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, $T=U+\varepsilon E$ on $\ell^2(\mathbb{Z}_{\ge 0})$, with $U$ the forward unilateral shift and $E=\langle e_0,\cdot\rangle e_0$. The Lie algebra $\mathcal{A}=\mathrm{span}\{U^aE(U^*)^b,\,U^n\}$ 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 $\omega_j(X,Y)=\langle e_j,[X,Y]e_j\rangle$; 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 $U^*$, the total cochain $\sum_{j}\omega_j(T_f,T_g)$ equals the symbol pairing $\frac{1}{2\pi i}\oint f\,dg$, which for conjugate symbols $g=1/f$ is the Fredholm index; the $\omega_j$ thus form a site-resolved index density, $\omega_j(U^n,(U^*)^n)=-\mathbf{1}_{\{j<n\}}$, localized at the edge. For modulated couplings with $\varepsilon_j\to c$, the index is fixed by the bulk limit and undergoes a topological transition as $|c|$ crosses~$1$, independently of the boundary profile.