The correlation length for the two-dimensional classical ising model goes as $$\xi_{ising}(T)\sim |T-T_c|^{-\nu};\qquad \nu=1$$ We can map the classical ising model to its quantum cousin, one-dimensional transverse field ising model or simply TIM. Jordan-Wigner transformation followed by Bogolibouv transformation gives free fermionic theory whose mass goes like $$m\sim |g-g_c|$$ Field theory interpretation (baiscally $m=\xi^{-1}$) gives $$\xi_{fermionic}\sim |g-g_c|^{-\nu};\qquad \nu=1 $$ Is this coincidence that both critical exponents $\nu$ pertaining to correlation length turns out to be same?
With this question, I also give a relation of $g$ with $T$: $$g(T)=T\frac{e^{-J/T}}{J}$$ when $T\gg T_c$ then $g$ goes linear as $g(T)\sim \frac{T}{J}-2$. But we intrested near criticality, i.e., $T$ near $T_c$.
