Movable seams in root-of-unity XXZ chains: Relative flux classes and antiunitary spectral pairing

Abstract

Starting from the cyclic duality tensors introduced by Vernier, Miao, and Yamazaki (VMY), whose endpoint-traced matrix-product operators obey ${\mathbb{Z}}_N$ Tambara--Yamagami fusion and realize topological defect lines of the compactified-boson conformal field theory, we retain the virtual endpoint as an $N$-state dynamical degree of freedom and construct an exactly movable seam in the spin-$\frac{1}{2}$ XXZ chain at $q={\mathrm{e}}^{\mathrm{i}\pi M/N}$ with $\gcd (M,N)=1$. The local movement identity holds for any unitary $q$-Weyl pair; in the finite cyclic realization, all output phases are locally gauge equivalent and share the same four seam eigenvalues. On a ring, the output gauge becomes a directed twist on a single bond, while endpoint conjugacy reduces the $2N$ labels to two relative ${\mathbb{Z}}_2$ flux classes. An explicit antiunitary symmetry protects the class $\eta \equiv M-1(\mathrm{mod}2)$: for even $N$ it pairs distinct charge sectors isospectrally, whereas for odd $N$ it fixes one sector and squares to $-I$ there. Hence every many-body energy eigenspace in the protected class has even multiplicity. The source value ${\eta }_{\mathrm{VMY}}=1-M$ satisfies this condition for every coprime root, whereas the ungauged value $\eta =0$ does so only for odd $M$. Exact finite-size controls show that the opposite class can contain simple levels, so a global flux invisible to local gauge equivalence distinguishes the two classes spectrally.

23 pages, 4 figures