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 $\mathrm{\ensuremath{\Delta}}{N}^{2}\ensuremath{\ge}|\mathcal{I}|/2$ and the entanglement entropy by $S\ensuremath{\ge}2|\mathcal{I}|ln2$. 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.
We derive exact expressions for the local entanglement entropy
E in the ground state of the one-dimensional Hubbard model at a quantum phase transition driven by a change in magnetic field
h or chemical potential $\ensuremath{\mu}$. The leading divergences of $\ensuremath{\partial}\mathcal{E}/\ensuremath{\partial}h$ and $\ensuremath{\partial}\mathcal{E}/\ensuremath{\partial}\ensuremath{\mu}$ are shown to be directly related to those of the zero-temperature spin and charge susceptibilities. Logarithmic corrections to scaling signal a change in the number of local states accessible to the system as it undergoes the transition.