  The production mechanism of repeating fast radio bursts (FRBs) is still a
mystery, and correlations between burst occurrence times and energies may
provide important clues to elucidate it. While time correlation studies of FRBs
have been mainly performed using wait time distributions, here we report the
results of a correlation function analysis of repeating FRBs in the
two-dimensional space of time and energy. We analyze nearly 7,000 bursts
reported in the literature for the three most active sources of FRB 20121102A,
20201124A, and 20220912A, and find the following characteristics that are
universal in the three sources. A clear power-law signal of the correlation
function is seen, extending to the typical burst duration ($\sim$ 10 msec)
toward shorter time intervals ($\Delta t)$. The correlation function indicates
that every single burst has about a 10-60% chance of producing an aftershock at
a rate decaying by a power-law as $\propto (\Delta t)^{-p}$ with $p =$ 1.5-2.5,
like the Omori-Utsu law of earthquakes. The correlated aftershock rate is
stable regardless of source activity changes, and there is no correlation
between emitted energy and $\Delta t$. We demonstrate that all these properties
are quantitatively common to earthquakes, but different from solar flares in
many aspects, by applying the same analysis method for the data on these
phenomena. These results suggest that repeater FRBs are a phenomenon in which
energy stored in rigid neutron star crusts is released by seismic activity.
This may provide a new opportunity for future studies to explore the physical
properties of the neutron star crust.
