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arXiv:1608.04696 · Physical review. B./Physical review. B

Compactly supported Wannier functions and algebraic K -theory

N. Read

Published Mar 20, 2017• 96 citations
arXivHTMLPDF

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; 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.

DOI
10.1103/physrevb.95.115309

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In a tight-binding lattice model with nnn orbitals (single-particle states) per site, Wannier functions are nnn-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 ddd 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 ddd 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 KKK-theory, based on a ring of polynomials in e\ifmmode±\else\textpm\fiikx,e\ifmmode±\else\textpm\fiiky,...,{e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{x}},\phantom{\rule{0.16em}{0ex}}{e}^{\ifmmode\pm\else\textpm\fi{}i{k}_{y}},...,e\ifmmode±\else\textpm\fiikx​,e\ifmmode±\else\textpm\fiiky​,..., which occur as entries in the Fourier-transformed Wannier functions.
We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., Proc. Natl. Acad. Sci. USA 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an O(NlogN)O(NlogN)O(NlogN) algorithm in the number of lattice sites NNN. We present benchmark results on one-dimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question of whether compact Wannier functions can exist for symmetry-protected topological states such as topological insulators in two dimensions. The existence of such functions would imply exact flat-band models with finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipative-state preparation and in devising variational sets of states for tensor network methods.
Trial wave functions that can be represented by summing over locally coupled degrees of freedom are called tensor network states (TNSs); they have seemed difficult to construct for two-dimensional topological phases that possess protected gapless edge excitations. We show it can be done for chiral states of free fermions, using a Gaussian Grassmann integral, yielding ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ and Chern insulator states, in the sense that the fermionic excitations live in a topologically nontrivial bundle of the required type. We prove that any strictly short-range quadratic parent Hamiltonian for these states is gapless; the proof holds for a class of systems in any dimension of space. The proof also shows, quite generally, that sets of compactly supported Wannier-type functions do not exist for band structures in this class. We construct further examples of TNSs that are analogs of fractional (including non-Abelian) quantum Hall phases; it is not known whether parent Hamiltonians for these are also gapless.
Abstract Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band Hamiltonians, it is possible to construct Wannier functions that are even more localized, so that they are compactly supported thus having zero support outside their corresponding locations. Under what general conditions is it possible to construct compactly supported Wannier functions? We answer this question in this paper. Specifically, we show that in 1 d non-interacting tight-binding models, strict locality of the projection operator is a necessary and sufficient condition for a subspace to be spanned by a compactly supported orthogonal basis, independent of lattice translation symmetry. For any strictly local projector, we provide a procedure for obtaining such a basis. For higher dimensional systems, we discuss some additional conditions under which an occupied subspace is spanned by a compactly supported orthogonal basis, and show that the corresponding projectors are topologically trivial in many cases. We also show that a projector in arbitrary dimensions is strictly local if and only if for any chosen axis, its image is spanned by hybrid Wannier functions that are compactly supported along that axis.