Almost commuting matrices, localized Wannier functions, and the quantum Hall effect

Abstract

On the oscillator realization of conformal U(2, 2) quantum particles and their particle-hole coherent states On a classification of irreducible almost commutative geometries, a second helping For models of noninteracting fermions moving within sites arranged on a surface in three-dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are K-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward and does not require translational invariance or introduce a flux torus. We further show that there is a Z 2 index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time-reversal or particle-hole conjugation. Finally, in the case of the sphere-mathematically speaking, three almost commuting Hermitians whose sum of square is near the identity-we give the first quantitative result, showing that this index is the only obstruction to finding commuting approximations. We review the known nonquantitative results for the torus.Given a list of bounded operators on infinite dimensional Hilbert space, it is often natural to seek a finite-rank projection P that almost commutes with that set. The C ‫ء‬ -algebraist would do so in the study of quasidiagonality. 1-3 In physics, we are interested in a projection onto a band of energy states separated from the rest of the spectrum by an energy gap; assuming the underlying Hamiltonian is local, this projection will itself be local due to the gap, and hence will approximately commute with a list of observables.Whatever exact relations might be known to hold for the original operators ͑H 1 , ... ,H r ͒ will generally hold only approximately for the compressions ͑PX 1 P , ... , PX r P͒. In a lattice model, the projection might be from a finite dimensional space to a space whose dimension is much lower, but still the outcome is finite-rank operators that approximately satisfy some relations. Can these be approximated by finite-rank operators that exactly satisfy those relations?For example, if X 1 and X 2 in B͑H͒ satisfy −I Յ X j Յ I and ͓X 1 , X 2 ͔ = 0, then P almost commuting with the X j implies − I Յ PX j P Յ I, a͒ Electronic We especially want to know if these almost commuting Hermitian operators are close to commuting Hermitian operators in the corner PB͑H͒P Х M k ͑C͒. It is sufficient to answer this question: can two almost commuting Hermitian matrices be approximated by commuting Hermitian matrices? The answer is yes. This is Lin's theorem. 4 The si...

Disordered topological insulators via C * -algebras

“…This index being −1 is first of all an obstruction to our being able to approximate the H r simultaneously by exactly commuting self-dual matrices. The analysis in §V.B of [9] is valid in the self-dual case, and it shows that this is also an obstruction to finding exponentially localized Wannier functions with time-reversal symmetry.…”

“…To see this, note that if P is local, then P almost commutes with exp(iΘ). More precise results for a system on a sphere are in [9]. Further, U and V almost commute if E F is in a mobility gap.…”

“…The physical importance of the Bott index is that when it is non-zero, it is not possible to find a complete, orthonormal basis of localized functions (so-called Wannier functions) spanning the occupied states [9]. This invariant can be computed very quickly.…”

“…-We now consider invariants in systems with time reversal symmetry. We begin with the invariant for matrices that almost represent a sphere constructed in [9]. This theory applies to all triples (H 1 , H 2 , H 3 ) of self-dual Hermitian matrices for which…”

“…-We describe first an integer invariant in the case of systems without time-reversal symmetry. We largely follow [9], which described an integer invariant for two-dimensional systems on the sphere or torus, and a Z 2 invariant in the case of a sphere. In the next section, we present a Z 2 invariant for systems on a torus.…”