Real-time dynamics of Chern-Simons fluctuations near a critical point

The real-time Chern-Simons diffusion rate is studied in $(1+1)$-dimensional massive Schwinger model with a $\theta$-term. We evaluate the real-time correlation function of electric field that represents the topological Chern-Pontryagin number density in $(1+1)$ dimensions. Near the parity-breaking critical point located at $\theta=\pi$ and fermion mass $m$ to coupling $g$ ratio of $m/g \approx 0.33$, we observe a sharp maximum in the Chern-Simons diffusion rate. We interpret this maximum in terms of the growth of critical fluctuations near the critical point, and draw analogies between the massive Schwinger model, QCD near the critical point, and ferroelectrics near the Curie point.