Beyond Calabrese-Cardy Scaling: Exceptional-Point Sensitivity and the de Sitter RT Surface

Entanglement entropy at one-dimensional criticality typically follows the Calabrese–Cardy scaling. In the non-Hermitian critical free-fermion systems studied here, however, we find an additional interval-independent contribution $S_{\mathrm{r}\mathrm{e}\mathrm{s}}=\log (\mathrm{\Delta }L)$, which allows the entropy to remain sensitive to energy gaps below the finite-size scale, $\mathrm{\Delta }<1\mathrm{/}L$. We refer to this term as the residual entropy. To understand its infrared origin, we examine finite-depth non-unitary continuous multiscale entanglement renormalization on a ring. Terminating the critical RG circuit before its singular final layer leaves a two-site infrared state. We calculate the entropy of this state directly from its correlation matrix and recover the same $\log (\mathrm{\Delta }L)$ dependence. The emergent de Sitter geometry provides a complementary geometric interpretation: extremal curves associated with different boundary intervals extend to a common infrared endpoint, suggesting how the entanglement retained there can contribute equally to their entropies.