  Hyperuniform states of matter are characterized by anomalous suppression of
long-wavelength density fluctuations. While most of interesting cases of
disordered hyperuniformity are provided by complex many-body systems like
liquids or amorphous solids, classical spin chains with certain long-range
interactions have been shown to demonstrate the same phenomenon. It is
well-known that the transverse field Ising model shows a quantum phase
transition (QPT) at zero temperature. Under the quantum effects of a transverse
magnetic field, classical hyperuniform spin chains are expected to lose their
hyperuniformity. High-precision simulations of these cases are complicated
because of the presence of highly nontrivial long-range interactions. We
perform extensive analysis of these systems using density matrix
renormalization group to study the possibilities of phase transitions and the
mechanism by which they lose hyperuniformity. We discover first-order QPTs in
the hyperuniform spin chains. An interesting feature of the phase transitions
in these disordered hyperuniform spin chains is that, depending on the
parameter values, the presence of transverse magnetic field may remarkably lead
to increase in the order of the ground state as measured by the "$\tau$ order
metric," even if hyperuniformity is lost. Therefore, it would be possible to
design materials to target specific novel quantum behaviors in the presence of
a transverse magnetic field. Our numerical investigations suggest that these
spin chains can show no more than two QPTs. We further analyze the long-range
interacting spin chains via the Jordan-Wigner mapping, showing that under the
pairwise interacting approximation and a mean-field treatment, there can be at
most two QPTs. Based on these numerical and theoretical explorations, we
conjecture that these spin chains can show a maximum of two QPTs at zero
temperature.
