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Thermal expansion carries crucial information about phase transitions. Inthis study, we employ an interferometer-based dilatometer to investigate thephase transition in BaFe$_2$As$_2$ with picometer-resolution. Thanks to theultra-high resolution and the differential nature of our measurement, wediscover a hysteresis behavior near the antiferromagnetic transitiontemperature $T_\mathrm{N}$ whose pattern reveals a propagation speed of thephase boundary between the antiferromagnetic and paramagnetic phases.Remarkably, this speed is only about 188 $\mu$m/s, seven orders of magnitudelower than that of acoustic waves. This discrepancy, despite the magnetoelasticnature of the transition at $T_\mathrm{N}$, reveals a hidden constraint relatedto the electronic degrees of freedom. Our findings highlight the potential ofhigh-resolution dilatometry as a powerful probe of electron-correlation effectsin quantum materials.

Electronic Phase Propagation Speed in BaFe$_2$As$_2$ Revealed by Dilatometry