Understanding Reentrance in Frustrated Magnets: The Case of the ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ Pyrochlore

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, yet its microscopic cause 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 ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. We use single crystal heat capacity measurements to expose that ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ 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 ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ are linked to soft modes. These soft modes arise from phase competition and enhance thermal fluctuations that entropically stabilize a specific ordered phase, resulting 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.

Figure 1

Example of sixfold degenerate states: (a) Palmer-Chalker [36] and (b) ${\psi }_2$ and (c) ${\psi }_3$ basis states of ${\mathrm{\Gamma }}_5$ [41]. The ${\psi }_2$ and ${\psi }_3$ states are connected by a rotation of the spins by an angle $\varphi$ within their local easy planes (yellow circles): $\varphi \equiv n\pi /3(+\pi /6)$ for $n=0,\dots ,5$ correspond to ${\psi }_2$ (${\psi }_3$) [42]. Panels (b) and (c) are for $\varphi =0$ and $\pi /2$, respectively. The manifold with $U(1)$ degeneracy, $\varphi \in [0,2\pi ]$, forms the so-called ${\mathrm{\Gamma }}_5$ states that appear in the [111] phase diagram. (d) Heat capacity, $C_p(T)$, vs temperature, $T$, of ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ with the magnetic field along [100], showing the reentrant nature of the transition. Curves at different fields are offset vertically for clarity. Similar data for the [110] and [111] field directions are included in the Supplemental Material [43]. (Inset) $C_p(T)$ ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ in zero field, with short and long pulse measurements on crystal samples overlaid. Powder data from Shirai et al. [48] is also overlaid to demonstrate agreement between sample types.

Figure 2

$B\text{−}T$ phase diagrams of ${\mathrm{Er}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ in the (a) [100], (b) [110], and (c) [111] field directions, comparing experimental data with sharp (red square box) and smooth (red filled square box) heat-capacity peaks [Fig. 1], to Monte Carlo results with first (square box) and second (black filled square box) order transitions. Experiments and simulations are notably similar, showing the same (multiple) reentrance. The degeneracy $Z_n$ found in simulations is given for each phase. The width of the red rectangles at 0 and 0.1 T represents the position of the double peaks. (d) In a [111] field, each of the six FEPC ground states has a ${\mathrm{\Gamma }}_5$ contribution described by an angle $\varphi$ [Figs. 1 and 1], that can be computed exactly by minimizing the energy of one tetrahedron as a function of $B$.

Figure 3

Classical spin-wave dispersions for $B=0$, 0.40, 0.82, and 1.20 T along the [100] direction, for a path in the FCC Brillouin zone. Note that $B_c=0.82\mathrm{T}$ is a critical field at $T=0$, as shown in Fig. 2. The gray boxes indicate energy scales below $T_c(B=0\mathrm{T})\approx 160\mathrm{mK}$ from Monte Carlo simulations; when modes occur within this region they are considered “soft.” Note that the dispersions for all Palmer-Chalker states are plotted, but may overlap at high-symmetry points or due to their degeneracies in a field.