Splitting-from-singularity-filling conjecture for long-range free-fermion chains

Let $H_{\operatorname{BDI}}$ be an open chain of size $L$, whose related bulk Hamiltonian has winding number $\omega>0$ and is described by

$$f(z)=z^\omega b_+(z)b_-(z).$$

Define

$$m(z)=\frac{b_+(1/z)}{b_-(1/z)}.$$

Suppose that $m(z)$ has singularities as described above, and let $\mathcal{E}'_s(n)=\Omega_{\theta_s}+2n$ for $n\in\mathbb{Z}_{\geq 0}$ denote the associated levels. Splitting-from-singularity-filling conjecture. The $\omega$ finite-size edge modes have splittings

$$L^{-\varepsilon_1},\dots,L^{-\varepsilon_\omega},$$

where the $\varepsilon_k$ correspond to the $\omega$ lowest levels $\mathcal{E}'_s(n)$. Given $f(z)$, there may exist an $\omega_{\max}$ such that this holds for $\omega<\omega_{\max}$. For $\omega<0$, the analogous statement holds after replacing $m(z)$ by

$$l(z)=\frac{b_-(z)}{b_+(z)}.$$

This conjecture predicts how singularities of the bulk symbol determine the finite-size splitting of multiple edge modes in long-range systems; the source notes that such systems with $|\omega|>1$ had not previously been studied in this context. The range of validity may be limited by an $\omega_{\max}$ depending on $f(z)$.