Low-temperature entropies and possible states in geometrically frustrated magnets

TL;DR

The paper investigates how the entropy released at low temperatures in geometrically frustrated magnets encodes the structure of their low-energy spin states. It numerically computes ground-state entropies for spin-1 Ising on the triangular lattice and spin-3/2 Ising on SCGO-type lattices using Wang-Landau sampling, and contrasts these with experimentally measured peak entropies in NiGa_2S_4, FeAl_2Se_4, SCGO, and BSZCGO. Key findings show $S_{\infty} = 0.435854$ per spin for the spin-1 triangular lattice and $S_{\mathrm{SCGO}} = 0.331991$ per spin for the SCGO lattice, with experimental data indicating effective spin-$\tfrac{1}{2}$ degrees of freedom governing the low-energy manifold in several triangular-lattice compounds; SCGO/BSZCGO analyses reveal nuanced contributions from additional spins and potential high-temperature entropy peaks. These results demonstrate how entropy measurements constrain the nature of low-energy states in GF magnets and motivate further thermodynamic measurements at both low and high temperatures to fully map the entropy landscape across related materials.

Abstract

The entropy that an insulating magnetic material releases upon cooling can reveal important information about the properties of spin states in that material. In many geometrically frustrated (GF) magnetic compounds, the heat capacity exhibits a low-temperature peak that comes from the spin states continuously connected to the ground states of classical models, such as the Ising model, on the same GF lattice, which manifests in the amount of entropy associated with this heat-capacity peak. In this work, we simulate numerically the values of entropy released by higher-spin triangular-lattice layered systems and materials on SCGO lattices. We also compare the experimentally measured values of entropy in several strongly GF compounds, $NiGa_2S_4$, $FeAl_2Se_4$ and SCGO/BSZCGO, with possible theoretical values inferred from the classical models to which the quantum states of those materials may be connected. This comparison suggests that the lowest-energy states of higher-spin layered triangular-lattice compounds can be described in terms of doublet states on individual magnetic sites. Our analyses demonstrate how the values of entropy can reveal the structure of low-energy magnetic states in GF compounds and call for more accurate thermodynamic measurement in GF magnetic materials.

Figures (6)

• Figure 1: The heat capacity and the entropy per spin (in units of the ideal gas constant $R$) as a function of temperature in a geometrically frustrated magnet. The heat capacity exhibits two peaks, near the "hidden energy scale" $T^*$, and near the Curie-Weiss temperature $\theta_{CW}$. The peaks may partially overlap in some GF compounds. The total entropy $S=\int_0^\infty C(T)/T dT$ per spin associated with the two peaks is given by the entropy $\ln(2s+1)$ of a free spin. The entropy of the lower peak matches the ground-state entropy $S_\text{classical}$ of a classical (e.g. Ising) spin model on the same lattice.
• Figure 2: Cr$^{3+}$ sites in SCGO. The labels on the right (4f$_{\mathrm{vi}}$, 12k, 2a) denote the crystallographic Wyckoff positions of the Cr$^{3+}$ ions in the magnetoplumbite structure. The kagome layers correspond to the 12k sites, the triangular layers to the 2a sites, and the interlayer ions to the 4f$_{\mathrm{vi}}$ sites. The red dashed outline highlights an hourglass unit, consisting of two corner-sharing tetrahedra (12k–2a–12k). The BCGO and BSZCGO compounds have a similar magnetic structure but do not contain the 4f$_{\mathrm{vi}}$ sites.
• Figure 3: The microcanonical entropy $h(E)=\ln g(E)/N$ per spin, where $g(E)$ is the degeneracy of the system's level with the energy $E$ (per spin), in the spin-3/2 Ising model on the kagome–triangle–kagome trilayer lattice, characteristic of SCGO and BCGO, consisting of $21\times 21$ hourglass units shown in Fig. \ref{['fig:SCGO']}.
• Figure 4: Non-linear fit results for the ground-state entropy per spin of the SCGO/BCGO lattice. Where $L$ is the linear dimension of the system, and $N = L^2$ is the total hourglass units. The entropy per spin converges rapidly and stabilizes around the thermodynamic limit of approximately $0.331991$ per spin. The numerical errors are smaller than the symbol size and therefore not shown.
• Figure S1: The microcanonical entropy $h(E)=\ln g(E)/N$ per spin, where $g(E)$ is the degeneracy of the system's level with the energy $E$ (per spin), in the spin-1 Ising model on the $47 \times 47$ triangular lattice.