Multistage spin correlations in the $s$ = 1/2 stuffed hyper-star lattice Li$_{2}$Cu$_{2}$(MoO$_{4}$)$_{3}$

Abstract

Star lattice, which can be visualized as a honeycomb network with each vertex replaced by a triangle, provides a rare platform for realizing exotic quantum states such as quantum spin liquids and disorder-driven random-singlet (RS) states. Herein, we investigate the ground-state properties of the three-dimensional (3D) stuffed hyper-star lattice Li$_2$Cu$_2$(MoO$_4$)$_3$, which exhibits a crossover from short-range spin correlations to a disorder-driven RS-like state below $T^{*}\sim$15.8 K. Thermodynamic and microscopic measurements capture this crossover through a change in the power-law behavior of various observables, from $\sim T^{0.25}$ for $T > T^{*}$ to $\sim T^{-0.50}$ for $T < T^{*}$. Upon further cooling, a quasi-frozen state emerges near $T_{\rm f} = 0.32$ K, likely associated with weakly coupled spin chains within the hyper-star spin network. Our results underscore the crucial role of orphan spins and weak residual interactions in stabilizing a disorder-driven quantum-disordered state in 3D.

Figures (7)

• Figure 1: (a) Synchrotron XRD pattern at 120 K together with the Rietveld refinement. Experimental data are shown as filled circles, the calculated profile as a black line, and their difference (obs.–cal.) as a blue line. The Bragg peak positions are marked by olive vertical bars. (b) Schematic of the 3D version of a distorted stuffed hyper-star lattice of Cu$^{2+}$ ions projected along the $b$-axis. The Cu1, Cu2, and Cu3 sites are interconnected via O ions, forming superexchange pathways of the type Cu–O–Cu. In contrast, the Cu4 site resides at the center of the polygons and nearly isolated from the other Cu sites. (c) The dashed region in Fig. \ref{['STFig']}(b) is shown with the $b$-axis perpendicular to the $ac$ plane, highlighting a distorted triangular spin network of Cu$^{2+}$ ions formed by the Cu1, Cu2, and Cu3 sites. These units are vertically connected by inter-triangular bonds, generating zigzag chains of Cu ions running along the $b$-axis. An example of a Cu3 chain is shown; similar chains are also formed by the Cu1 and Cu2 sites. In contrast, the Cu4 sites form a linear chain of Cu$^{2+}$ ions.
• Figure 2: (a) Temperature-dependent magnetic susceptibility $\chi(T)$ of Li$_{2}$Cu$_{2}$(MoO$_{4}$)$_{3}$ at $\mu_{0}H = 0.5$ T. (b) Temperature dependence of the inverse magnetic susceptibility after subtracting the diamagnetic contribution. The high-temperature Curie–Weiss fit is shown by a red line. (c) Log–log plot of $\chi(T)$ at various magnetic fields. The dashed vertical line (black) indicates the position of $T^{*} = 15.8$ K. Above and below $T^{*}$, $\chi(T)$ follows a power-law behavior, as indicated by the dashed lines (purple). (d) Isothermal magnetization at various temperatures. (e) High-field magnetization data decomposed into intrinsic magnetization $M_{\mathrm{int}}$ and orphan-spin contribution $M_{\mathrm{orp}}$ using Eq. (\ref{['eq1']}), as described in the text. (f) Real part of the ac magnetic susceptibility at various frequencies in the low-temperature range. The inset shows the data above 2 K for lower frequencies.
• Figure 3: (a) Temperature dependence of the specific heat at several magnetic fields, where the solid red line corresponds to the lattice contribution. (b) Magnetic entropy as a function of temperature at zero field. The dashed pink horizontal line indicates the expected entropy $R\ln2$ for an $s=1/2$ system. The inset shows the magnetic specific heat as a function of temperature. (c) Log–log plot of the magnetic specific heat as a function of temperature at various magnetic fields. In zero field, three distinct power-law regimes below the broad maximum are identified, as indicated in the legend, and separated by dashed vertical lines at $T^{*}=11.11$ K and $T_{\rm f}=0.32$ K.
• Figure 4: (a)Derivative of the ESR absorption spectra of Li$_{2}$Cu$_{2}$(MoO$_{4}$)$_{3}$ at selected temperatures where the Lorentzian line-shape fitting is shown by the solid red lines. (b) Temperature dependence of the ESR linewidth $\Delta H_{w}$ that shows a crossover in its power-law dependence across $T^{*}$ = 15.8 K (dashed vertical line). (c) Temperature dependence of the $g$ factor, exhibiting a non-monotonic variation.
• Figure 5: (a) Time evolution of normalized muon spin polarization in zero-field at three temperatures. (b) and (c) Temperature dependence of the muon spin relaxation rate of electronic origin ($\lambda_{\rm ZF}$) and nuclear origin ($\sigma_{\rm ZF}$). The dashed lines indicate the crossover in spin–spin correlations across $T^{*}$ = 15.8 K, highlighting distinct power-law regimes as discussed in the text. (d) Time evolution of normalized muon spin polarization in several longitudinal fields at 2 K. (e) Time dependence of muon spin asymmetry at a weak-transverse field $H_{\rm TF}$ = 46 Oe at several temperatures. (f) Temperature dependence of the transverse relaxation rate extracted from weak transverse-field measurements.