Evidence for Correlated Dynamics near the Berezinskii-Kosterlitz-Thouless-like Transition in Highly Underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$

A low-frequency resistance noise study in highly underdoped thick films of La$_{2-x}$Sr$_{x}$CuO$_{4}$ ($x=0.07$ and 0.08) reveals slow, correlated dynamics and breaking of ergodicity near the superconducting transition of the Berezinskii-Kosterlitz-Thouless type. The observed correlated behavior is strongly suppressed by disorder.