Magnetic monopole relaxation effects in spin ice Dy$_2$Ti$_2$O$_7$

Spin ice compounds enable the exploration of the dynamics of magnetic monopoles in condensed matter systems. In this study, we use ac calorimetry to probe the dynamical response of the heat capacity of the classical spin-ice compounds Dy$_2$Ti$_2$O$_7$ at low temperatures (0.5-5 K). Using frequencies of 0.01-500 Hz, we find a strong frequency dependence in the measured heat capacity and are able to study thermal relaxation effects on the corresponding timescales. The relaxation time $\tau$ is determined from the frequency dependence of the heat capacity as the characteristic frequency below which the heat capacity saturates. The extracted $\tau$ shows a divergent behavior below 1 K reaching $\sim$6 s at 0.65 K, similar to the relaxation time seen in previous studies. Corresponding specific heat shows a maximum around this temperature. Performing dynamic Monte Carlo simulations, we verify that the specific heat frequency response has its origin in the slow magnetic monopole dynamics indigenous to spin ice. We find a timescale of 20 ms per Monte Carlo step at 4 K in contrast to 2.5 ms mentioned in previous studies by other techniques.