Thermally Activated Long-Range Entanglement from Non-Abelian Conservation Laws

Thermal noise ordinarily suppresses quantum entanglement. We show that a strong non-Abelian conservation law can convert local thermal fluctuations into an unbounded operational resource. For a broad class of finite-range $SU(2)$-invariant spin chains restricted to the global-singlet sector, an explicit representation-space protocol yields $Y_N=\frac{1}{2}{\log }_{2}N+O_{\beta }(1)$, and hence $E_{\mathrm{D}}\ge Y_N$, throughout a finite high-temperature interval. Local thermal fluctuations produce subsystem spins $j\sim \sqrt{N}$, whose globally locked irreducible representations contain ${\log }_2(2j+1)\sim \frac{1}{2}{\log }_{2}N$ ebits. An exactly solvable dimer chain exhibits a sharper effect: its zero-temperature state is unentangled across the cut, whereas every fixed $T>0$ produces $E_{\mathrm{D}}=\frac{1}{2}{\log }_{2}N+C(T)+o(1)$, with crossover scale $T_{\ast }(N)\sim \mathrm{\Delta }\mathrm{/}\ln N$. Exact diagonalization of a nonintegrable chain is consistent with the predicted scaling. Thus heating can activate system-size-diverging distillable entanglement across a macroscopic bipartition when thermalization is confined by a non-Abelian conservation law.