Observing the nodal-line conversion determined by the relative homotopy

Directly identifying the non-Abelian nodal-line semimetals (NASM) is quite challenging because nodal-line semimetals typically do not possess topologically protected boundary modes. Here, by reconstructing the correspondence between the bulk states of Hermitian systems and circuit voltage modes through gauge scale potential, the temporal topolectrical circuits (TTC) for evidencing NASM are proposed. Following the logical progress of discovering NASM, we start by demonstrating the relative homotopy group of two-band models using TTC, which can faithfully determine the conversion rules between the nodes in and out of the non-local-symmetry invariant subspace. Next, we show that those rules dramatically change with the consideration of the additional band, historically leading to the arising of the NASM. Also, we demonstrate the unique non-Abelian constrained nodal configuration -- earring nodal lines. Our results established NASM for further investigating topological line degeneracies, and proposed TTC will be a versatile platform for exploring nodal-line semimetals.