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

Abstract

Entanglement entropy at one-dimensional criticality typically follows the Calabrese-Cardy scaling. In non-Hermitian critical chains near exceptional points, however, we show that the biorthogonal entropy of a finite system retains an additional sensitivity to a small energy gap \(Δ\) even when \(Δ< 1/L\). On top of the usual Calabrese-Cardy term, we find an interval-independent contribution \(S_{\rm res}=\log(ΔL)\), visible as a vertical offset and detectable even for a one-site subsystem. This behavior has no Hermitian analogue: a sub-finite-size gap is effectively invisible to entanglement in unitary critical chains, whereas here the entropy continues to resolve such a gap through its dependence on \(ΔL\). We interpret the result within the de Sitter geometry generated by non-unitary continuous multiscale entanglement renormalization: because the dS extremal surface reaches the IR endpoint, entanglement necessarily retains the endpoint contribution. On a finite ring, a regular circuit cannot terminate at a one-site product state and instead leaves an entangled two-site IR state. Computing the entanglement of the IR state recovers the same \(\log(ΔL)\) term, identifying this additional long-range entanglement as the residual entropy left after finite-depth disentangling.