The standard realization of crystal lattices and spectra of magnetic transition operators

晶格是有限图的复盖无限图。整数晶格、三角形晶格、六边形晶格都是晶格的例子。我们定义了磁性跃迁算符来描述周期性磁场作用下晶格上的电子转移。这一定义由中心极限定理证明：即，证明了由磁转移算子生成的半群收敛到由具有阿尔巴尼斯度量的欧氏空间的磁拉普拉斯生成的半群。晶格上的磁场用第二群上同调来定义。接下来，我们构造了一个与磁场相关的C^*-代数，并证明了磁转移算子属于C^*-代数。在此基础上，我们证明了磁场中磁跃迁算符的谱是Lipschitz连续函数，没有磁场，电子的行为就像随机游动。我们证明了大偏差原理适用于晶格上的随机游动。通过使晶格间距变小，晶格在Gromov-Hausdorff拓扑中收敛到具有Banach范数的有限维向量空间。这种Banach范数的特征是出现在大偏差中的速率函数。

A crystal lattice is an abelian covering infinite graph of a finite graph. The integer lattices, the triangular lattice, the hexagonal lattice are examples of crystal lattices. We define magnetic transition operators to describe electron transfer on a crystal lattice under periodic magnetic field. The definition is justified by the central limit theorem : Namely, we show that the semigroup generated by the magnetic transition operators converges to the semigroup generated by a magnetic Laplacian of the Euclidean space with the Albanese metric. Magnetic fields on a crystal lattice are defined in terms of the second group cohomology. Next we construct a C^*-algebra associated with the magnetic field and show the magnetic transition operator belongs to the C^*-algebra. By using this, we show the spectra of the magnetic transition operators is a Lipschitz continuous function in magnetic field.Without magnetic field, electrons behave like random walks. We show large deviation principle holds for random walks on a crystal lattice. By letting lattice spacing smaller, a crystal lattice converges to a finite dimensional vector space with a Banach norm in the Gromov-Hausdorff topology. This Banach norm is characterized in terms of the rate function appearing in the large deviation.