  Reentrance, the return of a system from an ordered phase to a previously
encountered less-ordered one as a controlled parameter is continuously varied,
is a recurring theme found in disparate physical systems, from condensed matter
to black holes. While diverse in its many incarnations and generally
unsuspected, the cause of reentrance at the microscopic level is often not
investigated thoroughly. Here, through detailed characterization and
theoretical modeling, we uncover the microscopic mechanism behind reentrance in
the strongly frustrated pyrochlore antiferromagnet Er$_2$Sn$_2$O$_7$. Taking
advantage of the recent advance in rare earth stannate single crystal
synthesis, we use heat capacity measurements to expose that Er$_2$Sn$_2$O$_7$
exhibits multiple instances of reentrance in its magnetic field $B$ vs.
temperature $T$ phase diagram for magnetic fields along three cubic high
symmetry directions. Through classical Monte Carlo simulations, mean field
theory and classical linear spin-wave expansions, we argue that the origins of
the multiple occurrences of reentrance observed in Er$_2$Sn$_2$O$_7$ are linked
to soft modes. Depending on the field direction, these arise either from a
direct $T=0$ competition between the field-evolved ground states, or from a
field-induced enhancement of the competition with a distinct zero-field
antiferromagnetic phase. In both scenarios, the phase competition enhances
thermal fluctuations which entropically stabilize a specific ordered phase.
This results in an increased transition temperature for certain field values
and thus the reentrant behavior. Our work represents a detailed examination
into the mechanisms responsible for reentrance in a frustrated magnet and may
serve as a template for the interpretation of reentrant phenomena in other
physical systems.
