Enhanced-Entropy Phases in Geometrically Frustrated Pyrochlore Magnets

Abstract

Frustrated magnets provide a platform for exploring exotic phases beyond conventional ordering, with potential relevance to functional materials and information technologies. In this work, we use Monte Carlo simulations to map the thermodynamic phase diagram of pyrochlore iridates R2Ir2O7 (R = Dy, Ho) with three stable magnetic ground-state stable phases: frustrated spin-ice 2 in 2 out (2I2O) phase, frustrated fragmented 3 in 1 out/1 in 3 out (3I1O/1I3O) phase, and antiferromagnetic all in all out (AIAO) phase without frustration. We discovered two additional emergent metastable phases at finite temperatures, located between the boundaries separating those stable phases. These metastable phases exhibit high magnetic susceptibility and high entropy without long-range order. Their stabilization arises from entropic minimization of the free energy, where the entropy dominates energetic competition near phase boundaries at finite temperatures. Our results demonstrate a platform to engineer highly susceptible and degenerated states through frustration and thermal activation, offering a foundation for entropy-based design of metastable phases in correlated systems.

Figures (5)

• Figure 1: Magnetic structure and frustration in a pyrochlore lattice. (a) Illustration of lattice and spin structure of pyrochlore $R_2$Ir$_2$O$_7$ magnets, where 3I1O/1I3O spin structures are sketched due to the intermediate strength of $d$-$f$ interaction as compared to $J_\text{eff}$ interaction. The tetrahedra associated with $R^{3+}$ moments are shown in blue (3I1O) and red (1I3O), while those associated with $\text{Ir}^{4+}$ moments are shaded white. (b) Magnetic order parameter $M$ defined as \ref{['Eq:2']}, as a function of $k_\text{B}T/J_\text{eff}$ and local static field $H_\text{loc}/J_\text{eff}$ (upper panel), and ground-state $M$ as a function of $H_\text{loc}/J_\text{eff}$ (lower panel). (c) Normalized entropy $S/S_\infty$ defined as \ref{['Eq:3']}, as a function of $k_\text{B}T/J_\text{eff}$ and $H_\text{loc}/J_\text{eff}$ (upper panel), and ground-state $S/S_\infty$ versus $H_\text{loc}/J_\text{eff}$ (lower panel). Three ground state phases, 2I2O, 3I1O/1I3O, AIAO, and the paramagnetic phase, are labeled as I, II, III, and PM, respectively. PB I-II and PB II-III stand for the phase boundaries between phases I/II and II/III, respectively.
• Figure 2: Magnetic and thermodynamic response of pyrochlore $R_2$Ir$_2$O$_7$ magnets. (a) Magnetic susceptibility $\chi$ (upper panel) and specific heat capacity $C_v$ (lower panel) for phases I, II, and III as a function of normalized temperature $k_\text{B}T/J_\text{eff}$ for different $H_\text{loc}/J_\text{eff}$ values. (b) and (c) Magnetic susceptibility $\chi$ (upper panel) and specific heat capacity $C_v$ (lower panel) near the vicinity of PB I-II and PB II-III respectively, as a function of $k_\text{B}T/J_\text{eff}$ for various $H_\text{loc}/J_\text{eff}$ values. Shaded regions indicate the peak regions in the heat capacity, which are associated with phase transitions. We tentatively label the phase immediately below PM as an enhanced-entropy (EE) phase.
• Figure 3: Magnetic and thermodynamic response phase diagrams. (a) Specific heat capacity $C_v$ and (b) Magnetic susceptibility $\chi$ as a map of $k_\text{B}T/J_\text{eff}$ and $H_\text{loc}/J_\text{eff}$. Small circles filled with colors represent the peak positions of $\chi$ and $C_v$ as a function of $k_\text{B}T/J_\text{eff}$ and $H_\text{loc}/J_\text{eff}$. Phases I (2I2O), II (3I1O/1I3O), and III (AIAO) are labeled. Two EE phases are observed between phases I/II and II/III, as defined in the $C_v$ map, and are labeled as $\text{EE}_{\text{I-II}}$ and $\text{EE}_{\text{II-III}}$, respectively.
• Figure 4: Magnetic frustration and entropy of EE phases. (a) 2D illustration of EE phases $\text{EE}_{\text{I-II}}$ (left panel) and $\text{EE}_{\text{II-III}}$ (right panel). Tetrahedral sites with magnetic charge $Q = 0$ (phase I), $Q = \pm q_i$ (phase II), and $Q = \pm 2q_i$ (phase III) are shown in green, yellow, and light red, respectively. (b) $M$ as a function of $k_\text{B}T/J_\text{eff}$ for $\text{EE}_{\text{I-II}}$ (left panel) and $\text{EE}_{\text{II-III}}$ (right panel). (c) $S/S_\infty$ as a function of $k_\text{B}T/J_\text{eff}$ for $\text{EE}_{\text{I-II}}$ (left panel) and $\text{M}_\text{II-III}$ (right panel). Observables are shown over the range of $H_\text{loc}/J_\text{eff}$ with interval 0.007: from $H_\text{loc}/J_\text{eff} = 1.91$ to $2.08$ for $\text{EE}_{\text{I-II}}$, and from $5.91$ to $6.08$ for $\text{EE}_{\text{II-III}}$.
• Figure 5: Free energy mechanism for entropic stabilization of EE phases. (a) Internal energy $U$ and (b) Free energy $F = U - TS$ as a function of static background field $H_{\text{loc}}/J_{\text{eff}}$, shown for regions surrounding the stable phase boundaries PB I--II (left panel) and PB II--III (right panel). Lowest and higher values of each $H_{\text{loc}}/J_{\text{eff}}$ parameter are represented by solid and dashed lines, respectively. (c) Comparison between predicted and simulated EE phase boundaries. The solid lines were obtained from the equilibrium of free energy across the phase boundary as described in \ref{['Eq:4']}. Small circles filled with color represent the observed peak positions of $C_v$ as a function of $k_\text{B}T/J_\text{eff}$ and $H_{\text{loc}}/J_{\text{eff}}$ near to phase transitions.