Boundary Cochains and the Toeplitz Index on the Half-Lattice

Abstract

We study the operator algebra of a rank-one boundary defect in a semi-infinite tight-binding chain, $T=U+\epsilon E$ on ${\ell }^2({\mathbb{Z}}_{\ge 0})$, with $U$ the forward unilateral shift and $E=⟨e_0,\cdot ⟩e_0$. The Lie algebra $\mathcal{A}=\mathrm{span}\{U^{a}E(U^{\ast }{)}^b,U^n\}$ has finitely supported, trace-zero commutators, the noncommutativity confined to the boundary and vanishing on the bulk. To each site we attach a $2$-cochain ${\omega }_j(X,Y)=⟨e_j,[X,Y]e_j⟩$; each is a Chevalley--Eilenberg coboundary, yet $H^2(\mathcal{A},\mathbb{C})$ is infinite-dimensional, carried by the abelian bulk and classifying the central extensions. On the polynomial Toeplitz algebra obtained by adjoining $U^{\ast }$, the total cochain ${\sum }_j{\omega }_j(T_f,T_g)$ equals the symbol pairing $\frac{1}{2\pi i}\oint fdg$, which for conjugate symbols $g=1/f$ is the Fredholm index; the ${\omega }_j$ thus form a site-resolved index density, ${\omega }_j(U^n,(U^{\ast }{)}^n)=-1_{\{j<n\}}$, localized at the edge. For modulated couplings with ${\epsilon }_j\to c$, the index is fixed by the bulk limit and undergoes a topological transition as $|c|$ crosses~$1$, independently of the boundary profile.