  Gauge theories are powerful tools in theoretical physics, allowing complex
phenomena to be reduced to simple principles, and are used in both high-energy
and condensed matter physics. In the latter context, gauge theories are
becoming increasingly popular for capturing the intricate spin correlations in
spin liquids, exotic states of matter in which the dynamics of quantum spins
never ceases, even at absolute zero temperature. We consider a spin system on a
three-dimensional pyrochlore lattice where emergent gauge fields not only
describe the spin liquid behaviour at zero temperature but crucially determine
the system's temperature evolution, with distinct gauge fields giving rise to
different spin liquid phases in separate temperature regimes. Focusing first on
classical spins, in an intermediate temperature regime, the system shows an
unusual coexistence of emergent vector and matrix gauge fields where the former
is known from classical spin ice systems while the latter has been associated
with fractonic quasiparticles, a peculiar type of excitation with restricted
mobility. Upon cooling, the system transitions into a low-temperature phase
where an entropic selection mechanism depopulates the degrees of freedom
associated with the matrix gauge field, rendering the system spin ice like. We
further provide numerical evidence that in the corresponding quantum model, a
spin liquid with coexisting vector and matrix gauge fields has a finite window
of stability in the parameter space of spin interactions down to zero
temperature. Finally, we discuss the relevance of our findings for non-Kramers
pyrochlore materials.
