Hidden Twisted Sectors and Exponential Degeneracy in Root-of-Unity XXZ Heisenberg Chains
Recently, product states have been identified as simple-structured eigenstates of XXZ Heisenberg spin models in arbitrary dimensions, occurring at anisotropy values corresponding to certain roots of unity. Yet, the product states typically only span parts of a larger degenerate eigenspace. Here, we classify this eigenspace in the one-dimensional periodic XXZ chain at all roots of unity $q$, where $q^2$ is an $\ell$-th primitive root of unity. For commensurate chain lengths $N$ with $q^N=1$, we prove that the minimal degeneracy is $2^{N/\ell}\ell$ using the representation theory of the affine Temperley-Lieb (aTL) algebra. For the incommensurate case, we derive analogous exponential lower bounds of $2^{2\lfloor\frac{N}{2\ell}\rfloor+1}$ if $N$ is even and $2^{2\lfloor \frac{N}{2\ell}+\frac{1}{2}\rfloor}$ if $N$ is odd and $q^\ell=1$. Our proof employs the morphisms between aTL modules discovered by Pinet and Saint-Aubin and emphasizes the importance of exact sequences and hidden twisted boundary condition sectors that mediate the degeneracy. In the case of commensurate chain lengths, we connect to the Fabricius-McCoy string construction of all Bethe roots of the degenerate subspace, which previously uncovered parts of our results. We corroborate our results numerically and demonstrate that the lower bound is saturated for chain lengths $N\leq20$. Our work demonstrates for a concrete system how the interplay of the Bethe ansatz, aTL representation theory, and twisted boundary conditions explains degeneracy connected to long-lived product states, stimulating research towards generalization to higher dimensions. Exponential degeneracy could boost applications of spin chains as quantum sensors.