Integer-Graded Reciprocal Skin Effect from Internal Rotation

Abstract

A prominent class of reciprocal non-Hermitian skin effects has a ${\mathbb{Z}}_2$ partner structure, in which a binary label records which of two partners occupies which edge. We show that conserved internal rotation extends this binary structure to an unbounded integer-graded hierarchy. Every angular momentum channel is an exactly solvable asymmetric chain with its own integer point-gap winding, while conjugate channels accumulate at opposite edges so that reciprocity is preserved globally. For conserved channels, the classification contains one integer per conjugate pair, with the number of independent integers determined by the internal multiplet. Half-integer spin realizes the symplectic class AII${}^{†}$, whose ${\mathbb{Z}}_2$ index is recovered as the parity of the grading. Conjugate-channel mixing destroys the grading and leaves a critical, scale-free skin effect. A cyclic three-site lattice and a passive array of lossy resonators provide minimal realizations.

10 pages, 3 figures