is much smaller than $L^2$. Therefore, one can restrict the sum (\[part\_funct\]) to the first few relevant terms.

Lifetime of the interacting model {#sec:lifetime_mf}
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From Eq. (\[leadingNmf\]) we obtain that, even though the number of anyons $N_{\rm mf}$ grows with the system size $L$, the anyon density $n_{\rm mf}$ goes to zero for long-range repulsive interactions with $0 \leq \alpha < 2$. Hence, the population of anyons is increasingly diluted and the system is essentially frozen in the ground state at large system size. This remarkable effect can be attributed to the divergence of the excitation energy $\epsilon_{\rm mf}\simeq T \ln L_\alpha$, which is self-consistently determined from the anyon population in the *whole* system due to the long-range nature of the interactions. Note also that, despite the fact that $\epsilon_{\rm mf}$ is diverging, the total excitation energy density $n_{\rm mf}\epsilon_{\rm mf}/2$ goes to zero for large $L$.

Secondly, the divergence of $\epsilon_{\rm mf}$ leads to a vanishing anyon pair creation rate at large $L$, $$\label{gamma_2emf}
\gamma(- 2\epsilon_{\rm mf}) \simeq \kappa_n T^n \frac{(2\ln L_\alpha)^{n+2}}{2 L_\alpha^2}.$$ This fact allows us to revise the lifetime for the non-interacting memory Eq. (\[T\_estimate\]), simply by substituting $J$ with the equilibrium value $\epsilon_{\rm mf}$, yielding 