紧支Wannier函数与代数$K$ -理论
Compactly-supported Wannier functions and algebraic $K$-theory
摘要：
在一个每个位置都有$n$轨道（单粒子态）的紧束缚晶格模型中，Wannier函数是位置的$n$ -分量向量函数，它从某个位置迅速下降，并且在某种意义上它们的集合跨越了给定能带或能带集合中的所有态;紧支撑的Wannier函数是这样的函数，它在有界区域之外消失。它们不仅出现在能带理论中，而且与非相互作用费米子系统的张量网络状态以及具有严格短程跳跃矩阵元素的平带哈密顿有关。在早期的工作中，证明了对于一般的复能带结构（向量丛）或一般的复哈密顿量--也就是哈密顿量和能带结构的十重分类中的A类--一组紧支撑的Wannier函数可以跨越向量丛，只有当丛在拓扑上是平凡的，在空间的任何维度$d$中，即使允许使用这样的函数的过完备集。这意味着，对于具有A类非平凡丛的自由费米子张量网络态，任何严格短程父哈密顿量必须是无隙的。在这里，这个结果被扩展到所有10个对称类的能带结构没有额外的晶体对称性，结果是，在一般情况下，非平凡的丛，可以产生从紧支撑的瓦尼尔型功能是那些可能拥有，在每个$d$方向，非平凡的缠绕，可以发生在同一对称类在一维，但没有别的。这些结果是从一个非常自然的使用代数$K$理论的基础上，环的多项式在$e^{\pm ik_x}$，$e^{\pm ik_y}$，。. .，其作为傅里叶变换的Wannier函数中的项出现。
英文摘要：
In a tight-binding lattice model with $n$ orbitals (single-particle states) per site, Wannier functions are $n$-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactly-supported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for non-interacting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians---that is, class A in the ten-fold classification of Hamiltonians and band structures---a set of compactly-supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension $d$ of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a non-trivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the non-trivial bundles that can arise from compactly-supported Wannier-type functions are those that may possess, in each of $d$ directions, the non-trivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic $K$-theory, based on a ring of polynomials in $e^{\pm ik_x}$, $e^{\pm ik_y}$, . . . , which occur as entries in the Fourier-transformed Wannier functions.
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