Boundary Cochains and the Toeplitz Index on the Half-Lattice
January 01, 1970
Abstract
We study the operator algebra induced by 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 (\(Ue_n = e_{n+1}\)), \(E = \langle e_0,\cdot\rangle e_0\), and \(\varepsilon \in \mathbb{C}\). The enlarged Lie algebra \(\mathcal{A} = \mathrm{span}\{U^a E (U^*)^b,\, U^n\}\) has finitely supported, trace-zero commutators, \([\mathcal{A},\mathcal{A}] = \mathfrak{sl}_{\mathrm{fin}}\); the noncommutativity is boundary-induced, vanishing on the bulk \(\mathrm{span}\{U^n\}\). We attach site-resolved \(2\)-cochains \(\omega_j(X,Y) = \langle e_j, [X,Y] e_j \rangle\), each a continuous Chevalley–Eilenberg cocycle and a coboundary (\(\omega_j=d\eta_j\)); the cohomology \(H^2(\mathcal{A},\mathbb{C})\) is nonetheless infinite-dimensional, carried by the abelian bulk and classifying central extensions (the simplest a Heisenberg extension), so the diagnostic role of \(\omega_j\) is site resolution, not cohomology.
Our main result gives these cochains a topological meaning. On the polynomial Toeplitz algebra obtained by adjoining \(U^*\), the total cochain computes a bilinear pairing of symbols that specializes, for conjugate symbols \(g=1/f\), to the Fredholm index: \[\sum_{j\ge0}\omega_j(T_f,T_g)=\mathrm{tr}\,[T_f,T_g]=\frac{1}{2\pi i}\oint_{\mathbb{S}^1} f\,dg, \qquad \sum_{j\ge0}\omega_j(U^n,(U^*)^n)=-n=\operatorname{index}(U^n).\] The \(\omega_j\) thus form a site-resolved index density: the bulk winding number is a sum of unit edge contributions \(\omega_j(U^n,(U^*)^n)=-\mathbf{1}_{\{j<n\}}\), and the trace relation \(\sum_j\omega_j=0\) on \(\mathcal{A}\) reflects the absence of winding for analytic symbols. For spatially modulated couplings \(U+\sum_j\varepsilon_j|e_j\rangle\langle e_j|\) with \(\varepsilon_j\to c\), the index is fixed by the bulk limit alone and undergoes a topological transition as \(|c|\) crosses \(1\), independently of the boundary profile.
1 Introduction↩︎
The operator \(T = U + \varepsilon E\) arises naturally in the study of open quantum systems with a boundary. Throughout this paper, \(U\) denotes the forward unilateral shift on \(\ell^2(\mathbb{Z}_{\ge 0})\), defined by \[\label{eq:forward-shift} U e_n = e_{n+1}, \quad n \ge 0.\tag{1}\] Thus \(U\) is an isometry (\(U^*U = I\)), and its adjoint \(U^*\) is the backward shift: \(U^* e_n = e_{n-1}\) for \(n \ge 1\), \(U^* e_0 = 0\). The operator \(T\) admits two complementary physical interpretations.
In the Hermitian case (\(\varepsilon \in \mathbb{R}\)), the related Hamiltonian \[\label{eq:spin-chain} H = U + U^* + \varepsilon E\tag{2}\] describes the single-particle sector of the spin-\(\tfrac{1}{2}\) XY chain with a boundary magnetic field (a setting in which boundary and surface critical behavior is classical [1]), or a tight-binding model with a boundary potential. In such models, a boundary defect may host an exponentially localized edge mode [2], [3].
In the non-Hermitian case (\(\varepsilon \in \mathbb{C}\)), the operator \(T = U + \varepsilon E\) models the single-step operator of a boundary-deformed discrete-time quantum walk (non-unitary for \(\varepsilon\ne0\), since \(T^*T = I + |\varepsilon|^2 E\)); related walks with boundary defects have been studied in photonic lattices [4] and trapped ions [5], with related programmable quantum simulators on optical-lattice and superconducting platforms [6], [7]. For the general spectral theory of non-selfadjoint operators we refer to [8]. The perturbation \(\varepsilon E\) is compact and rank-one, so the model does not exhibit the non-Hermitian skin effect.
In both settings, the rank-one term \(\varepsilon E\) models a boundary probe or defect. Our focus is on its algebraic consequences: we show that it induces a detectable deformation of the operator algebra whose noncommutativity is generated by the boundary defect. It connects to operator-algebraic work on the half-lattice [9] and on the rigidity of (quasi-)Lie brackets [10], and to the noncommutative-geometric viewpoint of [11].
The contrast between bulk and edge is already visible at the algebraic level. The polynomial algebra \(\langle T \rangle = \mathrm{span}\{T^n : n \ge 0\}\) is abelian, but the enlarged algebra \[\mathcal{A} := \mathrm{span}\{U^a E (U^*)^b,\, U^n : a,b,n \ge 0\}\] is non-abelian. Its commutators are finitely supported and of trace zero, and the noncommutativity is generated entirely by the boundary term \(E\) (it vanishes on \(\langle U\rangle=\mathrm{span}\{U^n\}\)); the site-resolved cochains below make this bulk–edge dichotomy precise. The corner operators \(U^a E (U^*)^b = |e_a\rangle\langle e_b|\) represent hopping processes that involve the boundary site.
The central objects are the site-localized \(2\)-cochains \[\omega_j(X,Y) = \langle e_j, [X,Y] e_j \rangle, \quad j \in \mathbb{Z}_{\ge 0}.\] We prove that each \(\omega_j\) is a \(2\)-cocycle and that each is a coboundary (\(\omega_j = d\eta_j\)), so \([\omega_j] = 0\). We also show that \(H^2(\mathcal{A},\mathbb{C})\) is not trivial (the abelian bulk contributes nonzero classes), so the diagnostic value of \(\omega_j\) comes from site resolution rather than cohomological nontriviality. In fact the bulk cocycles form an infinite-dimensional family classifying central extensions of \(\mathcal{A}\), the simplest of which is a Heisenberg central extension (with \([I,U]=c\) in the extended bracket; Section 3.3). Each finite subfamily of \(\{\omega_j\}\) is linearly independent, the full family obeying the single relation \(\sum_j \omega_j = 0\) in the diagonal cocycle space \(Z^2_{\mathrm{diag}}(\mathcal{A},\mathbb{C})\).
Section 4 turns these cochains into a topological invariant. On the polynomial Toeplitz algebra obtained by adjoining the backward shift \(U^*\), the total cochain computes a bilinear pairing of symbols that specializes, for conjugate symbols \(g=1/f\), to the Fredholm index: \[\sum_{j\ge0}\omega_j(T_f,T_g)=\frac{1}{2\pi i}\oint_{\mathbb{S}^1} f\,dg, \qquad \sum_{j\ge0}\omega_j(U^n,(U^*)^n)=-n=\operatorname{index}(U^n),\] so the \(\omega_j\) form a site-resolved density for the bulk winding number. The vanishing \(\sum_j\omega_j=0\) on \(\mathcal{A}\) reflects the absence of winding for analytic symbols.
Table 1 compares the construction with established index-theoretic boundary models.
| Model | Bulk invariant | Edge quantity | Algebraic structure |
|---|---|---|---|
| SSH [3] | \(\mathbb{Z}\) winding | Edge modes | Abelian bulk algebra |
| Kitaev [12] | \(\mathbb{Z}_2\) | Majorana modes | Clifford algebra |
| Floquet [13] | Dynamical winding | Anomalous modes | Driven |
| This work | Winding \(\mathrm{wind}(f)\) | \(\sum_j\omega_j=-\mathrm{wind}(f)\) | Toeplitz Lie algebra; site-resolved index density \(\{\omega_j\}\) |
The remainder is organized as follows. Section 2 introduces the operator-theoretic setting. Section 3 constructs \(\omega_j\), shows each is a coboundary, analyzes the diagonal cocycle space, and constructs the central extensions carried by the bulk. Section 4 proves the index identity: the total cochain computes a pairing of symbols, equal to the Fredholm index for the conjugate pairing \(g=1/f\), with the \(\omega_j\) a site-resolved index density. Section 5 presents finite-dimensional realizations. Section 6 gives spectral illustrations. Section 7 summarizes. Appendix ¿sec:app:numerics? describes numerical methods.
2 Operator-Theoretic Framework↩︎
2.1 Banach and Lie Algebraic Frameworks↩︎
Definition 1 (Banach algebra). A Banach algebra \((\mathfrak{A},\|\cdot\|)\) is a Banach space with associative bilinear multiplication satisfying \(\|xy\| \leq \|x\|\,\|y\|\).
Definition 2 (Banach–Lie algebra). A Banach–Lie algebra \((\mathfrak{g},[\cdot,\cdot])\) is a Banach space with a continuous antisymmetric bracket satisfying the Jacobi identity \([X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0\).
Remark 1 (Topology and non-closedness of \(\mathcal{A}\)). We equip \(\mathcal{A} \subset \mathcal{B}(\ell^2)\) with the operator norm. The subalgebra \(\mathcal{A}\) is not* norm-closed. To see this, consider the diagonal operator \(D = \mathrm{diag}(1, \tfrac{1}{2}, \tfrac{1}{3}, \ldots)\). Since its eigenvalues tend to zero, \(D\) is compact; hence \(D \in \mathcal{K}(\ell^2)\) (see [14] for trace ideals and compact operators). However, \(D\) is not a finite linear combination of corner operators \(|e_a\rangle\langle e_b|\) or powers \(U^n\), so \(D \notin \mathcal{A}\). The finite-rank truncations \(D_N := \mathrm{diag}(1,\tfrac{1}{2},\ldots,\tfrac{1}{N},0,0,\ldots)\) belong to \(\mathcal{A}\) and satisfy \[\|D_N - D\|_{\mathcal{B}(\ell^2)} = \sup_{k > N}\frac{1}{k} = \frac{1}{N+1} \xrightarrow{N\to\infty} 0.\] Thus \(D\) lies in the norm-closure of \(\mathcal{A}\) but not in \(\mathcal{A}\) itself, proving that \(\mathcal{A}\) is not norm-closed.*
**Remark on strong vs.norm topology.* The partial sums \(S_N = \sum_{k=0}^{N} |e_k\rangle\langle e_k|\) converge to the identity \(I\) in the strong operator topology (i.e., \(\|S_N f - f\| \to 0\) for each fixed \(f \in \ell^2\)), but not in operator norm: indeed, \(\|(I - S_N)e_{N+1}\| = 1\) for every \(N\), so \(\|I - S_N\| = 1\) for all \(N\). In particular, the sequence \((S_N)\) is not norm-Cauchy in \(\mathcal{A}\), and the above non-closedness argument relies on the sequence \((D_N)\), which is norm-Cauchy, not on \((S_N)\).*
We work throughout with continuous cochains satisfying \(|\omega_j(X,Y)| \leq \|[X,Y]\| \leq 2\|X\|\|Y\|\).
Definition 3 (Spatially localized commutators). A Lie subalgebra \(\mathfrak{g} \subset \mathcal{B}(\ell^2(\mathbb{Z}_{\ge 0}))\) exhibits boundary-localized noncommutativity* if there is a direct sum decomposition \(\mathfrak{g} = \mathfrak{g}_{\mathrm{bulk}} \oplus \mathfrak{g}_{\mathrm{edge}}\) (with \(\mathfrak{g}_{\mathrm{bulk}} \cap \mathfrak{g}_{\mathrm{edge}} = \{0\}\)) such that:*
\(\mathfrak{g}_{\mathrm{bulk}} = \mathrm{span}\{U^n : n \ge 0\}\) is abelian;
\(\mathfrak{g}_{\mathrm{edge}} = \mathrm{span}\{U^a E (U^*)^b : a,b \ge 0\}\) consists of finite-rank operators;
every commutator involving \(\mathfrak{g}_{\mathrm{edge}}\) is finite-rank and finitely supported.
Remark 2 (Triviality of the intersection and direct sum decomposition). The intersection \(\mathfrak{g}_{\mathrm{bulk}} \cap \mathfrak{g}_{\mathrm{edge}} = \{0\}\) holds because no nonzero power \(U^n\) (\(n \ge 1\)) is compact: \(U^n\) is an isometry on a separable infinite-dimensional Hilbert space, hence not compact. For \(n=0\), \(U^0 = I\) (the identity operator) belongs to \(\mathfrak{g}_{\mathrm{bulk}}\) but not to \(\mathfrak{g}_{\mathrm{edge}}\), since \(I\) is not compact. Thus \(\mathfrak{g}_{\mathrm{bulk}} \cap \mathfrak{g}_{\mathrm{edge}} = \{0\}\).
To verify the direct sum \(\mathcal{A} = \mathfrak{g}_{\mathrm{bulk}} \oplus \mathfrak{g}_{\mathrm{edge}}\), note that by definition every element of \(\mathcal{A}\) is a finite linear combination \(\sum_n \alpha_n U^n + \sum_{a,b} \beta_{ab} U^a E (U^*)^b\). Such a decomposition into a bulk part \(X_{\mathrm{bulk}} = \sum_n \alpha_n U^n\) and an edge part \(X_{\mathrm{edge}} = \sum_{a,b} \beta_{ab} U^a E (U^*)^b\) is unique by the intersection condition above. Thus \(\mathcal{A}\) satisfies Definition 3. We write \(\mathcal{A}_{\mathrm{bulk}} := \mathfrak{g}_{\mathrm{bulk}} = \mathrm{span}\{U^n : n\ge0\}\) and \(\mathcal{A}_{\mathrm{edge}} := \mathfrak{g}_{\mathrm{edge}} = \mathrm{span}\{U^a E (U^*)^b : a,b\ge0\} = \mathrm{span}\{|e_a\rangle\langle e_b| : a,b\ge0\}\), so that \(\mathcal{A} = \mathcal{A}_{\mathrm{bulk}} \oplus \mathcal{A}_{\mathrm{edge}}\); the elements of \(\mathcal{A}_{\mathrm{edge}}\) are exactly the operators with finitely many nonzero matrix entries.
2.2 Algebraic Structure on the Half-Lattice↩︎
Let \((e_n)_{n\ge0}\) be the canonical orthonormal basis of \(\ell^2(\mathbb{Z}_{\ge 0})\).
Definition 4 (Forward shift and boundary projector). The forward unilateral shift* \(U\) and boundary projection \(E\) are: \[\begin{align} U e_n &= e_{n+1}, \quad n \ge 0 \quad (U \text{ is an isometry}), \\ Ef &= \langle e_0, f \rangle e_0. \end{align}\] The backward shift \(U^*\) satisfies \(U^* e_n = e_{n-1}\) (\(n \ge 1\)), \(U^* e_0 = 0\). The boundary-deformed shift is \(T := U + \varepsilon E\).*
Lemma 1 (Key auxiliary identity). \(EU = 0\), i.e., \(E \circ U = 0\) as operators on \(\ell^2(\mathbb{Z}_{\ge 0})\).
Proof. For any \(f \in \ell^2(\mathbb{Z}_{\ge 0})\), \((Uf)(n) = f(n-1)\) for \(n \ge 1\) and \((Uf)(0) = 0\), since \(U\) is the forward shift (\(Ue_k = e_{k+1}\) means the zeroth component of \(Uf\) is always zero). Therefore \[(EUf) = \langle e_0, Uf \rangle e_0 = (Uf)(0) \cdot e_0 = 0.\] ◻
Remark 3. The identity \(EU = 0\) expresses the forward-shift boundary condition \((Uf)(0)=0\). It is what makes the \(n=0\) component of the eigenvalue equation decouple in the spectral analysis (Lemma 9).
Remark 4 (Corner operators). For \(a, b \ge 0\), the operator \(U^a E (U^*)^b \in \mathcal{A}\) acts by \[(U^a E (U^*)^b) f = \langle e_b, f \rangle e_a = (|e_a\rangle\langle e_b|) f.\] Indeed: \(((U^*)^b f)(n) = f(n+b)\), so \(\langle e_0, (U^*)^b f\rangle = f(b) = \langle e_b, f\rangle\). Then \(E(U^*)^b f = \langle e_b, f\rangle e_0\). Finally \(U^a e_0 = e_a\) (since \(U\) is the forward shift). Hence \(U^a E (U^*)^b f = \langle e_b, f\rangle e_a\), confirming \(U^a E (U^*)^b = |e_a\rangle\langle e_b|\).
Proposition 5 (Commutator of corner operators). \[[|e_a\rangle\langle e_b|,\, |e_c\rangle\langle e_d|] = \delta_{b,c}\, |e_a\rangle\langle e_d| - \delta_{d,a}\, |e_c\rangle\langle e_b|.\]
Proof. \((|e_a\rangle\langle e_b|)(|e_c\rangle\langle e_d|) = \delta_{b,c}|e_a\rangle\langle e_d|\). Subtract the reversed product. ◻
Lemma 2 (Structure of the derived algebra). Let \[\mathfrak{sl}_{\mathrm{fin}} := \bigl\{\, F \in \mathcal{A}_{\mathrm{edge}} : \mathrm{tr}\,F = 0 \,\bigr\}\] denote the finitely supported (i.e.finitely many nonzero matrix entries \(\langle e_a,Fe_b\rangle\)) trace-zero operators. Then \[[\mathcal{A},\mathcal{A}] = \mathfrak{sl}_{\mathrm{fin}}.\] In particular, every commutator of elements of \(\mathcal{A}\) is finitely supported with trace zero. Such commutators are not* diagonal in general; for instance \([U,E]=|e_1\rangle\langle e_0|\) and \([|e_1\rangle\langle e_0|,|e_0\rangle\langle e_2|]=|e_1\rangle\langle e_2|\) are off-diagonal.*
Proof. Inclusion \([\mathcal{A},\mathcal{A}] \subseteq \mathfrak{sl}_{\mathrm{fin}}\). By bilinearity of the bracket it suffices to treat commutators of generators. First, \([U^m,U^n]=0\) (powers of a single operator commute). For an edge–edge pair, Proposition 5 gives \[[|e_a\rangle\langle e_b|,\,|e_c\rangle\langle e_d|] = \delta_{b,c}\,|e_a\rangle\langle e_d| - \delta_{d,a}\,|e_c\rangle\langle e_b|,\] which has rank \(\le 2\) and trace \(\delta_{b,c}\delta_{a,d} - \delta_{d,a}\delta_{c,b} = 0\). For a bulk–edge pair, a direct computation using \(U^n e_k = e_{k+n}\) and \((U^*)^n e_k = e_{k-n}\) (zero if \(k<n\)) gives \[\label{eq:bulk-edge-comm} [U^n,\,|e_a\rangle\langle e_b|] = |e_{a+n}\rangle\langle e_b| - |e_a\rangle\langle e_{b-n}|,\tag{3}\] with the convention that the second term is \(0\) when \(b<n\); this is finite-rank with trace \(\delta_{a+n,b}-\delta_{a,b-n} = 0\). A general commutator \([X,Y]\) with \(X,Y\in\mathcal{A}\) is a finite linear combination of such corner terms, hence finitely supported with trace zero.
Inclusion \(\mathfrak{sl}_{\mathrm{fin}} \subseteq [\mathcal{A},\mathcal{A}]\). Fix \(N\ge 1\) and set \(M_N := \mathrm{span}\{|e_a\rangle\langle e_b| : 0 \le a,b \le N\} \cong \mathfrak{gl}_{N+1}(\mathbb{C})\), a Lie subalgebra of \(\mathcal{A}_{\mathrm{edge}}\). For \(a\ne b\), Proposition 5 gives the off-diagonal unit as a commutator, \[|e_a\rangle\langle e_b| = [\,|e_a\rangle\langle e_b|,\;|e_b\rangle\langle e_b|\,],\] and for \(a\ne b\) it gives the diagonal difference \[|e_a\rangle\langle e_a| - |e_b\rangle\langle e_b| = [\,|e_a\rangle\langle e_b|,\;|e_b\rangle\langle e_a|\,].\] The off-diagonal units together with the diagonal differences span the trace-zero matrices supported in \(\{0,\dots,N\}\), i.e.\(\mathfrak{sl}_{N+1}\). Hence every trace-zero operator supported in \(\{0,\dots,N\}\) lies in \([\mathcal{A},\mathcal{A}]\). Taking the union over all \(N\) yields \(\mathfrak{sl}_{\mathrm{fin}} \subseteq [\mathcal{A},\mathcal{A}]\). ◻
Remark 6 (\(\mathcal{A}_{\mathrm{edge}}\) is an ideal). Equation 3 shows \([U^n,\mathcal{A}_{\mathrm{edge}}]\subseteq \mathcal{A}_{\mathrm{edge}}\), and \([\mathcal{A}_{\mathrm{edge}},\mathcal{A}_{\mathrm{edge}}] \subseteq\mathcal{A}_{\mathrm{edge}}\) by Proposition 5. Hence \(\mathcal{A}_{\mathrm{edge}}\) is a Lie ideal of \(\mathcal{A}\), with abelian quotient \(\mathcal{A}/\mathcal{A}_{\mathrm{edge}}\cong\mathrm{span}\{U^n:n\ge0\}\). This fact is used in Theorem 12(3).
Lemma 3 (Cyclicity). For nonzero \(f \in \ell^2\) with \(\langle e_k,f\rangle \ne 0\), the span \(\{Xf : X \in \mathcal{A}_{\mathrm{edge}}\}\) is dense in \(\ell^2\).
Proof. \(|e_j\rangle\langle e_k| \in \mathcal{A}_{\mathrm{edge}}\) maps \(f\) to \(\langle e_k,f\rangle e_j\), reaching every \(e_j\). ◻
Remark 7 (Diagonal evaluation and the trace relation). The cochain \(\omega_j(X,Y) = \langle e_j,[X,Y]e_j\rangle\) records only the \((j,j)\) diagonal entry of the commutator \([X,Y]\), not the full operator. Since \([X,Y]\in\mathfrak{sl}_{\mathrm{fin}}\) by Lemma 2, only finitely many indices \(j\) contribute, and the trace-zero property gives the single linear relation \[\sum_{j\ge0}\omega_j(X,Y) = \mathrm{tr}[X,Y] = 0 \qquad (X,Y\in\mathcal{A}).\] Thus the family \(\{\omega_j\}\) reads off the diagonal of the commutator and is constrained by this one relation.
Lemma 4 (Finite support). For any \(X, Y \in \mathcal{A}\), there exists \(N\) such that \(\langle e_j,[X,Y]e_j\rangle = 0\) for all \(j \ge N\).
Proof. \(X\) and \(Y\) are finite linear combinations of generators; the commutator involves only finitely many indices. ◻
Proposition 8 (Abelian polynomial algebra). \([T^m, T^n] = 0\) for all \(m,n \ge 0\).
Proof. Powers of a single operator commute. ◻
2.2.1 Boundary-localized corrections↩︎
Lemma 5 (Telescoping identity). For \(m \ge 1\): \(T^m - U^m = \varepsilon \sum_{j=0}^{m-1} U^{m-1-j} E T^j\).
Proof. We proceed by induction on \(m\). For \(m = 1\): \(T - U = \varepsilon E\), which matches the formula (the single term \(j=0\) gives \(U^0 E T^0 = E\)). Assume the identity holds for \(m-1\). Then \[T^m = T \cdot T^{m-1} = (U + \varepsilon E) T^{m-1} = U T^{m-1} + \varepsilon E T^{m-1}.\] Now \(U^m = U \cdot U^{m-1}\), so \[T^m - U^m = U(T^{m-1} - U^{m-1}) + \varepsilon E T^{m-1}.\] By the induction hypothesis, \(T^{m-1} - U^{m-1} = \varepsilon \sum_{j=0}^{m-2} U^{m-2-j} E T^j\), hence \[T^m - U^m = \varepsilon \sum_{j=0}^{m-2} U^{m-1-j} E T^j + \varepsilon E T^{m-1} = \varepsilon \sum_{j=0}^{m-1} U^{m-1-j} E T^j.\] ◻
Lemma 6 (Support localization). \(T^m - U^m\) has rank \(\le m\) and range in \(\mathrm{span}\{e_0,\dots,e_{m-1}\}\).
Proof. By Lemma 5, \(T^m - U^m = \varepsilon \sum_{j=0}^{m-1} U^{m-1-j} E T^j\). Each term \(U^{m-1-j} E T^j\) has range in \(\mathrm{span}\{e_{m-1-j}\}\) (since \(E\) projects onto \(e_0\) and \(U^{m-1-j} e_0 = e_{m-1-j}\)), hence rank at most \(1\). The sum of \(m\) rank-one operators has rank at most \(m\), and its range is contained in \(\mathrm{span}\{e_0, e_1, \dots, e_{m-1}\}\). ◻
2.3 Quantitative bounds and essential spectrum↩︎
Proposition 9 (Norm and rank bounds). For all \(m, n \ge 1\): \[\|[T^m, U^n]\| \le 2\bigl((1+|\varepsilon|)^m - 1\bigr) \le 2|\varepsilon|\, m\, (1 + |\varepsilon|)^{m-1}, \quad \mathrm{rank}[T^m, U^n] \le m.\]
Proof. From Lemma 5, \([T^m,U^n] = [T^m - U^m, U^n]\), a sum of \(m\) rank-one terms. The norm bound \(\|[T^m,U^n]\| \le 2|\varepsilon|\sum_{j=0}^{m-1}(1+|\varepsilon|)^j = 2((1+|\varepsilon|)^m-1)\) follows from \(\|U\|=1\) and \(\|E\|=1\). For the second inequality, apply the mean value theorem to \(f(t) = (1+t)^m\) on \([0,|\varepsilon|]\): \(f(|\varepsilon|) - f(0) = f'(\xi) \cdot |\varepsilon|\) for some \(\xi \in (0,|\varepsilon|)\). Since \(f'(t) = m(1+t)^{m-1}\) is increasing, \(f'(\xi) \le f'(|\varepsilon|) = m(1+|\varepsilon|)^{m-1}\), giving \((1+|\varepsilon|)^m - 1 \le |\varepsilon| \cdot m (1+|\varepsilon|)^{m-1}\). ◻
Theorem 10 (Essential spectrum preservation). \(\sigma_{\mathrm{ess}}(T) = \mathbb{S}^1\). Since \(\varepsilon E\) is compact (rank one), Weyl’s theorem on the invariance of the essential spectrum under compact perturbations gives \(\sigma_{\mathrm{ess}}(T) = \sigma_{\mathrm{ess}}(U)\). The essential spectrum of the forward unilateral shift is \(\sigma_{\mathrm{ess}}(U) = \mathbb{S}^1\) (see, e.g., [15] or [16]), so \(\sigma_{\mathrm{ess}}(T) = \mathbb{S}^1\), and the model does not exhibit the non-Hermitian skin effect.
3 Cochains, Coboundaries, and the Boundary Cocycle System↩︎
3.1 Site-localized cochains↩︎
We work in the continuous Chevalley–Eilenberg complex of \(\mathcal{A}\), equipped with the operator norm, and trivial coefficients \(\mathbb{C}\). Here \(\mathcal{A}\) is a normed Lie subalgebra of the Banach–Lie algebra \((\mathcal{B}(\ell^2),[\cdot,\cdot])\) (Definition 2); it is not itself complete (Remark 1), so we use continuous (operator-norm bounded) cochains throughout. For the algebraic background on Lie-algebra cohomology see, e.g., [17], [18].
Definition 5 (Boundary cochains). For each \(j \in \mathbb{Z}_{\ge 0}\), define \[\omega_j : \mathcal{A} \times \mathcal{A} \to \mathbb{C}, \quad \omega_j(X,Y) = \langle e_j, [X,Y] e_j \rangle.\]
Proposition 11 (Cocycle property). Each \(\omega_j\) is a continuous antisymmetric Chevalley–Eilenberg \(2\)-cocycle:
**(Antisymmetry)* \(\omega_j(X,Y) = -\omega_j(Y,X)\) for all \(X,Y\in\mathcal{A}\).*
**(Cocycle identity) \(d\omega_j(X,Y,Z) := \omega_j([X,Y],Z) + \omega_j([Y,Z],X) + \omega_j([Z,X],Y) = 0\).
**(Continuity)* \(|\omega_j(X,Y)| \le \|[X,Y]\| \le 2\|X\|\|Y\|\).*
Proof. (1) \(\omega_j(X,Y) = \langle e_j,[X,Y]e_j\rangle = -\langle e_j,[Y,X]e_j\rangle = -\omega_j(Y,X)\).
(2) By linearity of \(A \mapsto \langle e_j, A e_j\rangle\): \[d\omega_j(X,Y,Z) = \langle e_j,\bigl([[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]\bigr)e_j\rangle = 0,\] where the bracket expression vanishes by the Jacobi identity.
(3) \(|\omega_j(X,Y)| = |\langle e_j,[X,Y]e_j\rangle| \le \|[X,Y]\| \le 2\|X\|\|Y\|\). ◻
Definition 6 (Diagonal cocycle space). Let \(\delta:\mathfrak{sl}_{\mathrm{fin}}\to\mathfrak{sl}_{\mathrm{fin}}\) be the diagonal projection \(\delta(F)=\sum_{j\ge0}\langle e_j,Fe_j\rangle\,|e_j\rangle\langle e_j|\). We define the diagonal cocycle space* as the locally finite linear span of the boundary cochains, \[Z^2_{\mathrm{diag}}(\mathcal{A},\mathbb{C}) := \Bigl\{\, \textstyle\sum_{j\ge0} c_j\,\omega_j : c_j\in\mathbb{C} \,\Bigr\},\] where the sum is required to be finite on each pair \((X,Y)\) (automatic, since by Lemma 4 only finitely many \(\omega_j(X,Y)\) are nonzero). Each element is a continuous \(2\)-cocycle, and depends on \((X,Y)\) only through the diagonal \(\delta([X,Y])=\sum_j\omega_j(X,Y)\,|e_j\rangle\langle e_j|\) of the commutator; this is the sense in which the cocycles of \(Z^2_{\mathrm{diag}}\) factor through the diagonal of the commutator.*
Example 1 (Non-triviality as cochains). Let \(X = |e_1\rangle\langle e_0|\), \(Y = |e_0\rangle\langle e_1|\). Then \([X,Y] = |e_1\rangle\langle e_1| - |e_0\rangle\langle e_0|\), so \(\omega_1(X,Y) = 1 \ne 0\). Each \(\omega_j\) is nontrivial as a cochain, though coboundary as a cohomology class (see Theorem 12).
| Algebra | Generators | Class \([\omega_j]\) | Cocycle structure |
|---|---|---|---|
| \(\langle T \rangle\) | \(\{T^n\}\) | \(0\) | abelian; \(\omega_j \equiv 0\); \(H^2\ne0\) (forms on bulk) |
| \(\mathcal{A}_{\mathrm{edge}}\) | \(\{|e_a\rangle\langle e_b|\}\) | \(0\) | \(\{\omega_j\}\) diagonal cochains, relation \(\sum_j\omega_j=0\) |
| \(\mathcal{A}\) | \(\langle U\rangle + \mathcal{A}_{\mathrm{edge}}\) | \(0\) | same \(\{\omega_j\}\); \(H^2(\mathcal{A},\mathbb{C})\ne0\) (bulk) |
3.2 Coboundary structure of the boundary cochains↩︎
Theorem 12 (Coboundary structure of the boundary cochains).
For each \(j \ge 0\), the continuous \(1\)-cochain \(\eta_j : \mathcal{A} \to \mathbb{C}\) defined by \(\eta_j(A) = \langle e_j, Ae_j\rangle\) satisfies \(\omega_j = d\eta_j\); hence each \(\omega_j\) is a coboundary and \([\omega_j] = 0\) in \(H^2(\mathcal{A},\mathbb{C})\).
Any finite subfamily \(\{\omega_j : 0 \le j \le M\}\) is linearly independent, while the full family obeys the single relation \(\sum_{j\ge0}\omega_j = 0\) (Remark 7). Consequently every element of \(Z^2_{\mathrm{diag}}(\mathcal{A},\mathbb{C})\) (Definition 6) is of the form \(\sum_{j}c_j\,\omega_j\), the coefficients \((c_j)\) being unique modulo addition of a common constant.
The cohomology \(H^2(\mathcal{A},\mathbb{C})\) is not* trivial: the abelian bulk \(\mathrm{span}\{U^n:n\ge0\}\) already supports an infinite-dimensional space of nontrivial continuous \(2\)-cocycle classes. The diagnostic role of \(\{\omega_j\}\) therefore rests on their site resolution, not on cohomological nontriviality.*
Proof. (1). For \(X,Y\in\mathcal{A}\), \[(d\eta_j)(X,Y) = \eta_j([X,Y]) = \langle e_j,[X,Y]e_j\rangle = \omega_j(X,Y),\] so \(\omega_j = d\eta_j \in B^2(\mathcal{A},\mathbb{C})\) and \([\omega_j]=0\).
(2). Independence. Suppose \(\sum_{j=0}^{M}\lambda_j\,\omega_j = 0\). Fix \(i\in\{0,\dots,M\}\), choose any \(k>M\), and set \(X=|e_i\rangle\langle e_k|\), \(Y=|e_k\rangle\langle e_i|\). By Proposition 5, \([X,Y]=|e_i\rangle\langle e_i|-|e_k\rangle\langle e_k|\), so \(\omega_j(X,Y)=\delta_{j,i}-\delta_{j,k}\). Since \(k>M\), evaluating the relation at \((X,Y)\) gives \(\lambda_i=0\). As \(i\) was arbitrary, the subfamily is linearly independent. Relation. By Lemma 2, \(\sum_{j}\omega_j(X,Y)=\mathrm{tr}[X,Y]=0\) for all \(X,Y\), i.e.\(\sum_j\omega_j=0\). Spanning of \(Z^2_{\mathrm{diag}}\). By Definition 6 every \(\Omega\in Z^2_{\mathrm{diag}}\) is, by construction, of the form \(\Omega=\sum_j c_j\,\omega_j\) (finite on each pair by Lemma 4). The coefficients \((c_j)\) are determined by \(\Omega\) up to adding a common constant to all of them: if \(\sum_j c_j\omega_j=\sum_j c_j'\omega_j\) then \(\sum_j(c_j-c_j')\omega_j=0\), and evaluating at the pairs \((|e_i\rangle\langle e_k|, |e_k\rangle\langle e_i|)\) above forces \(c_i-c_i'=c_k-c_k'\) for all \(i,k\), i.e.\(c_j-c_j'\) is constant in \(j\).
(3). It suffices to exhibit continuous \(2\)-cocycles that are not coboundaries. Each \(X\in\mathcal{A}\) has a unique decomposition \(X=p_X(U)+X_{\mathrm{edge}}\) with \(p_X(z)=\sum_{m\ge0}\alpha_m(X)z^m\) a polynomial and \(X_{\mathrm{edge}}\in\mathcal{A}_{\mathrm{edge}}\). For an antisymmetric array \((B_{mn})\) set \[\Omega_B(X,Y):=\sum_{m,n}\alpha_m(X)\,\alpha_n(Y)\,B_{mn},\] so that \(\Omega_B\) is bilinear, antisymmetric, and vanishes whenever \(X\) or \(Y\) lies in \(\mathcal{A}_{\mathrm{edge}}\). For \(k\ge1\) let \(\Omega_k:=\Omega_{B^{(k)}}\) with \(B^{(k)}_{0k}=-B^{(k)}_{k0}=1\) and all other entries zero, i.e. \(\Omega_k(X,Y)=\alpha_0(X)\alpha_k(Y)-\alpha_k(X)\alpha_0(Y)\).
Cocycle. Since \(\mathcal{A}_{\mathrm{edge}}\) is an ideal with abelian quotient \(\mathrm{span}\{U^n\}\) (Remark 6), each term of \(d\Omega_B(X,Y,Z)=\Omega_B([X,Y],Z)+\Omega_B([Y,Z],X)+\Omega_B([Z,X],Y)\) vanishes: if all three arguments are bulk every bracket is \(0\); if at least one is in \(\mathcal{A}_{\mathrm{edge}}\) then in each term either the bracket lies in \(\mathcal{A}_{\mathrm{edge}}\) or the remaining slot does, and \(\Omega_B\) kills it. Hence \(\Omega_B\in Z^2(\mathcal{A},\mathbb{C})\).
Continuity. The coefficient functionals are bounded in operator norm. Passing to the Calkin algebra (whose quotient norm is the essential norm \(\|\cdot\|_{\mathrm{ess}}\)) annihilates the finite-rank part \(X_{\mathrm{edge}}\) and sends \(U\) to a unitary of spectrum \(\mathbb{S}^1\) (Theorem 10; see also [19], [20]); since a polynomial in a unitary is normal, \(\|p_X\|_\infty=\|p_X(U)\|_{\mathrm{ess}}=\|X\|_{\mathrm{ess}}\le\|X\|\). By Cauchy’s estimate, \(|\alpha_m(X)|=|p_X^{(m)}(0)|/m!\le\|p_X\|_\infty\le\|X\|\). Therefore \(|\Omega_k(X,Y)|\le 2\|X\|\,\|Y\|\), so each \(\Omega_k\) is a continuous cocycle.
Non-triviality and infinite dimension. With \(I=U^0\) one has \(\Omega_j(I,U^k)=\delta_{jk}\). If \(\sum_k\lambda_k\Omega_k\) (a finite combination) were a coboundary \(d\phi\), then for each \(k\), \(\lambda_k=\bigl(\sum_i\lambda_i\Omega_i\bigr)(I,U^k)=\phi([I,U^k])=\phi(0)=0\). Hence the classes \(\{[\Omega_k]\}_{k\ge1}\) are linearly independent and nonzero, so \(H^2(\mathcal{A},\mathbb{C})\) is infinite-dimensional; in particular it is nonzero. ◻
Remark 13 (On terminology and topology). We avoid the term basis for \(\{\omega_j\}\): because of the relation \(\sum_j\omega_j=0\), the family is linearly independent only in finite subfamilies and spans \(Z^2_{\mathrm{diag}}\) only modulo that relation. We work with algebraic (not topological) spanning; a topological statement would require fixing a Banach or Fréchet topology on \(Z^2_{\mathrm{diag}}\), which we do not pursue here.
Remark 14 (Diagnostic value despite triviality of the classes). Although each \(\omega_j\) is a coboundary (and although \(H^2(\mathcal{A},\mathbb{C})\) is itself nontrivial through the bulk), the map \(j \mapsto \omega_j(X,Y)\) is an intrinsic site-resolved quantity: it reads the \(j\)-th diagonal entry of \([X,Y]\), vanishing for \(j \gg 0\) (bulk) and being nonzero near \(j = 0\) (edge). This yields a bulk–edge dichotomy at the cochain level, independent of cohomological considerations. Table 2 collects the cocycle structure of the three algebras \(\langle T\rangle\), \(\mathcal{A}_{\mathrm{edge}}\), and \(\mathcal{A}\).
Remark 15 (Distinction from Virasoro cocycle). The Virasoro cocycle is translation-invariant and generates a nontrivial central extension. Our cochains \(\omega_j\) are spatially localized, coboundaries in the full complex, and reflect geometric inhomogeneity rather than global symmetry breaking.
3.3 Central extensions from the bulk↩︎
The non-vanishing of \(H^2(\mathcal{A},\mathbb{C})\) found in Theorem 12(3) has a concrete structural meaning: it classifies the nontrivial one-dimensional central extensions of \(\mathcal{A}\). Recall that a continuous \(2\)-cocycle \(\Omega\in Z^2(\mathcal{A},\mathbb{C})\) determines a central extension \[0 \longrightarrow \mathbb{C}c \longrightarrow \widetilde{\mathcal{A}}_\Omega \longrightarrow \mathcal{A}\longrightarrow 0,\] where \(\widetilde{\mathcal{A}}_\Omega := \mathcal{A}\oplus\mathbb{C}c\) carries the bracket \[\label{eq:central-bracket} [\,X\oplus s c,\; Y\oplus t c\,]_\Omega := [X,Y]\oplus \Omega(X,Y)\,c, \qquad c \text{ central}.\tag{4}\] The bracket 4 satisfies the Jacobi identity precisely because \(\Omega\) satisfies the cocycle identity, and \(\widetilde{\mathcal{A}}_\Omega\) is again a normed Lie algebra with continuous bracket when \(\Omega\) is continuous, since both \([\cdot,\cdot]\) and \(\Omega\) are. The extension splits, i.e.is equivalent to the trivial extension \(\mathcal{A}\oplus\mathbb{C}c\), if and only if \(\Omega\) is a coboundary; equivalence classes of central extensions are in bijection with \(H^2(\mathcal{A},\mathbb{C})\).
By Theorem 12(1) the diagnostic cochains \(\omega_j\) are coboundaries and hence yield only split extensions. The nontrivial extensions originate entirely in the abelian bulk. Recall from the proof of Theorem 12(3) that each \(X\in\mathcal{A}\) has a unique bulk symbol \(p_X(z)=\sum_m\alpha_m(X)z^m\), that \(|\alpha_m(X)|\le\|X\|\), and that for an antisymmetric array \((B_{mn})\) the form \[\Omega_B(X,Y)=\sum_{m,n}\alpha_m(X)\,\alpha_n(Y)\,B_{mn}\] is a \(2\)-cocycle vanishing on \(\mathcal{A}_{\mathrm{edge}}\).
Proposition 16 (A continuous family of nontrivial classes). Let \(B=(B_{mn})_{m,n\ge0}\) be antisymmetric with \(\sum_{m,n}|B_{mn}|<\infty\) (in particular, any finitely supported \(B\)). Then \(\Omega_B\) is a continuous \(2\)-cocycle, with \[|\Omega_B(X,Y)|\le\Bigl(\textstyle\sum_{m,n}|B_{mn}|\Bigr)\|X\|\,\|Y\|,\] and the assignment \(B \mapsto [\Omega_B]\in H^2(\mathcal{A},\mathbb{C})\) is linear and injective. Consequently \(H^2(\mathcal{A},\mathbb{C})\) is infinite-dimensional, and the central extensions \(\widetilde{\mathcal{A}}_{\Omega_B}\) are pairwise inequivalent for distinct \(B\).
Proof. That \(\Omega_B\) is a cocycle was shown in Theorem 12(3). Using \(|\alpha_m(X)|\le\|X\|\), \[|\Omega_B(X,Y)|\le\sum_{m,n}|B_{mn}|\,|\alpha_m(X)|\,|\alpha_n(Y)| \le\Bigl(\sum_{m,n}|B_{mn}|\Bigr)\|X\|\,\|Y\|,\] so \(\Omega_B\) is continuous. The map \(B\mapsto\Omega_B\) is clearly linear. If \(\Omega_B=d\phi\) for some \(1\)-cochain \(\phi\), then \(B_{mn}=\Omega_B(U^m,U^n)=\phi([U^m,U^n])=\phi(0)=0\); hence \([\Omega_B]=0\) forces \(B=0\), and \(B\mapsto[\Omega_B]\) is injective. The antisymmetric finitely supported arrays form an infinite-dimensional space, so \(\dim H^2(\mathcal{A},\mathbb{C})=\infty\). Finally, two central extensions are equivalent iff the defining cocycles are cohomologous, so injectivity gives the inequivalence of the \(\widetilde{\mathcal{A}}_{\Omega_B}\) for distinct \(B\). ◻
The simplest nontrivial class has a familiar shape.
Proposition 17 (Heisenberg central extension). For \(\Omega_1(X,Y)=\alpha_0(X)\alpha_1(Y)-\alpha_1(X)\alpha_0(Y)\) (the case \(B_{01}=-B_{10}=1\)), the elements \(\widetilde{I}:=I\oplus0\), \(\widetilde{U}:=U\oplus0\) and \(c\) span, inside \(\widetilde{\mathcal{A}}_{\Omega_1}\), a copy of the three-dimensional Heisenberg Lie algebra \(\mathfrak{h}_3\): \[[\widetilde{I},\widetilde{U}]_{\Omega_1}=c,\qquad [\widetilde{I},c]=[\widetilde{U},c]=0.\]
Proof. Since \(\alpha_0(I)=\alpha_1(U)=1\) and \(\alpha_1(I)=\alpha_0(U)=0\), we have \(\Omega_1(I,U)=1\), while \([I,U]=0\) in \(\mathcal{A}\). Hence by 4 , \([\widetilde{I},\widetilde{U}]_{\Omega_1}=[I,U]\oplus\Omega_1(I,U)c=c\). As \(c\) is central, \(\mathrm{span}\{\widetilde{I},\widetilde{U},c\}\) is a Lie subalgebra with exactly the Heisenberg relations. ◻
Remark 18. The algebra \(\mathcal{A}\) thus carries Heisenberg-type central extensions, arising from its abelian bulk \(\mathrm{span}\{U^n\}\) independently of the boundary defect* \(\varepsilon E\), in contrast to the diagnostic cochains \(\omega_j\), which are coboundaries (Theorem 12(1)). The nontrivial cohomology of \(\mathcal{A}\) thus sits in the abelian bulk rather than at the edge: the boundary defect manifests in the site-resolved (yet cohomologically trivial) cochains \(\{\omega_j\}\) at the edge, while the nontrivial cohomology is carried by the bulk. Unlike the Virasoro cocycle, the classes \([\Omega_B]\) are not tied to a single translation-invariant central charge but form an infinite-dimensional family indexed by antisymmetric symbols \(B\).*
4 The Boundary Cochains as a Site-Resolved Index↩︎
The trace relation \(\sum_j\omega_j=0\) on \(\mathcal{A}\) (Remark 7) is the vanishing of a Fredholm index. We make this precise by adjoining the backward shift \(U^*\), passing from the analytic algebra \(\mathcal{A}\) to the polynomial Toeplitz Lie algebra. There the total cochain \(\sum_j\omega_j\) computes the symbol pairing \(\frac{1}{2\pi i}\oint f\,dg\), equal to the Fredholm index \(-\mathrm{wind}(f)\) for conjugate symbols \(g=1/f\), and the individual \(\omega_j\) resolve that invariant over the boundary sites.
4.1 The Toeplitz extension↩︎
For a trigonometric polynomial \(f(z)=\sum_k\hat{f}_k z^k\) on \(\mathbb{S}^1\), let \(T_f\) be the associated Toeplitz operator on \(\ell^2(\mathbb{Z}_{\ge 0})\cong H^2(\mathbb{S}^1)\); on generators \(T_{z^k}=U^k\) for \(k\ge0\) and \(T_{z^{-k}}=(U^*)^k\) for \(k\ge0\). Set \[\mathcal{A}^\sharp := \mathrm{span}\{T_f : f \text{ a trigonometric polynomial}\} + \mathcal{A}_{\mathrm{edge}},\] the polynomial Toeplitz Lie algebra. Then \(\mathcal{A}\subset\mathcal{A}^\sharp\) is its analytic part (symbols \(f\in\mathbb{C}[z]\)), and the cochains \(\omega_j(X,Y)=\langle e_j,[X,Y]e_j\rangle\) extend verbatim. As is classical, \(\mathcal{A}^\sharp\) is commutative modulo \(\mathcal{A}_{\mathrm{edge}}\), realizing at the polynomial level the Toeplitz extension \[0 \longrightarrow \mathcal{K} \longrightarrow \mathcal{T} \longrightarrow C(\mathbb{S}^1) \longrightarrow 0\] of the compacts by the symbol algebra [15], [20], [21]. In this dictionary the edge ideal \(\mathcal{A}_{\mathrm{edge}}\) is the (finitary) compact part and the bulk quotient is the symbol \(f\) on the circle.
4.2 The index identity↩︎
Lemma 7 (Powers of the shift). For \(n\ge1\), \[(U^*)^n U^n = I,\qquad U^n(U^*)^n = I-P_n,\qquad [U^n,(U^*)^n] = -P_n,\] where \(P_n=\sum_{k=0}^{n-1}|e_k\rangle\langle e_k|\) is the orthogonal projection onto the first \(n\) sites. Consequently \[\omega_j\bigl(U^n,(U^*)^n\bigr) = -\,\mathbf{1}_{\{j<n\}},\qquad \sum_{j\ge0}\omega_j\bigl(U^n,(U^*)^n\bigr) = -n.\]
Proof. For all \(k\ge0\), \((U^*)^n U^n e_k=(U^*)^n e_{k+n}=e_k\), so \((U^*)^n U^n=I\). For the other order, \((U^*)^n e_k=e_{k-n}\) when \(k\ge n\) and \((U^*)^n e_k=0\) when \(k<n\); applying \(U^n\) gives \(U^n(U^*)^n e_k=e_k\) for \(k\ge n\) and \(0\) for \(k<n\), i.e.\(U^n(U^*)^n=I-P_n\). Subtracting, \([U^n,(U^*)^n]=(I-P_n)-I=-P_n\). The \((j,j)\) entry of \(-P_n\) is \(-1\) for \(j<n\) and \(0\) otherwise, and \(\mathrm{tr}(-P_n)=-n\). ◻
Theorem 19 (Total cochain as a symbol pairing and Fredholm index). For trigonometric polynomials \(f,g\), the commutator \([T_f,T_g]\) is finite-rank, and \[\sum_{j\ge0}\omega_j(T_f,T_g)\;=\;\mathrm{tr}\,[T_f,T_g]\;=\;\frac{1}{2\pi i}\oint_{\mathbb{S}^1} f\,dg .\] In particular, with \(f(z)=z^n\) and \(g(z)=z^{-n}\) this recovers \(\sum_j\omega_j(U^n,(U^*)^n)=-n=\operatorname{index}(U^n)\), and the cochain profile \(\omega_j(U^n,(U^*)^n)=-\mathbf{1}_{\{j<n\}}\) of Lemma 7 displays the index \(-n\) as a sum of \(n\) unit contributions localized at the boundary sites \(j=0,\dots,n-1\).
Proof. By bilinearity and antisymmetry it suffices to evaluate on monomials \(f=z^k\), \(g=z^l\), i.e. \(T_f=U^{(k)}\) and \(T_g=U^{(l)}\), where \(U^{(m)}:=U^m\) for \(m\ge0\) and \(U^{(m)}:=(U^*)^{-m}\) for \(m<0\). Each commutator \([U^{(k)},U^{(l)}]\) is a difference of corner operators (Proposition 5 together with Lemma 7), hence finite-rank; therefore \([T_f,T_g]\) is finite-rank and \(\sum_j\omega_j(T_f,T_g)=\sum_j\langle e_j,[T_f,T_g]e_j\rangle=\mathrm{tr}[T_f,T_g]\). For the trace, note that \([U^{(k)},U^{(l)}]\) maps each \(e_b\) into \(\mathbb{C}\,e_{b+k+l}\), so it is a combination of corner operators \(|e_{b+k+l}\rangle\langle e_b|\) and its diagonal entries vanish unless \(k+l=0\); hence \(\mathrm{tr}[U^{(k)},U^{(l)}]=0\) for \(k+l\ne0\). By Lemma 7, \(\mathrm{tr}[U^{(n)},U^{(-n)}]=\mathrm{tr}[U^n,(U^*)^n]=-n\) and \(\mathrm{tr}[U^{(-n)},U^{(n)}]=+n\) for \(n\ge1\). Hence \(\mathrm{tr}[T_{z^k},T_{z^l}]=l\,\delta_{k+l,0}\). On the other hand \[\frac{1}{2\pi i}\oint_{\mathbb{S}^1} z^k\,d(z^l) =\frac{l}{2\pi i}\oint_{\mathbb{S}^1} z^{k+l-1}\,dz =l\,\delta_{k+l,0}.\] The two expressions agree on monomials, and bilinearity extends the identity to all trigonometric polynomials. ◻
Example 2 (The index density for \(n=2\)). Take \(f(z)=z^2\) and \(g(z)=z^{-2}\), so \(T_f=U^2\) and \(T_g=(U^*)^2\). By Lemma 7, \([U^2,(U^*)^2]=-P_2\), the negative of the projection onto \(\mathrm{span}\{e_0,e_1\}\), whence \[\omega_0=\omega_1=-1,\qquad \omega_j=0\;(j\ge2),\qquad \sum_{j\ge0}\omega_j=-2=\operatorname{index}(U^2).\] The index \(-2\) is carried one unit per boundary site over the first two sites and is invisible in the bulk. Figure 1 shows this step profile for \(n=1,2,3\).
Remark 20 (Bulk–edge correspondence). Theorem 19 expresses the bulk–edge correspondence through the cochains. The winding number \(\mathrm{wind}(f)=\frac{1}{2\pi i}\oint_{\mathbb{S}^1} f^{-1}\,df\) is a bulk* quantity (it depends only on the symbol \(f\) on the circle), whereas the cochains \(\omega_j\) live at the edge, on the boundary sites \(j\). For an invertible symbol the Noether–Gohberg–Krein theorem gives \(\operatorname{index}(T_f)=-\mathrm{wind}(f)\) [8], [15], and the trace form of Theorem 19 extends from polynomials to smooth symbols [22], so that \[\operatorname{index}(T_f)\;=\;-\,\mathrm{wind}(f)\;=\;\sum_{j\ge0}\omega_j\bigl(T_f,T_{f^{-1}}\bigr).\] Thus \(\{\omega_j\}\) is a site-resolved index density: the bulk invariant is the sum of edge contributions, which are localized near \(j=0\). The vanishing \(\sum_j\omega_j=0\) on the analytic algebra \(\mathcal{A}\) (Remark 7) follows because analytic symbols \(f\in\mathbb{C}[z]\) extend holomorphically to the disk and have winding number zero, so no index appears until the backward shift \(U^*\) is adjoined; cf.the index-theoretic treatments of the bulk–edge correspondence in [21], [23]–[25].*
Remark 21 (Relation to the boundary defect). The deformed operator \(T=U+\varepsilon E\) has the same symbol \(z\) as \(U\) (the defect \(\varepsilon E\) is compact), hence \(\operatorname{index}(T)=\operatorname{index}(U)=-1\) for the Fredholm range of parameters, independently of \(\varepsilon\). The boundary defect therefore cannot change the bulk winding number; its effect is confined to the point spectrum (Corollary 25) and to the site-resolved profile of the cochains, consistent with the dichotomy of Remark 20.
4.3 Spatially modulated couplings and a topological transition↩︎
The single defect \(\varepsilon E\) is the case \(\varepsilon_j=\varepsilon\,\delta_{j0}\) of a spatially modulated on-site coupling \[T_\varepsilon := U + D_\varepsilon,\qquad D_\varepsilon := \sum_{j\ge0}\varepsilon_j\,|e_j\rangle\langle e_j|,\quad (\varepsilon_j)\in\ell^\infty.\] The index of §4.2 is insensitive to a boundary-localized profile (if \(\varepsilon_j\to0\) then \(D_\varepsilon\) is compact and \(\operatorname{index}(T_\varepsilon)=\operatorname{index}(U)=-1\)), but it jumps once the coupling tends to a nonzero bulk value.
Lemma 8 (Uniform coupling). Let \(T_c:=U+cI\) with \(c\in\mathbb{C}\). Then \(\ker T_c=\{0\}\), while \(\ker T_c^*\) is spanned by the vector \(v=(v_n)_{n\ge0}\), \(v_n=(-\bar c)^n\), which lies in \(\ell^2(\mathbb{Z}_{\ge 0})\) if and only if \(|c|<1\). Hence \(T_c\) is Fredholm for \(|c|\ne1\), with \[\operatorname{index}(T_c)=\dim\ker T_c-\dim\ker T_c^*=-\,\mathbf{1}_{\{|c|<1\}}=-\,\mathrm{wind}(z+c).\]
Proof. The equation \(T_c v=0\) reads \(cv_0=0\) at \(n=0\) and \(v_{n-1}+cv_n=0\) for \(n\ge1\); for \(c\ne0\) this forces \(v_0=0\) and then \(v_n=0\) for all \(n\), while for \(c=0\), \(T_0=U\) is injective. Thus \(\ker T_c=\{0\}\). The adjoint equation \(T_c^* v=(U^*+\bar c I)v=0\) reads \(v_{n+1}+\bar c v_n=0\), i.e.\(v_n=(-\bar c)^n v_0\), which is square-summable iff \(|c|<1\). Since \(\sigma_{\mathrm{ess}}(T_c)=\sigma_{\mathrm{ess}}(U)+c=\mathbb{S}^1+c\) (the unit circle centered at \(c\)), \(T_c\) is Fredholm iff \(0\notin\mathbb{S}^1+c\), i.e.\(|c|\ne1\). The index formula follows; it equals \(-\mathrm{wind}(z+c)\) because the symbol \(z+c\) winds once about the origin exactly when \(|c|<1\) (Theorem 19, Remark 20). ◻
Theorem 22 (Bulk-driven topological transition). Let \(\varepsilon_j\to c\) with \(|c|\ne1\). Then \(T_\varepsilon=U+D_\varepsilon\) is Fredholm with symbol \(z+c\), and \[\operatorname{index}(T_\varepsilon)=\operatorname{index}(U+cI)=-\,\mathbf{1}_{\{|c|<1\}} =-\,\mathrm{wind}(z+c)=\sum_{j\ge0}\omega_j\bigl(T_{z+c},T_{1/(z+c)}\bigr).\] The value is independent of the boundary profile* \((\varepsilon_j)\) and is fixed by the bulk limit \(c\) alone. As \(|c|\) crosses \(1\) the index jumps from \(-1\) to \(0\): a topological transition carried by the bulk.*
Proof. The operator \(T_\varepsilon-(U+cI)=\sum_j(\varepsilon_j-c)|e_j\rangle\langle e_j|\) is a norm-limit of finite-rank operators (since \(\varepsilon_j-c\to0\)), hence compact; therefore \(T_\varepsilon\) and \(U+cI=T_{z+c}\) share the symbol \(z+c\) and, by the invariance of the Fredholm index under compact perturbations, the same index. Lemma 8 evaluates it, and the cochain expression is Remark 20 applied to the smooth invertible symbol \(z+c\) (\(|c|\ne1\)). Figure 2 plots the resulting index as a function of the bulk modulus \(|c|\). ◻
Remark 23. The boundary profile \((\varepsilon_j)\) controls only where* the index density \(j\mapsto\omega_j\) sits (its localization length and shape near the edge), never its integer total, which is the bulk invariant \(-\mathrm{wind}(z+c)\): the integer changes only when the bulk symbol crosses the unit circle. The threshold \(|c|=1\) echoes the bound-state threshold of the single defect (Lemma 9), though the mechanisms differ: there a rank-one defect creates a point eigenvalue for \(|\varepsilon|>1\); here a uniform bulk coupling creates a cokernel, hence a nonzero index, for \(|c|<1\).*
5 Finite-Dimensional Realizations↩︎
Example 3 (Four-site truncation). On \(\mathbb{C}^4\) with basis \(\{e_0,e_1,e_2,e_3\}\), the forward shift is lower bidiagonal: \[U = \begin{pmatrix} 0&0&0&0\\1&0&0&0\\0&1&0&0\\0&0&1&0 \end{pmatrix}, \quad T = U + \varepsilon E = \begin{pmatrix} \varepsilon&0&0&0\\1&0&0&0\\0&1&0&0\\0&0&1&0 \end{pmatrix}.\] One verifies \([T,T^2]=0\) (abelian subalgebra). Nontrivial commutators arise only from \(\mathcal{A}_{\mathrm{edge}}\). In finite dimension the shift is nilpotent and every operator has Fredholm index \(0\); the index identity of Section 4 therefore has no finite-volume counterpart, confirming that it is a half-infinite, boundary phenomenon. It is recovered only in the limit \(N\to\infty\), where \(U^*U=I\) but \(UU^*=I-P_0\ne I\).
6 Applications↩︎
6.1 Tight-binding edge states↩︎
For \(\varepsilon \in \mathbb{R}\), the Hamiltonian \(H = U + U^* + \varepsilon E\) has nearest-neighbor hopping and boundary potential \(\varepsilon\) at site 0. Under the forward shift convention, site-localized spectral features arise from the rank-one perturbation.
6.2 Spectral structure↩︎
Lemma 9 (Eigenvectors of \(T\)). Let \(T = U + \varepsilon E\) with \(U\) the forward shift. The eigenvalue equation \(Tv = \lambda v\) reads, in components: \[\begin{align} n=0&: \quad \varepsilon v(0) = \lambda v(0), \\ n \ge 1&: \quad v(n-1) = \lambda v(n). \end{align}\] If \(v(0) \ne 0\), then \(\lambda = \varepsilon\) and \(v(n) = \varepsilon^{-n} v(0)\). The vector \(v \in \ell^2(\mathbb{Z}_{\ge 0})\) if and only if \(|\varepsilon| > 1\). If \(v(0) = 0\), then the recurrence \(v(n-1) = \lambda v(n)\) with \(v(0) = 0\) forces \(v = 0\), so there are no eigenvectors with \(v(0) = 0\).
Proof. With \((Uv)(n) = v(n-1)\) for \(n \ge 1\) and \((Uv)(0) = 0\) (equivalently \(EU=0\), Lemma 1), the eigenvalue equation at \(n=0\) gives \(0 + \varepsilon v(0) = \lambda v(0)\), so \(\lambda = \varepsilon\) (if \(v(0)\ne0\)). For \(n \ge 1\): \(v(n-1) = \lambda v(n)\), giving \(v(n) = \lambda^{-1} v(n-1) = \lambda^{-n} v(0)\). Then \(\sum_n |v(n)|^2 = |v(0)|^2 \sum_n |\lambda|^{-2n} < \infty\) iff \(|\lambda^{-1}|<1\), i.e., \(|\varepsilon|>1\). If \(v(0) = 0\), then \(v(n-1) = \lambda v(n)\) with \(v(0)=0\) gives \(v(1) = \lambda^{-1}v(0) = 0\), and by induction \(v(n) = 0\) for all \(n\). Thus no nonzero eigenvector exists with \(v(0) = 0\). ◻
Remark 24 (Bound state versus index). The boundary bound state appears for \(|\varepsilon|>1\), yet it does not alter the Fredholm index: since \(\varepsilon E\) is finite-rank, hence compact, \(\operatorname{index}(T)=\operatorname{index}(U)=-1\) for every \(\varepsilon\) (Remark 21). The defect thus reshapes the spectrum without touching the topological invariant, in contrast to the bulk coupling \(U+cI\) of Section 4, whose change is not compact and does move the index.
Corollary 25 (Spectral structure). Let \(T = U + \varepsilon E\) with \(U\) the forward shift. Then:
Remark 26 (Physical interpretation). The edge state exists for large boundary coupling \(|\varepsilon|>1\) under the forward shift convention. For the complementary physical picture of a decaying mode \(v(n) = \varepsilon^n\) for small \(|\varepsilon|<1\), one uses the adjoint operator \(T^* = U^* + \bar\varepsilon E\), where \(U^*\) is the backward shift.
6.3 Measurement of boundary cochains↩︎
For \(X = |e_1\rangle\langle e_0|\), \(Y = |e_0\rangle\langle e_1|\): \[\omega_0(X,Y) = -1, \quad \omega_1(X,Y) = +1, \quad \omega_j(X,Y) = 0 \text{ for } j \ge 2.\]
6.4 Algebraic exactness of the boundary cochains↩︎
For fixed \(X,Y \in \mathcal{A}_{\mathrm{edge}}\) the value \(\omega_j(X,Y)=\langle e_j,[X,Y]e_j\rangle\) depends only on the commutator \([X,Y]\), computed within the edge ideal. Adding any bulk operator \(\sum_n\alpha_n U^n\) to \(X\) or \(Y\) changes the commutator only through the bulk–edge brackets 3 , whose diagonal entries are fixed integers independent of the coefficients \(\alpha_n\). In this precise sense the site profile \(j\mapsto\omega_j(X,Y)\) is an exact algebraic quantity, not an approximate one. The listing in Appendix ¿sec:app:numerics? (Code 5) merely illustrates that the computed value of \(\omega_0\) equals its exact algebraic value \(-1\); it adds synthetic measurement noise and does not model physical disorder in \(T\), and should be read only as a display of the exact constant.
7 Conclusion↩︎
We have analyzed the operator-algebraic and cohomological structure of \(T = U + \varepsilon E\) on \(\ell^2(\mathbb{Z}_{\ge 0})\), with \(U\) the forward shift \(Ue_n = e_{n+1}\) (isometry) fixed as the unique convention throughout.
The key structural result (Lemma 2) is that every commutator in \(\mathcal{A}\) is finitely supported with trace zero, with \[[\mathcal{A},\mathcal{A}] = \mathfrak{sl}_{\mathrm{fin}} = \{\,F : F \text{ finitely supported},\;\mathrm{tr}\,F = 0\,\}\] (the finitely supported trace-zero operators; these are not diagonal in general). The site-resolved cochains \(\omega_j(X,Y) = \langle e_j,[X,Y]e_j\rangle\) read off the \(j\)-th diagonal entry of each commutator, subject to the trace relation \(\sum_j\omega_j=0\).
Theorem 12 establishes that each \(\omega_j\) is a coboundary (with bounding cochain \(\eta_j(A) = \langle e_j,Ae_j\rangle\)), so \([\omega_j]=0\). The cohomology \(H^2(\mathcal{A},\mathbb{C})\) is nonetheless nontrivial: the abelian bulk contributes an infinite-dimensional family of \(2\)-cocycles \(\Omega_B\) (Proposition 16) that classify central extensions of \(\mathcal{A}\), the simplest being a Heisenberg extension \([\widetilde{I},\widetilde{U}]=c\) (Proposition 17). Thus the nontrivial cohomology lives in the bulk, while the diagnostic value of \(\omega_j\) is its site resolution rather than any cohomological nontriviality: the map \(j \mapsto \omega_j(X,Y)\) vanishes in the bulk and is nonzero at the edge, a bulk–edge dichotomy at the cochain level.
Theorem 19 makes this dichotomy topological. On the polynomial Toeplitz algebra \(\mathcal{A}^\sharp\) obtained by adjoining \(U^*\), the total cochain is a Fredholm index, \[\sum_{j\ge0}\omega_j(T_f,T_g)=\frac{1}{2\pi i}\oint_{\mathbb{S}^1} f\,dg, \qquad \sum_{j\ge0}\omega_j(U^n,(U^*)^n)=-n=\operatorname{index}(U^n)=-\mathrm{wind}(z^n),\] and the profile \(\omega_j(U^n,(U^*)^n)=-\mathbf{1}_{\{j<n\}}\) exhibits \(\{\omega_j\}\) as a site-resolved density for the bulk winding number. The trace relation \(\sum_j\omega_j=0\) on \(\mathcal{A}\) is thereby explained: analytic symbols do not wind [21], [23], [24].
For spatially modulated couplings \(T_\varepsilon=U+\sum_j\varepsilon_j|e_j\rangle\langle e_j|\) with \(\varepsilon_j\to c\), the index is fixed by the bulk limit alone (Theorem 22): it equals \(-\mathbf{1}_{\{|c|<1\}}\), independent of the boundary profile, and undergoes a topological transition as \(|c|\) crosses \(1\). The boundary modulation sets only the localization of the index density, never its integer total.
Future directions include higher-dimensional half-lattices, the \(K\)-theory of operator extensions [25], and position-dependent symbols with several transition points. A companion study examines the profile of the index density \(j\mapsto\omega_j(T_f,T_{1/f})\) beyond its integer total: for an inner (Blaschke) symbol with zero at \(a\), this profile is the modulus-squared of the edge state, \(-(1-|a|^2)|a|^{2j}\), exponentially localized with a length \(\xi=(2\log|a|^{-1})^{-1}\) that diverges as the symbol’s zero approaches the unit circle, a geometric refinement of the integer index.
Declarations↩︎
Funding↩︎
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Ethical approval↩︎
Not applicable.
Informed consent↩︎
Not applicable.
Data Availability Statement↩︎
No datasets were generated or analyzed during the current study.
Conflict of Interest↩︎
The author declare that they have no conflict of interest.
Appendix A: Numerical Methods↩︎
All figures were generated using NumPy, SciPy, and Matplotlib.
Operator construction (forward shift)
\(U\) is represented as the lower bidiagonal matrix \(U_{i+1,i}=1\), consistent with \(Ue_i = e_{i+1}\).
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Figure 3: Commutator rank growth: rank of \([T^m, U^n]\) as a function of \(m\) (bound is linear in \(m\), independent of \(n\))..
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Figure 4: Site-resolved cochain profile..
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Figure 5: Exact algebraic value \(\omega_0(X,Y)=-1\). The listing displays this exact constant; the parameter \(W\) adds synthetic measurement noise only and does not model physical disorder in \(T\) (error bars \(=\) noise model, not algebraic uncertainty)..