushing the analysis on Sobolev domains. An assessment of possible generalizations to the many-body setting is given.
