1 Introduction↩︎

The simplest example of bulk-edge correspondence (BEC) occurs in discrete 1D model Hamiltonians, such as the Su–Schrieffer–Heeger model [1]. A rough statement is: the chiral-symmetry condition disconnects the manifold of spectrally gapped model Hamiltonians, with winding numbers labelling the connected components; furthermore, this winding number counts the number of edge-localized zero-energy eigenstates for the Hamiltonian truncated to a half-line. The reader may recognize this claim as an index theorem, \[\text{Analytic index} = \text{Topological index},\] guaranteeing a certain stability of the above-mentioned edge states. It should come as no surprise that the above BEC is really a rephrasing of a classic index theorem, that of Toeplitz operators (see Section 4.3).

Still, some refinements are desirable, because the actual parameter space of models under consideration is often strictly smaller than the “universal” one used in the abstract index theory. Concretely, physicists often restrict to finite hopping range, or even nearest-neighbour models, and would like to understand the BEC without the need to “borrow” arbitrarily long-range hopping terms and/or extra internal degrees of freedom.

In the literature, there are some “elementary” approaches to the 1D BEC, particularly for the minimal two-band case, which are based on linear algebra and basic complex function theory, e.g.[2], [3]. In view of this, the reader might wonder what the benefits of an index-theoretic perspective are. Here are some answers:

• In “elementary” approaches, one assumes generic cases within the full parameter manifold, and apply certain ansätze, for a simplifed analysis. There are subtleties involved in handling the submanifold of non-generic cases, particularly when the full manifold is already constrained by symmetries. See Appendix 5.2 for an example of how ansätze can fail. Conversely, non-generic subclasses of models, such as the Su–Schrieffer–Heeger models (see Example [ex:dimerized]), are used to exhibit BEC phenomena. Then extrapolation to the generic case needs justification. “Non-elementary” methods such as operator/spectral perturbation theory are generally needed to resolve these issues.

• For finite-range Hamiltonians, the eigenvalue problem can be rephrased as a linear recurrence equation. Purely “elementary” methods would suggest that the BEC is a feature of the latter, independent of extra analytic structures in quantum theory. In fact, the role of self-adjointness is important, even if the spectral theory is left implicit in the use of informal terminology (e.g., “bulk spectrum”, “edge spectrum”, “edge states”, “bulk gap”). BEC seems to be restricted to settings where a good Fredholm operator theory is available.

• BECs for differential operators (e.g.[4]) involves understanding the (in)sensitivity to choice of self-adjoint boundary conditions. The idea of propagating initial data subject to boundary conditions could be more closely mimicked in the case of BEC for discrete models. This is the approach that we take, and it highlights the dependence of the 1D BEC on boundary specifications.

• The classic Toeplitz index theory and Bott periodicity proof are themselves illuminated by their connection to our “elementary” approach to the 1D BEC; see the proof of Theorem 14 and the discussion in Section 4.3.

In this paper, we shall provide a general “elementary” approach to the 1D BEC. We devote special attention to isolating the mathematical structure essential to the phenomenon, as well as identifying points where “non-elementary” arguments enter. We cover the cases of arbitrary hopping range, arbitrary unit cell dimension, and possibly singular hopping terms. The main correspondence theorems appear in Section 4. Some special versions of these results are well-known, but the precise statements, “elementary” proofs, and handling of non-generic cases in this paper are new.

Section 2 introduces generalities about vector-valued recurrence equations without self-adjointness requirements. This is with a view towards “non-Hermitian topological phases” and other physical situations outside of a standard quantum mechanical setup. Then we study the role of self-adjointness in Section 3, followed by the BEC of chiral-symmetric Hamiltonians in Section 4.

2 Hamiltonians and vector-valued recurrence equations↩︎

2.1 General setup and terminology↩︎

Let \(\mathcal{V}\) be a finite-dimensional complex vector space of dimension \(d_\mathcal{V}\), and \[\mathcal{S}:=\{\psi\equiv (\psi_n)_{n\in\mathbb{Z}}\,:\,\psi_n\in \mathcal{V}\}\] be the linear space of \(\mathcal{V}\)-valued sequences. Consider the bulk parameters \[V,A_r,B_r\in{\rm End}(\mathcal{V}),\quad r=1,\ldots,R.\] Here, \(V\) is the on-site potential, \(A_r,B_r\) are respectively left-hopping and \(\emph{right-hopping}\) endomorphisms with hopping range \(r\), and \(R\in \mathbb{N}\) denotes the maximal hopping range. Unless otherwise stated, we assume that \(A_R,B_R\in{\rm GL}(\mathcal{V})\) are invertible, but the singular case will also be handled.

We will abbreviate \[A:=\{A_r\}_{r=1,\ldots,R},\qquad B:=\{B_r\}_{r=1,\ldots,R}.\] The endomorphisms \(V,A,B\) determine a linear operator \(H\equiv H(V,A,B)\) on \(\mathcal{S}\), \[(H\psi)_n=V\psi_n+\sum_{r=1}^R\left(B_r\psi_{n-r}+A_r\psi_{n+r}\right),\qquad n\in\mathbb{Z},\] called the bulk Hamiltonian. For later use, we also define the Bloch Hamiltonian at exponentiated momentum \(\lambda\in\mathbb{C}^*\) to be \[\mathbf{H}(\lambda)\equiv \mathbf{H}(\lambda;V,A,B):=V+\sum_{r=1}^R\left(\lambda^{-r}B_r+\lambda^r A_r\right)\in{\rm End}(\mathcal{V}).\]

At each energy \(E\in\mathbb{C}\), we are interested in the space of energy-\(E\) modes of \(H\), \[\mathcal{M}^E\equiv \mathcal{M}^E(V,A,B):=\{\psi\in\mathcal{S}\;\,|\;\,H\psi=E\psi\}=\ker(H-E).\] Let us write \(V_E:=V-E\). Observe that \[\psi\in\mathcal{M}^E\;\Leftrightarrow\; V_E\psi_n+\sum_{r=1}^R\left(B_r\psi_{n-r}+A_r\psi_{n+r}\right)=0,\qquad n\in\mathbb{Z}. \label{eqn:bulk46recurrence}\tag{1}\] So \(\mathcal{M}^E\) is equivalently the solution space of the order-\(2R\) \(\mathcal{V}\)-valued linear recurrence equation, Eq. 1 .

2.2 Energy-\(E\) modes from initial data↩︎

To solve Eq.@eq:eqn:bulk46recurrence , one typically invokes the ansatz \(\psi_n=\lambda^nu_\lambda, \lambda\in\mathbb{C}^*\) (e.g.[2], [3]), see Eq.@eq:eqn:basic46ansatz below. Strictly speaking, this is not sufficient (see Appendix 5.1), and a general conceptual viewpoint is desirable.

The invertibility of \(A_R\) implies that the data in any \(2R\) consecutive unit cells, \[\psi_n,\psi_{n+1},\ldots,\psi_{n+2R-1}\] uniquely and linearly determines the subsequent unit cell data, \(\psi_{n+2R}\). Explicitly, \[\scriptsize \underbrace{\begin{pmatrix} 0 & \mathbf{1}_\mathcal{V} & 0 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \vdots & 0 & \mathbf{1}_\mathcal{V} & \ddots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \vdots & \vdots & 0 & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \mathbf{1}_\mathcal{V} & 0 & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots & 0 & \mathbf{1}_\mathcal{V} & 0 & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & 0 & \mathbf{1}_\mathcal{V} & \ddots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & 0\\ 0 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & 0 & \mathbf{1}_\mathcal{V}\\ -A_R^{-1}B_R & -A_R^{-1}B_{R-1} & -A_R^{-1}B_{R-2} & \cdots & -A_R^{-1} B_1 & -A_R^{-1}V_E & -A_R^{-1} A_1 & \cdots & -A_R^{-1}A_{R-1} \end{pmatrix}}_{=: C_E\equiv C_E(V,A,B)} \begin{pmatrix} \psi_n\\ \psi_{n+1}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \psi_{n+2R-1} \end{pmatrix} =\begin{pmatrix} \psi_{n+1}\\ \psi_{n+2}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \psi_{n+2R} \end{pmatrix}\label{eqn:vector46companion46matrix} \normalsize\tag{2}\] The above \(2R\times 2R\) block matrix \(C_E\in {\rm End}(\mathcal{V}^{\oplus 2R})\) is called the companion matrix associated to the recurrence equation 1 , and it advances data one step to the right. Up to a sign, \(\det C_E=\det (A_R^{-1}B_R)\). Invertibility of \(B_R\) implies that \(C_E^{-1}\) exists with a similar form as \(C_E\), and \(C_E^{-1}\) advances data one step to the left. Here, the reader may anticipate difficulties when \(A_R\) and/or \(B_R\) is singular (see Example [ex:dimerized]).

For convenience, we write \(\mathcal{V}_n\) for the copy of \(\mathcal{V}\) at the \(n\)-th unit cell. The above discussion says that \(\mathcal{M}^E\) is a \((2R\cdot d_\mathcal{V})\)-dimensional linear space identifiable with the initial data space, \[\mathcal{D}:=\mathcal{V}_1\oplus\ldots\oplus\mathcal{V}_{2R}.\label{eqn:initial46data46space}\tag{3}\] Eq.@eq:eqn:bulk46recurrence is formally solved by applying \(C_E, C_E^{-1}\) iteratively to initial data in \(\mathcal{D}\), thereby obtaining energy-\(E\) modes in \(\mathcal{M}^E\).

The invertible companion matrix \(C_E\in{\rm GL}(\mathcal{V}^{\oplus 2R})={\rm GL}(\mathcal{D})\) has a (unordered) list of nonzero generalized eigenvalues \(\lambda_i\equiv \lambda_i(E)\in\mathbb{C}^*\), with respective algebraic multiplicities \(m_i\geq 1\). The corresponding generalized eigenspaces \[\mathcal{D}_i^E:=\{v\in \mathcal{D}\,:\,(C_E-\lambda_i)^{m_i}v=0\}\] are invariant subspaces for \(C_E\) (see [5], Chapter 1, §5.4). So the initial data space admits a canonical decomposition \[\mathcal{D}=\bigoplus_i \mathcal{D}_i^E.\label{eqn:dynamical46splitting}\tag{4}\] The energy-\(E\) mode space \(\mathcal{M}^E\cong\mathcal{D}\) inherits a corresponding decomposition into normal mode spaces labelled by the \(\lambda_i\). Note that the “dynamical splitting”, Eq.@eq:eqn:dynamical46splitting , may not be related to the unit cell splitting, Eq.@eq:eqn:initial46data46space .

Depending on the size of the generalized eigenvalue \(\lambda_i\), there is a further hierarchy, \[\mathcal{D}=\bigoplus_{i:|\lambda_i|<1} \mathcal{D}_i^E\bigoplus_{i:|\lambda_i|=1} \mathcal{D}_i^E\bigoplus_{i:|\lambda_i|>1}\mathcal{D}_i^E=:\mathcal{D}_{\downarrow}^E\oplus \mathcal{D}_{\rm Bloch}^E\oplus \mathcal{D}_{\uparrow}^E,\] with each type of initial data generating the energy-\(E\) decrease modes, Bloch modes, and increase modes, respectively.

2.2.1 Companion matrices versus Bloch Hamiltonians↩︎

The characteristic polynomial of \(C_E(V,A,B)\) can be computed as follows, \[\begin{align} P_E(\lambda)&:=\det(\lambda-C_E)\nonumber\\ &= (-1)^{d_\mathcal{V}^2\cdot(2R-1)}\det\left(\begin{array}{ccccc|c} -\mathbf{1} & 0 &\cdots &\cdots & \cdots & \lambda\\ \lambda & -\mathbf{1} & 0 & \cdots & \vdots & 0 \\ 0 & \ddots &\ddots &\ddots & \vdots& \vdots\\ \vdots &\ddots & \lambda & -\mathbf{1} & 0 & \vdots\\ \vdots & \cdots & 0 & \lambda & - \mathbf{1} & 0 \\ \hline A_R^{-1}B_{R-1} & \cdots & \cdots & A_R^{-1}A_{R-2} & A_R^{-1}A_{R-1} + \lambda & A_R^{-1}B_R \end{array}\right) \normalsize\nonumber\\ &= \scriptsize \frac{1}{\det(A_R)}\cdot\det\left( B_R- \begin{pmatrix} B_{R-1} & \cdots & A_{R-2} & A_{R-1}+\lambda A_R\end{pmatrix} \left( \begin{array}{ccccc} -\mathbf{1} & 0 &\cdots &\cdots & \cdots \\ -\lambda & -\mathbf{1} & 0 & \cdots & \vdots \\ -\lambda^2 & \ddots &\ddots &\ddots & \vdots\\ \vdots &\ddots & -\lambda & -\mathbf{1} & 0 \\ -\lambda^{2R-2} & \cdots & -\lambda^2 & -\lambda & - \mathbf{1} \end{array} \right) \begin{pmatrix}\lambda \\ 0 \\ \vdots \\ \vdots \\ 0\end{pmatrix} \right) \normalsize\nonumber\\ &=\frac{\lambda^{R\cdot d_\mathcal{V}}}{\det(A_R)}\cdot\det\left( \lambda^{-R}B_R+\lambda^{-(R-1)}B_{R-1}+\ldots +V_E+\ldots +\lambda^{R-1}A_{R-1}+\lambda^R A_R \right)\nonumber\\ &=\frac{\lambda^{R\cdot d_\mathcal{V}}}{\det(A_R)}\cdot\det\left(\mathbf{H}(\lambda;V,A,B)-E\right)\label{eqn:char46poly46calculation}, \end{align}\tag{5}\] where in the third equality, we used a standard identity for the determinant of a \(2\times 2\) block matrix. Thus, \[\lambda\;\textrm{is eigenvalue of}\;C_E\quad\Leftrightarrow\quad E\;\textrm{is eigenvalue of}\;\mathbf{H}(\lambda).\] The companion matrices and the Bloch Hamiltonians are complementary ways of investigating the operator \(H(V,A)\). The former is useful when varying the energy \(E\in \mathbb{C}\), while the latter is useful when varying the (exponentiated) momentum \(\lambda\in\mathbb{C}^*\).

2.2.1.1 Exponential ansatz.

Suppose the initial data happens to an eigenvector of \(C_E\), \[v_i\in\ker(C_E-\lambda_i)\subset\mathcal{D}_i^E.\] By considering Eq.@eq:eqn:vector46companion46matrix , it is readily seen that \(v_i\) must have a nice form with respect to the unit cell splitting, \[v_i=(\lambda_i u_i, \lambda_i^2 u_i,\ldots, \lambda_i^{2R}u_i),\qquad \textrm{for some}\; u_i\in\mathcal{V}. \label{eqn:companion46to46Bloch46eigenvector}\tag{6}\] Furthermore, the last entry of Eq.@eq:eqn:vector46companion46matrix for \(n=0\) becomes \[-B_Ru_i-B_{R-1}(\lambda_i u_i)-\ldots -V_E(\lambda_i^R u_i)-\ldots -A_{R-1}(\lambda_i^{2R-1}u_i)=A_R(\lambda_i^{2R}u_i).\] Equivalently, \(u_i\) is an \(E\)-eigenvector for the Bloch Hamiltonian \(\mathbf{H}(\lambda_i)\), \[0=\left(V_E+\sum_{r=1}^R\left(\lambda_i^{-r}B_r+\lambda_i^r A_r\right)\right)u_i\;\Leftrightarrow\;u_i\in\ker(\mathbf{H}(\lambda_i)-E).\] Thus the \(\lambda_i\)-eigenvector \(v_i\) of \(C_E\) generates the following energy-\(E\) normal mode, \[\psi_n=\lambda_i^nu_i,\qquad u_i\in \ker(\mathbf{H}(\lambda_i)-E).\label{eqn:basic46ansatz}\tag{7}\] Eq.@eq:eqn:basic46ansatz is the exponential ansatz.

A normal mode of the form in Eq.@eq:eqn:basic46ansatz is already generated by \(u_i\in\mathcal{V}\) — the initial data can be reduced to a single unit cell \(\mathcal{V}\), albeit supplemented by data of the geometric factor \(\lambda_i\).

However, when \(\lambda_i\) is a non-simple eigenvalue of \(C_E\), the generalized \(\lambda_i\)-eigenspace of \(C_E\) is not easily related to the \(E\)-eigenspace of \(\mathbf{H}(\lambda_i)\). The exponential ansatz is not applicable when the initial data \(v_i\) is only a generalized eigenvector, see Appendix 5.1 for a discussion.

Finally, we mention that when \(A_R\) and/or \(B_R\) is singular, the above algebraic arguments break down, and it is even possible for normal modes (and edge states) to be compactly-supported, see Example [ex:dimerized].

2.3 Half-space Hamiltonians and initial data for edge modes↩︎

Define the right-half truncated sequence space, \[\check{S}:=\{\check{\psi}\equiv (\check{\psi}_n)_{n\geq 1}\,:\,\check{\psi}_n\in\mathcal{V}\}.\] The half-space Hamiltonian \(\check{H}\) is typically defined to be the truncation of the bulk Hamiltonian \(H\) to \(\check{\mathcal{S}}\). Explicitly,

Eq.[eqn:half46space46bc] says that for the first \(R\) unit cells, any right-hopping term arriving from \(n^\prime\leq 0\) is set to zero. The half-space energy-\(E\) mode spaces are defined as \[\check{\mathcal{M}}^E:=\{\check{\psi}\in\check{\mathcal{S}}\,:\,\check{H}\check{\psi}=E\check{\psi}\}=\ker(\check{H}-E),\qquad E\in\mathbb{C}.\] As was the case with the bulk Hamiltonian, initial data in \(\mathcal{D}=\mathcal{V}_1\oplus\ldots\oplus\mathcal{V}_{2R}\) gets propagated to the right by the companion matrix \(C_E\) to generate energy-\(E\) modes. However, Eq.[eqn:half46space46bc] imposes extra conditions on the allowed initial data.

It is more efficient to think of elements of \(\check{S}\) as sequences \((\check{\psi}_n)_{n\geq {1-R}}\) subject to the Dirichlet boundary condition on the initial \(R\) unit cells, \[\check{\psi}_{1-R}=\ldots=\check{\psi}_{-1}=\check{\psi}_0=0.\label{eqn:Dirichlet46condition}\tag{8}\] Then an equivalent specification of \(\check{H}\) is \[(\check{H}\check{\psi})_n=\sum_{r=1}^{R}B_r\check{\psi}_{n-r}+V\check{\psi}_n+\sum_{r=1}^{R}A_r\check{\psi}_{n+r}, \qquad n\geq 1,\] with \(\check{\psi}\equiv (\check{\psi})_{n\geq 1-R}\) subject to Eq.@eq:eqn:Dirichlet46condition . In other words, we now consider the initial data space to be \[\mathcal{D}=\mathcal{V}_{1-R}\oplus\ldots\oplus\mathcal{V}_0\oplus \mathcal{V}_1\oplus\ldots\oplus\mathcal{V}_{R},\] but restrict to the “Dirichlet” subspace \[\mathcal{D}_{\rm Dir}:=0\oplus\ldots\oplus 0\oplus\mathcal{V}_1\oplus\ldots\oplus\mathcal{V}_R\;\subset\;\mathcal{D}.\] Then we have the identification \[\check{\mathcal{M}}^E\cong\mathcal{D}_{\rm Dir}\] provided by the iterated application of \(C_E\) on the Dirichlet initial data in \(\mathcal{D}_{\rm Dir}\). Thus \[\dim\check{\mathcal{M}}^E=\dim \mathcal{D}_{\rm Dir}=R\cdot d_\mathcal{V}.\label{eqn:Dirichlet46dimension}\tag{9}\]

Within \(\check{M}^E\cong \mathcal{D}_{\rm Dir}\), we are interested in characterizing the subspace of so-called energy-\(E\) edge modes.

By definition, an edge mode \(\check{\psi}\) belongs to \(\check{\mathcal{M}}^E\cong\mathcal{D}_{\rm Dir}\), and the decay condition just means that the initial data of \(\check{\psi}\) must also come from \(\mathcal{D}^E_{\downarrow}\) (i.e., initial data for decrease modes). Thus the space of energy-\(E\) edge modes is precisely the linear space \[\check{\mathcal{M}}^E_{\rm edge}\cong \mathcal{D}_{\rm Dir}\cap \mathcal{D}^E_{\downarrow}.\]

3 Self-adjoint Hamiltonians↩︎

Now, let \(\mathcal{V}\) have an inner product, then the sequence space \(\mathcal{S}\) contains the Hilbert space \(\ell^2(\mathbb{Z};\mathcal{V})\) of square-summable sequences. In quantum-mechanical problems¹, we require \(H(V,A,B)\) to be self-adjoint on \(\ell^2(\mathbb{Z};\mathcal{V})\). This is equivalent to taking \[V=V^*,\qquad B_r=A_r^*,\quad r=1,\ldots,R.\] From now on, we consider \(H\equiv H(V,A)\equiv H(V,A,A^*)\) as a self-adjoint operator on \(\ell^2(\mathbb{Z};\mathcal{V})\), \[(H\psi)_n=V\psi_n+\sum_{r=1}^R\left(A_r^*\psi_{n-r}+A_r\psi_{n+r}\right),\qquad n\in\mathbb{Z}.\] Similarly, the half-space Hamiltonian is the self-adjoint operator \(\check{H}(V,A)\) on \(\ell^2(\mathbb{N};\mathcal{V})\) given by \[(H\check{\psi})_n=V\check{\psi}_n+\sum_{r=1}^{R}\left(A_r^*\check{\psi}_{n-r}+A_r\check{\psi}_{n+r}\right), \qquad n\geq 1,\] with \(\check{\psi}\in\ell^2(\mathbb{N};\mathcal{V})\) extended to \(\check{\psi}=(\check{\psi}_n)_{n\geq 1-R}\) but subjected to the Dirichlet condition, Eq.@eq:eqn:Dirichlet46condition . (Here, \(\mathbb{N}\) denotes positive natural numbers.)

The spectrum of the self-adjoint \(H\) and \(\check{H}\) are closed subsets of \(\mathbb{R}\), and stability results of their spectra [5] are available.

3.1 Mode duality from self-adjointness↩︎

Self-adjointness imposes a fundamental symmetry on the generalized eigenvalues of \(C_E\).

3.2 Real energy bands and band gap↩︎

For self-adjoint Hamiltonians, the Bloch Hamiltonians at \(\lambda=e^{ik}\in{\rm U}(1)\) are of special interest. Note that \(\lambda\) is self-dual in this case. The parameter \(k\in[-\pi,\pi]/_{-\pi\sim\pi}\) is called the (real) quasimomentum. For the bulk Hamiltonian \(H\), only Bloch modes with \(|\lambda|=1\) are bounded and contribute to the spectrum of \(H\), whereas decrease modes blow up to the left, and increase modes blow up to the right.

The Bloch Hamiltonians with \(\lambda=e^{ik}\) are² Hermitian, \[\mathbf{H}(e^{ik};V,A)=\mathbf{H}(e^{ik};V,A)^*,\] and each of them has \(d_\mathcal{V}\) real eigenvalues (counted with multiplicities), labelled in increasing order as \[E_j(k),\qquad j=1,\ldots, d_\mathcal{V},\quad e^{ik}\in{\rm U}(1).\] The \(E_j\) define the energy band functions on the Brillouin zone \({\rm U}(1)\). They are continuously defined, although there is the complicated issue of smoothness at band crossings (when \(E_j(k)=E_{j+1}(k)\) occurs), which we will not address. What is important is the presence of a band gap.

3.2.1 Isolated edge states in band gap↩︎

The total spectrum \(\sigma(H)\) of \(H(V,A)\) is obtained as the union of the ranges of the energy band functions \(E_j\). It is also called the bulk spectrum of \(H(V,A)\). If \(A_R\) is invertible, \(H(V,A)\) has no point spectrum (see Appendix 5.1; whether or not \(|\lambda_i|=1\), there are no bounded modes in \(\mathcal{M}^E\)).

In general, the spectrum of the half-space Hamiltonian \(\check{H}(V,A)\) and that of \(H(V,A)\) are different. For example, it is certainly possible that \(\check{H}(V,A)\) acquires point spectrum at certain eigenvalues \(E\). These point spectra are due to the edge states of Definition 1; note that the latter are square-summable, see Appendix 5.1.

A rather non-trivial spectral theory result is that the essential spectrum of \(\check{H}\) and \(H\) coincide. For example, one might construct Weyl sequences from the Bloch modes, supported on the right-half Hilbert space. Alternatively, one can invoke Toeplitz operator theory, see Chapter 4.3 of [8]. Thus \(\sigma(\check{H})\cap (E_-,E_+)\) can only comprise discrete spectrum, i.e., isolated eigenvalues with finite multiplicity.

4 Chiral-symmetric Hamiltonians↩︎

A chiral symmetry operator satisfies \(\Gamma^2=1\), and it orthogonally decomposes the unit cell Hilbert space into \[\mathcal{V}=\mathcal{V}_+\oplus \mathcal{V}_-,\] according to its \(\pm\) eigenspaces, called the \(\pm\)-graded components. Similarly, \(\ell^2(\mathbb{Z};\mathcal{V})\cong\ell^2(\mathbb{Z};\mathcal{V}_+)\oplus \ell^2(\mathbb{Z};\mathcal{V}_-)\) is split into “sublattice” Hilbert spaces. In any basis adapted to this splitting, chiral symmetry of \(H\) is equivalent to \(V\) and \(A=\{A_1,\ldots,A_R\}\) being off-diagonal matrices, \[\Gamma=\begin{pmatrix} \mathbf{1}_{\mathcal{V}_+} & 0 \\ 0 & -\mathbf{1}_{\mathcal{V}_-} \end{pmatrix},\qquad V=\begin{pmatrix} 0 & v^*\\ v & 0\end{pmatrix},\qquad A_r=\begin{pmatrix}0 & a_{r,-+}\\ a_{r,+-} & 0\end{pmatrix},\] where \(v,a_{r,+-}:\mathcal{V}_{+}\to\mathcal{V}_{-}\) and \(a_{r,-+}:\mathcal{V}_-\to\mathcal{V}_+\). We also have \(a_{R,+-}, a_{R,-+}\) invertible iff \(A_R\) is invertible. Consequently, the Bloch Hamiltonians are off-diagonal operators, \[\begin{align} \mathbf{H}(\lambda)&=\begin{pmatrix} 0 & v^*+\sum_{r=1}^R\left(\lambda^{-r}a^*_{r,+-}+\lambda^r a_{r,-+}\right)\\ v+\sum_{r=1}^R\left(\lambda^{-r}a^*_{r,-+}+\lambda^r a_{r,+-}\right) & 0 \end{pmatrix}\nonumber\\ &=:\begin{pmatrix} 0 & h_{-+}(\lambda)\\ h_{+-}(\lambda) & 0\end{pmatrix},\label{eqn:chiral46Bloch46Hamiltonian} \end{align}\tag{10}\] and \(\Gamma \mathbf{H}(\lambda)\Gamma=-\mathbf{H}(\lambda)\) holds.

When restricted to \(\lambda=e^{ik}\in{\rm U}(1)\), the Hermitian Bloch Hamiltonians \(\mathbf{H}(e^{ik})\) have the property that \(h_{-+}(e^{ik})=h_{+-}^*(e^{ik})\). Therefore \[\begin{align} H\; \textrm{is gapped}\;&\Leftrightarrow\;\mathbf{H}(e^{ik})\;\textrm{invertible} &&\forall\,e^{ik}\in{\rm U}(1),\\ &\Leftrightarrow\; \mathbf{H}(e^{ik})^2=\begin{pmatrix} h_{+-}^*(e^{ik})h_{+-}(e^{ik}) & 0 \\ 0 & h_{+-}(e^{ik})h_{+-}^*(e^{ik})\end{pmatrix} >0 &\;\;&\forall\,e^{ik}\in{\rm U}(1),\\ &\Leftrightarrow\; h_{+-}(e^{ik})\;\;\textrm{invertible} && \forall\,e^{ik}\in{\rm U}(1). \end{align}\] Then the following is well-defined:

Here, we recall that the winding number of a differentiable curve \(\gamma:{\rm U}(1)\to \mathbb{C}^*\) about the origin can be defined via the contour integral \[{\rm Wind}(\gamma)=\frac{1}{2\pi i}\oint_{\gamma} \frac{dz}{z}.\]

Note that a continuous change of \((V,A)\), thus of \(H(V,A)\) by Remark 7, induces a homotopy of the map \(\det h_{+-}:{\rm U}(1)\to\mathbb{C}^*\). So Definition 9 is a topological invariant of \(H(V,A)\). We also mention that traditional solid-state physics constructions such as Berry connection on the Bloch bundle, Berry/geometric phase etc., are not needed to define \(\mathcal{W}(H)\).

Next, we examine the half-space Hamiltonian \(\check{H}=\check{H}(V,A)\), which is also of the off-diagonal form, \[\check{H}=\begin{pmatrix} 0 & \check{H}_{-+}\\ \check{H}_{+-} & 0 \end{pmatrix},\qquad \check{H}_{-+}=\check{H}_{+-}^*:\ell^2(\mathbb{N};\mathcal{V}_-)\to\ell^2(\mathbb{N};\mathcal{V}_+).\] The zero-energy mode space, i.e., \(\ker\check{H}\), is special because it admits a splitting according to the \(\Gamma\)-grading, \[\check{\mathcal{M}}_{\rm edge}^0\equiv \ker\check{H}=\ker\check{H}_{+-}\oplus\ker\check{H}_{-+}=:\check{\mathcal{M}}^0_{{\rm edge},+}\oplus\check{\mathcal{M}}^0_{{\rm edge},-}.\]

The bulk-edge correspondence refers to claims of the form \[\mathcal{W}(H)\overset{?}{=}{\rm Ind}_e(\check{H}).\]

4.1 Two-band nearest-neighbour case↩︎

The base case for the bulk-edge correspondence has \(R=1,d_\mathcal{V}=2\), where it is the sharpest.

For later use, we highlight three examples with singular \(A\), which all other chiral-symmetric gapped Hamiltonians will be deformed to, as explained in Section 4.2. See Appendix 5.2 for a family of examples whose \(C_0(V,A)\) have non-simple eigenvalues.

Example 1. Consider \[V=0,\qquad A=\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix},\] so \(H(V,A)\) is just the operator which hops between the \(\mathcal{V}_+\) and \(\mathcal{V}_-\) of adjacent unit cells. \(\check{H}(V,A)\) is the same, except for a “cut-off” at the boundary unit cell,

Other than the boundary unit cell, each adjacent \(\mathcal{V}_-, \mathcal{V}_+\) pair is coupled by the Hermitian matrix \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), which has eigenvalues \(\pm 1\). At the boundary unit cell, the left-hopping term \(A\) is set to zero, so there is a zero eigenvalue. In total, \(\check{H}(V,A)\) has spectrum \(\{-1,0,+1\}\), with \(\{-1,+1\}\) being infinitely-degenerate (thus essential spectrum³), while \(\{0\}\) is discrete spectrum arising from the compactly-supported edge state \[\left(\binom{1}{0},\binom{0}{0},\binom{0}{0},\ldots\right).\] So \({\rm Ind}_e(\check{H})=1\). The Bloch Hamiltonians are \[\check{H}(\lambda)=\begin{pmatrix} 0 & \lambda^{-1} \\ \lambda & 0\end{pmatrix},\] and so \(\mathcal{W}(H)=+1\). These calculations are consistent with Theorem 11.

A winding number \(-1\) example is obtained by swapping the roles of \(\mathcal{V}_-\) and \(\mathcal{V}_+\).

The basic “trivial” winding 0 example is given by \[V=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\qquad A=0,\] and it pairs up adjacent \(\mathcal{V}_+, \mathcal{V}_-\) within the same unit cell:

This \(\check{H}(V,A)\) clearly has spectrum \(\{-1,+1\}\) with no edge states at all.

Notice that the bulk Hamiltonians in these three examples are unitarily related to each other by a change of convention in the unit cell labelling and/or grading \(\mathcal{V}_+\leftrightarrow\mathcal{V}_-\). These conventions are implicitly specified via boundary conditions, and the intrinsic meaning of the “bulk winding number invariant” is actually quite subtle, see [6], [9].

The above examples are “dimerized limits” inside the (singular) subclass of Su–Schrieffer–Heeger models [1], \[V=\begin{pmatrix} 0 & t_1 \\ t_1 & 0 \end{pmatrix},\qquad A=\begin{pmatrix} 0 & 0 \\ t_2 & 0 \end{pmatrix},\qquad t_1, t_2 \in\mathbb{R}.\] Here, the winding number is \(+1\) when \(|t_2|>|t_1|\), and \(0\) when \(|t_2|<|t_1|\), with a “topological phase transition” at \(|t_1|=|t_2|\). For \(|t_2|>|t_1|>0\), it is easily checked that the edge state is \[\check{\psi}_n=\left(-t_1/t_2\right)^n\binom{1}{0},\qquad n\geq 1.\] Note that this edge state does not satisfy the Dirichlet condition at \(n=0\) (instead, \(\check{\psi}_0\in\ker A^*\)). It is also not a superposition of two normal modes with different decay rates.

4.2 More bands and longer range models↩︎

In the general case, \(R\geq 1, d_\mathcal{V}\geq 2\), a chiral-symmetric \(H(V,A)\) has Bloch Hamiltonian with off-diagonal term \(h_{+-}\) being an \({\rm End}(\mathcal{V}_+,\mathcal{V}_-)\)-valued Laurent polynomial with zeroes/poles of highest order \(R\).

Suppose \(A_R\) is invertible. Then \(\det h_{+-}\) is a Laurent polynomial, whose highest-order zeroes/poles have degree \(R\cdot d_\mathcal{V}/2\). The argument principle calculation, Eq.@eq:eqn:argument46principle , now gives \[\mathcal{W}(H)=|I_+|-\frac{R\cdot d_\mathcal{V}}{2}=\frac{R\cdot d_\mathcal{V}}{2}-|I_-|.\] However, instead of Eq.@eq:eqn:246band46chiral46eigenvectors , \(u_i\in\ker(\mathbf{H}(\lambda_i))\) now only gives \[i\in I_+\;\Rightarrow\; u_i\propto\binom{w_i}{0},\quad w_i\in\mathcal{V}_+,\qquad\;\;\; i\in I_-\;\Rightarrow\; u_i\propto\binom{0}{w_i},\quad w_i\in\mathcal{V}_-.\] Thus \[\mathcal{D}^0_{\downarrow,+}={\rm span}\left\{\begin{pmatrix} \lambda_i\binom{w_i}{0}\\ \vdots\\ \lambda_i^{2R}\binom{w_i}{0} \end{pmatrix}:i\in I_+\right\},\qquad w_i\in\mathcal{V}_+,\] and similarly for \(\mathcal{D}^0_{\downarrow,-}\). The Dirichlet condition on the initial \(R\) unit cells “generically” imposes the maximal number, \(R\cdot d_\mathcal{V}/2\), of constraints, but possibly fewer. So instead of Eq.@eq:eqn:even46kernel –13 , we can only say that \[\begin{align} |I_+|\geq\dim\ker \check{H}_{+-}&\geq\max\{0,|I_+|-R\cdot d_\mathcal{V}/2\}=\max \{0,\mathcal{W}(H)\},\\ |I_-|\geq\dim\ker \check{H}_{-+}&\geq\max\{0,|I_-|-R\cdot d_\mathcal{V}/2\}=\max \{0,-\mathcal{W}(H)\}. \end{align}\] We learn from this “elementary” analysis that

• \(\check{H}(V,A)\) has at least \(|\mathcal{W}(H)|\) and at most \(R\cdot d_\mathcal{V}\) energy-\(0\) edge modes.

• “Generically”, it has exactly \(|\mathcal{W}(H)|\) “robust” energy-\(0\) edge states, all lying in one graded component or the other.

Generally, the dimension of \(\ker \check{H}_{+-}\) and/or \(\ker \check{H}_{-+}\) could exceed \(|\mathcal{W}(H)|\), and vary wildly with \((V,A)\), so what is the precise formulation of “generic” and “robust”?

Remarkably, the number of extra zero-energy states in each graded component is always the same. Thus the weaker statement of Theorem 11 continues to hold even when \(R>1\) and/or \(d_\mathcal{V}>2\):

Returning to the \(d_\mathcal{V}=2\) case, we can improve Theorem 11.

Thus, for two-band models, \(d_\mathcal{V}=2\), there are always exactly \(|\mathcal{W}(H)|\leq R\) edge modes at \(0\)-energy, regardless of the hopping range \(R\). However, when \(R\geq 2\), the possibility of edge states with non-zero in-gap energy is not excluded (unlike Prop.12).

4.3 Toeplitz index theory discussion↩︎

The off-diagonal operator \(\check{H}_{+-}:\ell^2(\mathbb{N};\mathcal{V}_+)\to\ell^2(\mathbb{N};\mathcal{V}_-)\) is an example of a Toeplitz operator. The \({\rm End}(\mathcal{V}_+,\mathcal{V}_-)\)-valued symbol of this Toeplitz operator is precisely the off-diagonal part of the Bloch Hamiltonian function, \(h_{+-}|_{{\rm U}(1)}\). The Toeplitz index theorem ([14], [8] Theorem 4.4.3) reads \[\underbrace{\textrm{Fredholm index}(\check{H}_{+-})}_{{\rm Ind}_e(\check{H})}\equiv \dim\ker\check{H}_{+-}-\dim\ker\underbrace{\check{H}_{+-}^*}_{=\check{H}_{-+}}=\underbrace{{\rm Wind}(\det h_{+-}|_{{\rm U}(1)})}_{\equiv\mathcal{W}(H)},\label{eqn:Toeplitz46index46theorem}\tag{16}\] which is precisely Eq.@eq:eqn:BEC46many46bands . Here, there is implicit use of the Fredholm property, that \(0\) is avoided in the essential spectrum of \(\check{H}_{+-}\) (see [8] Chapter 4.3), in order that the left-hand-side is well-defined.

Eq.@eq:eqn:Toeplitz46index46theorem holds for Toeplitz operators whose invertible symbol functions only need to be continuous, not necessarily analytic/smooth. The winding number then refers to the homotopy/covering space definition, generalizing the complex analytic one. Hamiltonians which are not strictly finite-range, but are norm-approximated by finite-range ones, have continuous \(h_{+-}\). In this sense, the Toeplitz index theorem subsumes Theorem 14, as is known in the mathematical physics literature, e.g.Chapter 1.2 of [7], Sec.3B of [9].

Another technical advantage of the Toeplitz index method is that there is no need to stabilize in the sense of adding an extra \(\mathcal{V}^\prime\) to \(\mathcal{V}\). This is because \(\pi_1({\rm GL}(d))\cong\mathbb{Z}\) for all dimensions \(d\), via the determinant map. (Note: this does not require the strength of the Bott periodicity of \(\pi_n({\rm GL}(\infty))\).) So given any \(H(V,A)\) with maximal range \(R\) and fixed \(d_\mathcal{V}\), its \(h_{+-}|_{{\rm U}(1)}\) would be homotopic within continuous \({\rm GL}(d_\mathcal{V})\)-valued symbols, to the Hamiltonian with \(h_{+-}^\prime(\lambda)={\rm diag}(\lambda^{\mathcal{W}(H)},1,\ldots,1)\). The latter Hamiltonian is basically a dimerized 2-band model, whose half-space version is easily seen to have exactly \(|\mathcal{W}(H)|\) edge modes. The catch is that we generally have to “borrow” longer range terms, \(A_r, r>R\), to achieve this homotopy, since general continuous loops cannot be expressed as Laurent ones with finite expansion. This should be compared to the proof of Theorem 14, where \(R\) is never exceeded, but the extra \(\mathcal{V}^\prime\) may need to be very large.

Acknowledgments↩︎

The author thanks H.Zhang for stimulating discussions on discrete interface models, which led to a rethinking of the 1D BEC.

Author declarations↩︎

The author has no conflicts to disclose.

Data availability statement↩︎

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

5 Appendix↩︎

5.1 Validity of exponential ansatz for normal modes↩︎

Generally, the companion matrix \(C_E\) to \(H(V,A)\) may not be diagonalizable, and we must consider its generalized eigenvectors as \(\mathcal{D}\)-valued initial data for the energy-\(E\) modes, \[v_i\not\in\ker(C_E-\lambda_i),\qquad v_i\in\ker(C_E-\lambda_i)^{m_i},\qquad m_i\geq 2.\] In the more familiar case of scalar-valued (i.e.\(\mathcal{V}=\mathbb{C}\)) recurrence equations, one derives, for multiplicity \(m=2\) say, that a generalized eigenvector \(v_i\) has the form \[v_i=(\lambda_i,2\lambda_i^2,3\lambda_i^3,\ldots,2R\cdot\lambda_i^{2R}),\] up to adding some genuine eigenvector. This \(v_i\) generates the familiar “polynomial-exponential ansatz” for the normal modes, \[\psi_n=n\lambda_i^n,\qquad n\in\mathbb{Z}.\label{eqn:poly46exp46ansatz}\tag{17}\] Notice that this \(\psi\) is able to satisfy \(\psi_0=0\), whereas a normal mode generated from a genuine eigenvector, Eq.@eq:eqn:basic46ansatz , is always nowhere-vanishing.

For our \(\mathcal{V}\)-valued case, a generalized eigenvector \(v_i\in\mathcal{D}_i^E\) need not have the nice form \[v_i=(0,\lambda_i u_i,2\lambda_i^2 u_i,\ldots, 2R\cdot\lambda_i^{2R}u_i),\qquad u_i\in \ker(\mathbf{H}(\lambda_i)-E).\qquad \textrm{(fails)}\] This means that the “polynomial-exponential ansatz” analogous to Eq.@eq:eqn:poly46exp46ansatz may not work, \[\psi_n=n\lambda_i^nu_i,\qquad u_i\in \ker(\mathbf{H}(\lambda_i)-E).\qquad \textrm{(fails)}\label{eqn:poly46exp46fail}\tag{18}\] Nevertheless, we can still compute \[\begin{align} C_E^{2R}v_i&=(C_E-\lambda_i+\lambda_i)^{2R}v_i\\ &=\underbrace{(C_E-\lambda_i)^{2R}v_i}_{0}+\binom{2R}{1}\lambda_i(C_E-\lambda_i)^{2R-1}v_i+\ldots+\lambda_i^{2R}v_i\\ &=0+\ldots+0+\binom{2R}{m_i-1}\lambda_i^{2R-m_i+1}\underbrace{(C_E-\lambda_i)^{m_i-1}v_i}_{\textrm{eigenvector}}+\ldots+\lambda_i^{2R}v_i. \end{align}\] This says that after \(2R\) advancements by \(C_E\), the initial data \(v_i\in\mathcal{D}_i^E\) is turned into the above combination of (linearly independent) vectors in the Jordan chain generated by \(v_i\). It follows that the normal mode generated by \(v_i\) will still decay/blow-up as \(n\to\infty\), according to whether \(|\lambda_i|<1\) or \(|\lambda_i|>1\). The decay/blow-up rate is \(\lambda_i\) per-unit-cell, up to polynomial correction factors.

5.2 Edge states which are not built from exponentials↩︎

For \(d_\mathcal{V}=2, R=1\), consider the 1-parameter family of chiral-symmetric models, \[V=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix},\qquad A^{(\theta)}=A^{(\theta)}_1=e^{i\theta}\begin{pmatrix}0 & \frac{1}{4} \\ 1 & 0\end{pmatrix},\qquad \theta\in [-\pi,\pi].\] The Bloch Hamiltonians are \[\mathbf{H}(\lambda;V,A^{(\theta)})=\begin{pmatrix} 0 & e^{-i\theta}\lambda^{-1}+1+\frac{1}{4}e^{i\theta}\lambda \\ \frac{1}{4}e^{-i\theta}\lambda^{-1}+1+e^{i\theta}\lambda & 0 \end{pmatrix}.\] To calculate the winding number of the off-diagonal element \(h_{+-}|_{{\rm U}(1)}\), reparametrize by \(\tilde{\lambda}=e^{i\theta}\lambda\), so that \(h_{+-}:\tilde{\lambda}\mapsto \frac{1}{4}\tilde{\lambda}^{-1}+1+\tilde{\lambda}\). As \(\tilde{\lambda}\) is varied in \({\rm U}(1)\), \(h_{+-}\) traces out an ellipse with winding number \(+1\) (around the origin). So, by Theorem 11, there should be one zero-energy edge state for \(\check{H}(V,A^{(\theta)})\). Let us calculate this edge state.

The characteristic polynomial for \(C_0\) is \[\begin{align} \det(\lambda-C_0)=\lambda^2\det \mathbf{H}(\lambda)&=(\frac{1}{4}e^{-i\theta}+\lambda+e^{i\theta}\lambda^2)(e^{-i\theta}+\lambda+\frac{1}{4}e^{i\theta}\lambda^2)\\ &=\frac{1}{4}\cdot e^{i2\theta}(\lambda+\frac{1}{2}e^{-i\theta})^2(\lambda+2e^{-i\theta})^2, \end{align}\] which has the repeated root \(-\frac{1}{2}e^{-i\theta}\) inside the unit circle. It is easy to check that \((0,0,1,0)\) is a generalized eigenvector of \(C_0(V,A^{(\theta)})\), \[\begin{align} \left(C_0+\frac{1}{2}e^{-i\theta}\right)\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix} &=\begin{pmatrix} \frac{1}{2}e^{-i\theta} & 0 & 1 & 0 \\ 0 & \frac{1}{2}e^{-i\theta} & 0 & 1 \\ -\frac{1}{4} e^{-i2\theta} & 0 & -\frac{1}{2}e^{-i\theta} & 0 \\ 0 & -4e^{-i2\theta} & 0 & \frac{9}{2}e^{-i\theta} \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix}=\begin{pmatrix} 1 \\ 0 \\ -\frac{1}{2}e^{-i\theta} \\ 0 \end{pmatrix},\\ \left(C_0+\frac{1}{2}e^{-i\theta}\right)^2\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix}&=0. \end{align}\] This generalized eigenvector is Dirichlet in the zeroth-unit cell, and it provides the initial data \((\check{\psi}_0, \check{\psi}_1)=(0,0,1,0)\) for the following state, \[\check{\psi}=(\check{\psi}_1,\check{\psi}_2,\ldots)=\left(\binom{1}{0},\binom{-e^{-i\theta}}{0},\binom{\frac{3}{4}e^{-i2\theta}}{0},\binom{-\frac{1}{2}e^{-i3\theta}}{0},\ldots\right).\label{eqn:poly46exp46example}\tag{19}\] Note that \(\check{\psi}\) is not a linear combination of exponential ansätze states, Eq.@eq:eqn:basic46ansatz . Yet, it is readily verified that \(\check{\psi}\) is precisely the zero-energy edge state of \(\check{H}(V,A^{(\theta)})\), \[\begin{align} (A^{(\theta)})^*\check{\psi}_1+V\check{\psi}_2+A^{(\theta)}\check{\psi}_3&=0,\\ V\check{\psi}_1+A^{(\theta)}\check{\psi}_2&=0. \end{align}\]

We may transform this family of examples by \({\rm U}(1)\times {\rm U}(1)\) unitaries, and also scale them. So there is at least a 4-parameter space of gapped chiral-symmetric Hamiltonians for which the ansatz, Eq.@eq:eqn:basic46ansatz will fail to give the edge state.

The point is that the submanifold of those \((V,A)\) for which \(C_E(V,A)\) has non-simple eigenvalues, is generally not just a set of points, and is not automatically negligible. In the current case, the full parameter space is sufficiently large: \[V=\begin{pmatrix} 0 & \overline{v}\\ v & 0\end{pmatrix},\qquad A=\begin{pmatrix} 0 & a_{-+} \\ a_{+-} & 0 \end{pmatrix},\qquad v,a_{+-},a_{-+}\in\mathbb{C},\] subject to the gapped constraint. Then the submanifold of “bad” \((V,A)\) does not introduce disconnections.

A final remark is that the edge state in these examples, Eq.@eq:eqn:poly46exp46example , happen to have polynomial-exponential form. It can be viewed as a certain limiting case, \(\lambda_1,\lambda_2\to-\frac{1}{2}e^{-i\theta}\), as \(E\to 0\). As mentioned in Eq.@eq:eqn:poly46exp46fail , the story is more complicated in general models, with more \(\lambda_i\) and possibly higher multiplicities, and we cannot guarantee the polynomial-exponential ansatz.

References↩︎