We consider the topological protection of entanglement and particle fluctuations for a general one-dimensional chiral topological insulator with winding number I. We prove, in particular, that when the periodic system is divided spatially into two equal halves, the single-particle entanglement spectrum has 2|I| protected eigenvalues at $1/2$. Therefore the number fluctuations are bounded from below by ΔN²≥|I|/2 and the entanglement entropy by S≥2|I|ln2. We note that our results are obtained by applying directly an index theorem to the microscopic model and do not rely on an equivalence to a continuum model or a bulk-boundary correspondence for a slowly varying boundary.
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Monkman et al. (2023) studied this question.