Determine the disorder order at which temperature dependence enters the disorder variance

Determine at which order in the disorder strength \(\delta\) the temperature dependence first enters the disorder variance of the infinite-data effective coupling in the two-dimensional random-bond Ising model, given that its leading linear-response contribution is temperature-independent.

Background

The paper expands the infinite-data effective coupling $J^*$ around the clean Ising model in powers of the bond-disorder offsets. The leading contribution to the disorder variance is $O(\delta^2)$ and is independent of temperature, producing a self-averaging floor proportional to $1/L^2$.

The authors then identify temperature-dependent contributions at $O(\delta^4)$, including the variance of the quadratic response and the covariance between the linear and cubic responses. The latter is numerically associated with the critical minimum of the disorder variance. The passage explicitly frames determining the first disorder order at which temperature enters as an unresolved question, although subsequent analysis provides an explanation through the fourth-order expansion.