  Although numerous white dwarf stars host dusty debris disks, the temperature
distribution of these stars differs significantly from the white dwarf
population as a whole. Dusty debris disks exist exclusively around white dwarfs
cooler than 27,000 K. This is all the more enigmatic given that the formation
processes of dusty debris disks should favor younger, hotter white dwarfs,
which likely host more dynamically unstable planetary systems. Here we apply a
sophisticated material sublimation model to white dwarf systems to show that
these statistics are actually a natural result of the interplay of thermal and
tidal forces, and show how they define the circumstellar regions where dusty
debris disks can form. We demonstrate that these processes tend to prevent
stability against both sublimative destruction and reaccretion into
planetesimals for rocky materials until white dwarfs cool to below
~25,000-32,000 K, in agreement with the observed limit of ~27,000 K. For pure
water ice, this critical temperature is less than 2,700 K (requiring a cooling
age older the universe); this precludes pure water ice-rich debris disks
forming through the accepted two-step mechanism. The critical temperature is
size-dependent; more massive white dwarfs could potentially host dusty debris
disks at warmer temperatures. Our model suggests that the location of the disks
within the PG 0010+280, GD 56, GD 362, and PG 1541+651 systems are consistent
with a forsterite-dominated olivine composition. We also find that very cool
white dwarfs may simultaneously host multiple, independently formed dusty
debris disks, consistent with observations of the LSPM J0207+3331 system.
