  We report results of low-temperature heat capacity, magnetocaloric effect and
neutron diffraction measurements of TmVO$_{4}$, an insulator that undergoes a
continuous ferroquadrupolar phase transition associated with local
partially-filled $4f$ orbitals of the thulium (Tm$^{3+}$) ions. The
ferroquadrupolar transition, a realization of Ising nematicity, can be tuned to
a quantum critical point using a magnetic field oriented along the $c$-axis of
the tetragonal crystal lattice, which acts as an effective transverse field for
the Ising-nematic order. In small magnetic fields, the thermal phase transition
can be well-described using a semi-classical mean field treatment of the
transverse field Ising model. However, in higher magnetic fields, closer to the
field-tuned quantum phase transition, subtle deviations from this
semi-classical behavior are observed due to quantum fluctuations. Although the
phase transition is driven by the local $4f$ degrees of freedom, the crystal
lattice still plays a crucial role, both in terms of mediating the interactions
between the local quadrupoles, and in determining the critical scaling
exponents, even though the phase transition itself can be described via mean
field. In particular, bilinear coupling of the nematic order parameter to
acoustic phonons changes the spatial and temporal fluctuations of the former in
a fundamental way, resulting in different critical behavior of the nematic
transverse-field Ising model as compared to the usual case of the magnetic
transverse-field Ising model. Our results establish TmVO$_{4}$ as a model
material, and electronic nematicity as a paradigmatic example, for quantum
criticality in insulators.
