Compactly supported Wannier functions and algebraicK-theory

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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 noninteracting 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 tenfold 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 nontrivial 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 nontrivial bundles that can arise from compactly supported Wannier-type functions are those that may possess, in each of $d$ directions, the nontrivial 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 ${e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{x}},\phantom{\rule{0.16em}{0ex}}{e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{y}},...,$ which occur as entries in the Fourier-transformed Wannier functions.