Frustration and chirality in three-dimensional trillium lattices: Insights and Perspectives

Abstract

Condensed matter physics continues to seek new frustrated quantum materials that not only deepen our understanding of fundamental physical phenomena but also hold promise for transformative technologies. In this review article, we highlight the unique features of chiral spin topology and review the topological phenomena recently identified in trillium lattice compounds. Based on the unique spin states realized in these systems, we explore the potential for realizing various theoretically proposed chiral quantum phases. We examine representative materials including the magnetic insulating compound K2Ni2(SO4)3 and and the intermetallic EuPtSi discussing both experimental findings and theoretical predictions, while outlining several key questions. Finally, we offer a perspective on promising research directions aimed at uncovering novel emergent behavior in chiral trillium lattice-based materials.

Figures (6)

• Figure 1: (a) Schematic projection of the chiral trillium lattice (space group $P$2$_1$3), consisting of a three-dimensional network of corner-sharing equilateral triangles, onto the (111) plane, viewed along the $C_3$ rotational axis. (b) Schematic of an interpenetrating double trillium lattice. (c) Geometry of a hypertrillium lattice, arising from the specific bond lengths between magnetic ions in the double trillium lattice.
• Figure 2: (a) Temperature dependence of the inverse magnetic susceptibility of Na[Mn(HCOO)$_3$] at 100 Oe with a Curie-Weiss (CW) fit. The pale sky-blue shaded region indicates the classical spin liquid (CSL) regime, where the inverse magnetic susceptibility (red line) is obtained from classical Monte Carlo calculations, as discussed in the text. (b) Dipolar-to-Heisenberg interaction ratio ($D/J$) versus reduced temperature ($T/J$) phase diagram of a trillium lattice, with phase boundaries determined by classical Monte Carlo simulations. The intensity of red and blue shading reflects the strength of magnetic scattering associated with the 2-K and Y phases, respectively, while the empty circle indicates the position of Na[Mn(HCOO)$_3$] (c) Magnetic field--temperature phase diagram with a background contour map of $\chi_{m} = d(M/M_{\mathrm{sat}})/d(\mu_{0}H)$. The inset shows the isothermal magnetization at 100 mK, with experimental data in black and a low-field linear fit in red. The field-induced UUD ordered phase, the 120$^{\circ}$ Y phase, and the CSL phase are separated by a dotted curve that closely follows the expected phase boundary from the antiferromagnetic Heisenberg model with an additional Zeeman term. Adapted from PhysRevLett.128.177201 with permission APS.
• Figure 3: (a) Schematic of dimer-singlet coverings (shown as dark thin ellipsoids) on the pyrochlore lattice, with one singlet per tetrahedron. It also shows a hole (light-colored sphere) moving through the lattice by binding two $\pi$-fluxes to it. Adapted from Glittum2025 with permission NPG.
• Figure 4: (a) Schematic of the double trillium lattice formed by two sublattices of Ni$^{2+}$ ions in K$_{2}$Ni$_{2}$(PO$_{4}$)$_{3}$. Other atoms are omitted for clarity to highlight the two interpenetrating trillium networks. Among the five exchange interactions, only $J_4$ and $J_5$ dominate, jointly forming a hypertrillium lattice that induces strong frustration in the system. (b) Temperature dependence of the magnetic susceptibility measured in an applied field of 0.1 T along the three orthogonal directions of the cubic lattice (left $y$-axis), with the inverse magnetic susceptibility for $H$$//$[111] plotted on the right $y$-axis. (c) Temperature dependence of the specific heat, showing two anomalies at $T^{*} = 1.14$ K and $T^{**} = 0.74$ K. The solid line represents the specific heat of the nonmagnetic analogue K$_{2}$Mg$_{2}$(SO$_{4}$)$_{3}$. (d) Magnetic field--temperature phase diagram illustrating the suppression of the specific heat anomaly with increasing field. (e) Temperature dependence of the specific heat at low temperatures under several magnetic fields: 0 T, 0.5 T, 1.5 T, 7 T, and 14 T (top to bottom). The solid and dashed lines indicate the nature of the magnetic excitations as described in the text. (f) Schematic phase diagram showing temperature ($T$) versus tuning parameter ($g$), illustrating the proximity of K$_2$Ni$_2$(SO$_4$)$_3$ to a quantum critical point $g_c$. The material lies in a regime where quantum fluctuations suppress magnetic order ($T_{\rm N} \ll |\theta_{\rm CW}|$), leading to the emergence of a quantum spin liquid phase. The shaded region marks the quantum critical regime. Adapted from PhysRevLett.127.157204 and Gonzalez2024 with permission APS and NPG, respectively.
• Figure 5: The first row presents the structural and thermodynamic data of KSrFe$_{2}$(PO$_{4}$)$_{3}$, while the second row shows the corresponding data for K$_{2}$FeSn(PO$_{4}$)$_{3}$. (a) Schematic illustration of randomly oriented dimers formed by Fe1 and Fe2 sites, coupled via intra-dimer exchange interaction $J_1$, which are further linked through inter-dimer coupling $J_2$. (b) Two independent trillium lattices coupled via exchange couplings $J_3$ and $J_5$ and linked through inter-trillium coupling $J_4$, forming a hypertrillium lattice. (c) Temperature dependence of the $ac$ magnetic susceptibility at 100 Hz, with the inset displaying an image of a single crystal of KSrFe$_{2}$(PO$_{4}$)$_{3}$. (d) Low-temperature magnetic specific heat of KSrFe$_{2}$(PO$_{4}$)$_{3}$ measured in zero magnetic field. The solid line corresponds to the power-law fitting. (e--h) Corresponding figures for K$_{2}$FeSn(PO$_{4}$)$_{3}$, in which two Fe sites are partially substituted by Sn ions. Panel (h) additionally shows the magnetic specific heat under different applied magnetic fields. Adapted from 10.1063/5.0096942 and lmsf73hn with permission from AIP and APS, respectively, and from Ref. ChoiPrivateComm.