Compactly supported Wannier functions and algebraicK-theory

Abstract

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; compactlysupported 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 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 non-trivial bundles that can arise from compactlysupported 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 ±ikx , e ±iky , . . . , which occur as entries in the Fourier-transformed Wannier functions.

Nontrivial topological flat bands in a diamond-octagon lattice geometry

“…interesting flat band optimization techniques for shortrange hopping models have also been proposed in recent times [37,38]. It is worth mentioning that, one cannot have nontrivial topology, finite-range hopping and exactly flat bands simultaneously -only two of these three criteria can be realized simultaneously [39,40]. In the present paper, we propose and study a discrete 2D diamond-octagon lattice model with short-ranged hopping and an external magnetic flux piercing through the diamond plaquettes.…”

Strictly local tensor networks for topological insulators with short-range interactions

“…24 are "exponentially local": the importance of their links decays exponentially with distance. This is distinct from the "strictly local" tensor networks in the no-go theorems [14][15][16] mentioned earlier; there links are entirely absent beyond a certain finite distance. Such strictly local tensor networks also naturally arise in applying the approach of Ref.…”

“…(3). The parent Hamiltonians here, however, are Hamiltonians with respect to which nonzero temperature expectation values are calculated using the tensor networks, thus are understood in a slightly different sense than parent Hamiltonians in earlier works [14][15][16] where tensor networks encode the corresponding ground states.…”

“…Sec. III): the impossibility for the flat band Hamiltonian h to be both strictly short-range and topologically nontrivial [16,35].…”

Fermionic tensor networks for higher-order topological insulators from charge pumping

“…However, Gaussian PEPSs with chiral topological properties do not adequately describe the bulk of a gapped phase, since their correlation functions decay algebraically [40,41]. Indeed, the existence of gapped PEPSs with finite bond dimension for SPT phases of free fermions is forbidden in dimensions d > 1 [41,42]. The no-go theorem states that only Wannierizable phases corresponding to the product of one-and zero-dimensional systems can be exactly represented as a TNS.…”