Topologically protected negative entanglement

The entanglement entropy encodes fundamental characteristics of quantum many-body systems, and is particularly subtle in non-Hermitian settings where eigenstates generically become non-orthogonal. In this work, we find that negative biorthogonal entanglement generically arises from topologically protected non-orthogonal edge states in free fermion systems, especially for flat-band edge states. Departing from previous literature which associated negative entanglement with exceptional gapless points, we show that robustly negative entanglement can still occur in gapped systems. Gapless 2D flat-band edge states, however, exhibit novel $S_A\sim -\frac{1}{2}L_y^2\log L$ entanglement behavior which scales quadratically with the transverse dimension $L_y$, independent of system parameters. This dramatically negative scaling can be traced to a new mechanism known as non-Hermitian critical skin compression (nHCSC), where topological and skin localization in one direction produces a hierarchy of extensively many probability non-conserving entanglement eigenstates across a cut in another direction. Our discovery sheds light on new avenues where topology interplays with criticality and non-Hermitian localization, unrelated to traditional notions of topological entanglement entropy. This topologically protected negative entanglement also manifests in the second R\'enyi entropy, which can be measured through SWAP operator expectation values.