Sub-extensive non-stabilizerness in the Dyck-Fredkin spin chain

The stabilizer Rényi entropy is a quantitative measure of non-stabilizerness, or magic, and has typically been found to scale extensively with system size $N$ (i.e., $\mathrm{\Theta }(N)$) for a variety of many-body quantum states. In this note, we study the stabilizer Rényi entropy of the ground state of the spin-$\frac{1}{2}$ Dyck-Fredkin chain and its $t$-deformation, a local frustration-free model with unusual spectral-gap scaling. Exploiting the combinatorial structure, we carry out numerically exact finite-size calculations, which indicate asymptotic behavior depending on $t$: $\mathrm{\Theta }(N)$ for $t<1$, $\mathrm{\Theta }(\log N)$ at $t=1$, and $\mathrm{\Theta }(1)$ for $t>1$. The scaling at $t=1$ could be another manifestation of the unconventional criticality of the model, while the contrast with the behavior of the entanglement entropy suggests that non-stabilizerness might provide a new window into quantum many-body systems.