The thermal conductivity $\kappa$ of underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$ was measured in the $T\to 0$ limit as a function of hole concentration $p$ across the superconducting critical point at $p_{\mathrm{SC}}\equiv 5.0\%$. The evolution of bosonic and fermionic contributions to $\kappa$ was tracked as the doping level evolved continuously in each of our samples. For $p\le p_{\mathrm{SC}}$, we observe a $T^3$ component in $\kappa$ which we attribute to the boson excitations of a phase with long-range spin or charge order. Fermionic transport, observed as a $T$-linear term in $\kappa$ which persists unaltered through $p_{\mathrm{SC}}$, violates the Wiedemann-Franz law, since the electrical resistivity varies as $\log (1/T)$ and grows with decreasing $p$.

• Received 26 June 2006

DOI:https://doi.org/10.1103/PhysRevLett.97.207001