Geometrically frustrated magnets, characterized by incompatible magnetic interactions, exhibit strong quantum fluctuations and lack trivial order parameters. This susceptibility to perturbations makes them ideal candidates for hosting exotic ground states like quantum spin liquids (QSLs). QSLs are distinguished by the absence of long-range order even at absolute zero temperature, and the presence of exotic fractionalized excitations instead of conventional spin waves. The Kitaev model, proposed for S=1/2 spins on a honeycomb lattice, exemplifies a QSL with Majorana fermions. However, experimental realizations often deviate from the ideal theoretical models, with disorder playing a significant role. Several examples of frustrated magnets exhibiting quantum disordered ground states due to randomness exist, such as YbMgGaO₄ and Sr₂CuTe₁₋ₓWₓO₆. The interplay of quantum fluctuations and randomness can lead to novel states, including the random-singlet phase, characterized by the formation of singlets due to a random distribution of antiferromagnetic exchange interactions, resulting in unconventional scaling behaviors in susceptibility and specific heat. This paper focuses on the study of Li₄CuTeO₆ (LCTO), a frustrated antiferromagnet with inherent anti-site disorder between Li⁺ and Cu²⁺ ions, to investigate the emergence of a randomness-induced spin-liquid-like state.

The research on quantum spin liquids (QSLs) is extensive, with numerous candidate materials studied, including those based on triangular, kagome, and honeycomb lattices. Triangular lattice compounds like κ-(ET)₂Cu₂(CN)₃ and YbMgGaO₄, kagome compound ZnCu(OH)Cl₂, and hyperkagome compound PbCuTe₂O₅ have been investigated for their unusual properties. Significant efforts have focused on honeycomb lattice materials since Kitaev's model, though many, even those with 5d iridates or 4d ruthenates, exhibit long-range order at low temperatures. Materials like H₃LiIr₂O₆ and α-Ru₁₋ₓIrₓCl₃, created by substituting non-magnetic ions into ordered magnets, show no long-range order. Studies on systems with disorder, such as A₂LiIr₂O₆ and Lu₂Mo₂O₅N₂, reveal scaling behavior indicative of disorder's impact on QSL states. The random-singlet phase, theoretically proposed and observed in 1D organic and inorganic spin-chain compounds (e.g., BaCu₂(Si₁₋ₓGeₓ)₂O₇), represents another avenue of exploring disorder's influence. This phase is characterized by a power-law distribution of exchange energies and density of states, leading to distinctive scaling behavior in thermodynamic properties. Higher-dimensional materials with intrinsic disorder are also of growing interest.

Polycrystalline samples of Li₄CuTeO₆ were synthesized using a conventional solid-state reaction. Room-temperature powder X-ray diffraction confirmed phase purity, while neutron diffraction provided detailed structural information, including the determination of Li/Cu anti-site disorder. Rietveld refinement was used to analyze both X-ray and neutron diffraction data. Ab initio density functional theory (DFT) calculations, employing the LSDA+U scheme, were performed to determine the exchange couplings between Cu²⁺ moments. Exact diagonalization (ED) calculations were then conducted on a random spin model derived from the DFT results to calculate thermodynamic quantities (magnetic susceptibility, specific heat, and magnetic entropy) as a function of temperature and magnetic field. Experimental measurements included magnetic susceptibility using a SQUID magnetometer, electron spin resonance (ESR) spectroscopy, specific heat using a Physical Properties Measurement System (PPMS) and a dilution refrigerator, and muon spin relaxation (µSR) using the GPS spectrometer at the Paul Scherrer Institute. The lattice contribution was subtracted from the measured specific heat to obtain the magnetic specific heat. Data fitting and analysis were performed using appropriate models (e.g., Curie-Weiss law, damped Gaussian Kubo-Toyabe relaxation function, and power-law scaling).

Rietveld refinement of neutron diffraction data revealed that Li₄CuTeO₆ crystallizes in a monoclinic structure with partial occupancy of Cu²⁺ ions at two crystallographic sites (2d and 4g), indicating anti-site disorder. DFT calculations confirmed strong antiferromagnetic exchange interactions (J and J'). The high-temperature magnetic susceptibility followed a Curie-Weiss law with a large negative Curie-Weiss temperature (-154 K), consistent with strong antiferromagnetic interactions. However, no long-range magnetic order was observed down to 45 mK, as evidenced by the absence of any anomalies in the temperature dependence of magnetic susceptibility, specific heat, or muon spin relaxation. Specific heat data showed a broad maximum indicative of short-range spin correlations, and a low-temperature power-law behavior (Cmag/T ~ T⁻⁰·⁵). Analysis of the magnetization data revealed data collapse behavior (M⁰·⁸⁵ ~ f(H/T⁰·⁸⁵)), characteristic of a random-singlet phase. Zero-field muon spin relaxation (µSR) experiments showed no signs of static magnetic order down to 1.55 K, confirming a dynamic ground state. Transverse-field µSR measurements revealed a Knight shift that scaled as T⁻⁰·⁷⁵ at low temperatures, further supporting the random-singlets picture.

The experimental findings strongly suggest that Li₄CuTeO₆ exhibits a randomness-induced spin-liquid-like state. The absence of long-range order at low temperatures, despite the presence of strong antiferromagnetic interactions, is a hallmark of frustrated magnetism. The observed data collapse behavior in both magnetization and specific heat, along with the power-law dependencies and the dynamic ground state revealed by µSR, are all consistent with a random-singlet phase. The anti-site disorder within the material is the likely source of the randomness driving the formation of this unconventional ground state. The presence of an underlying spin-chain fragment structure with random connectivity might add complexities not fully captured by typical random-singlet models, warranting further investigation.

This study provides compelling evidence for a randomness-driven spin-liquid state in the frustrated antiferromagnet Li₄CuTeO₆. The combination of structural analysis, thermodynamic measurements, and µSR spectroscopy, supported by DFT and ED calculations, points to a dynamic ground state dominated by a network of random singlets arising from Li/Cu anti-site disorder. Future research could explore the interplay between the random-singlet behavior and the underlying spin-chain structure, and investigate the effects of tuning the degree of disorder on the system's properties.

The study is limited to polycrystalline samples, which may affect the interpretation of certain results, particularly those related to anisotropy. The analysis of the random-singlet state relies on scaling behavior and fits to theoretical models; direct observation of fractionalized excitations would provide stronger confirmation of the spin-liquid nature. The determination of exchange coupling constants relies on DFT calculations and subsequent fitting, so uncertainties may exist.