  We report a comprehensive specific heat and inelastic neutron scattering
study to explore the possible origin of multiferroicity in HoCrO$_3$. We have
performed specific heat measurements in the temperature range 100 mK - 290 K
and inelastic neutron scattering measurements were performed in the temperature
range 1.5 - 200 K. From the specific heat data we determined hyperfine
splitting at 22.5(2) $\mu$eV and crystal field transitions at 1.379(5) meV,
10.37(4) meV, 15.49(9) meV and 23.44(9) meV, indicating the existence of strong
hyperfine and crystal field interactions in HoCrO$_3$. Further, an effective
hyperfine field is determined to be 600(3) T. The quasielastic scattering
observed in the inelastic scattering data and a large linear term
$\gamma=6.3(8)$ mJmol$^{-1}$K$^{-2}$ in the specific heat is attributed to the
presence of short range exchange interactions, which is understood to be
contributing to the observed ferroelectricity. Further the nuclear and magnetic
entropies were computed to be, $\sim$$17.2$ Jmol$^{-1}$K$^{-1}$ and $\sim$34
Jmol$^{-1}$K$^{-1}$, respectively. The entropy values are in excellent
agreement with the limiting theoretical values. An anomaly is observed in peak
position of the temperature dependent crystal field spectra around 60 K, at the
same temperature an anomaly in the pyroelectric current is reported. From this
we could elucidate a direct correlation between the crystal electric field
excitations of Ho$^{3+}$ and ferroelectricity in HoCrO$_3$. Our present study
along with recent reports confirm that HoCrO$_3$, and $R$CrO$_3$ ($R=$ Rare
earth) in general, possess more than one driving force for the ferroelectricity
and multiferroicity.
