Quantitative K-Theory Related to Spin Chern Numbers

Abstract

Abstract. We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant.

Theory of spin Bott index for quantum spin Hall states in nonperiodic systems

“…The Bott index measures the commutativity of a pair of almost commuting and almost unitary matrices, which can distinguish the pairs of commuting matrices close to commuting pairs from those are far from commuting pairs. 18,20,29 Adding the complementary projectors Q ± into Eq. ( 8) and Eq.…”

“…17 Loring and Hastings extended the definition of Z 2 index as the Kitaev's Z/2 index based on the theory of almost commuting matrices. [18][19][20] Loring further derives formulas and algorithms for Kitaev's invariants of different classes in the periodic table for topological insulators and superconductors [21][22][23] for finite disordered systems on lattices with boundaries. 24 Meanwhile, the concept of spin Chern number is also extended to disordered system based on the non-commutative theory of Chern number.…”

Bott periodicity and almost commuting matrices

“…in particular, if u and v happen to commute, this shows that e is an idempotent. Similarly, one can show that if }uv´vu} is suitably small, then }e 2´e } is also small, so in particular the spectrum of e misses 1{2: indeed, this is done qualitatively in [4, Proposition 3.5], while a quantitative result for a specific choice of f , g, and h can be found in [5,Theorem 3.5]; the latter could be used to make the conditions on t that are implicit in our results more explicit. Thus if χ is the characteristic function of r1{2, 8q, then χ is continuous on the spectrum of e, and so χpeq is a well-defined projection in B. Loring shows that if e n P M 2n pCq is the Loring element associated to the matrices u n , v n P M n pCq as in line (1), then for all suitably large n, rankpe n q´n " 1.…”

Almost commuting matrices, cohomology, and dimension

“…Striking applications can be found in Operator Theory (e.g. [FR01], [BD91], [KS15], [HZ16]), Quantum Physics and Condensed Matter Physics ( [HL11], [HLor10], [Has08], [Lor14], [Lor15], [FPL16], [LS13], [HL10], [Has09]) and even computer science ( [CS96], [GFS03]). Variations in which matrices almost commute with respect to various other norms have also found applications, e.g.…”