Unique magnetic transition process demonstrating the effectiveness of bond percolation theory in a quantum magnet

Abstract

Like the crystallization of water to ice, magnetic transition occurs at a critical temperature after the slowing down of dynamically fluctuating short-range correlated spins. Here, we report a unique type of magnetic transition characterized by a linear increase in the volume fraction of unconventional static short-range-ordered spin clusters, which triggered a transition into a long-range order at a threshold fraction perfectly matching the bond percolation theory in a new quantum antiferromagnet of pseudo-trigonal Cu₄(OH)₆Cl₂. Static short-range order appeared in its Kagome lattice plane below ca. 20 K from a pool of coexisting spin liquid, linearly increasing its fraction to 0.492(8), then all Kagome spins transitioned into a stable two-dimensional spin order at T_N = 5.5 K. Inspection on the magnetic interactions and quantum magnetism revealed an intrinsic link to the spin liquid material Herbertsmithite, ZnCu₃(OH)₆Cl₂. The unconventional static nature of the short-range order was inferred to be due to a pinning effect by the strongly correlated coexisting spin liquids. This work presents a unique magnetic system to demonstrate a complete bond percolation process toward the critical transition. Meanwhile, the unconventionally developed magnetic order in this chemically clean system should shed new light on spin-liquid physics.

Introduction

Ubiquitous phase transitions are exemplified paradigmatically in magnetic transitions, where the system at the transition point reaches a critical state characterized by long-range order. Like the crystallization process of water to ice, magnetic transition occurs at a critical temperature after the slowing down of dynamically fluctuating short-range correlated spin clusters. A universal theory for phase transition, known as percolation theory, describes the behavior of a network when nodes or links are added¹. It has been applied to materials science, biology, biochemistry, physical virology, ecology, etc., with an accelerated increasing speed. In materials science, the percolation theory has been recently used to explain anomalous high-temperature superconductivity² and quantum anomalous Hall effects^3,4,5. However, as to the paradigmatical magnetic transition, experimental evidence of the universal effectiveness of percolation theory has not been available. Here, we report a unique type of magnetic transition in a new quantum spin magnet of triclinic (pseudo-trigonal) Cu₄(OH)₆Cl₂ characterized by a linear increase in the fraction of an unconventional static short-range-ordered spin clusters, which triggered a transition into long-range order at a critical threshold fraction perfectly matching the bond percolation theory: 1. The short-range order is unconventional by being static like a long-range order. 2. It is a unique magnetic system demonstrating a complete percolation process. 3. The unconventional static nature of the short-range order has been inferred to arise from a pinning effect by coexisting spin liquids. The unconventionally developed quantum spin order in this chemically clean system thus should shed light on the hotly debated spin liquid physics.

The triclinic Cu₄(OH)₆Cl₂ is an unknown polymorphous structure of the orthorhombic Atacamite, monoclinic Clinoatacamite, and Botallackite^6,7,8,9. The Clinoatacamite has a deformed pyrochlore lattice for the magnetic Cu⁺², wherein selective substitution of the Cu⁺² in its triangular-lattice plane by nonmagnetic Zn led to the rigorously investigated spin liquid material Herbertsmithite, ZnCu₃(OH)₆Cl₂^10,11. The latter is viewed as an exemplary realization of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model on the Kagome lattice: a prototypical theoretical example of a quantum spin liquid. However, a considerable amount of site mixing between Zn/Cu inevitably existed in both its Kagome and triangular layers. It remains an open issue whether chemical disorders play a key role in the experimentally observed spin liquid behaviors. For example, some representative theoretical studies suggested that defects and bonding randomness directly induced spin liquid behaviors^12,13.

The magnetism of Clinoatacamite has been viewed as “a devil compound” for the mysterious nature of its intermediate T_N2 < T < T_N1 phase. Although magnetic susceptibility and specific heat measurements showed a very small anomaly at T_N1 = 18.1 K⁷, μSR clearly demonstrated a full magnetic order developing below T_N1¹⁴. All previous neutron diffraction experiments did not find any magnetic reflections for the T_N2 < T < T_N1 phase^15,16,17. Surprisingly, stacking faults were even hypothesized to explain the puzzling contradiction of neutron diffraction with μSR¹⁸. Now with the identification of the new triclinic structure and its unconventional magnetic transition, this puzzling issue has finally been solved. Furthermore, the unconventional transition and partial spin liquid state in the triclinic Cu₄(OH)₆Cl₂, which shows an interesting structural relation to the spin liquid material Herbertsmithite, sheds new light on spin-liquid physics.

Results and discussion

The crystal structure of triclinic Cu₄(OH)₆Cl₂

The X-ray diffraction pattern for the triclinic Cu₄(OH)₆Cl₂ (Supplementary Fig. 1) prepared and identified as described in “Methods” is quite like that of Clinoatacamite, as shown in Supplementary Fig. 2. The structural information is given in Table 1. The Cu²⁺ ions in this triclinic structure constitute nearly regular triangles with a standard deviation 0.36°–0.945° (Table 2) forming nearly ideal Kagome rings in the (111) lattice plane (Fig. 1a). The Cu²⁺ ions in the (111) lattice plane, which formed a Kagome spin order (Fig. 1b) at low temperatures as will be seen, are bonded to each other via a single Cu-O-Cu super-exchange path with bonding angles around 117°–123°. Meanwhile, the inter-Kagome-layer Cu²⁺ ions are bonded to the Kagome-layer Cu²⁺ via two Cu-O-Cu paths with bonding angles around 90°. This kind of double 90° bridge was previously reported in triangular lattice compounds LiNiO₂ and NaNiO₂, wherein the magnetic couplings for these double bridges were shown to be highly frustrated¹⁹. Similar properties also existed in Clinoatacamite (Fig. 1c), wherein its Kagome layers were weakly coupled²⁰. The present triclinic structure is pseudo-trigonal as shown by the close values of its lattice constants of a = 9.175(8) Å, b = 9.123(2) Å, c = 9.177(8) Å; α = 96.38(5)°, β = 96.05(2)°, γ = 96.23(5)° at 100 K. Le-Bail analysis clearly supported the triclinic model over the trigonal model or any related monoclinic models (Supplementary Figs. 3–5). It is worth noting that the present pseudo-trigonal structure agrees well with a theoretically predicted triclinic structure in the Material Project (mp-1201073). Neutron diffraction at 30 K (Supplementary Fig. 6) suggested the same triclinic structure with more precise positions for the H/D (Table 3). The pseudo-trigonal nature was stable over a wide temperature range from 300 K to 1 K (Fig. 2).

Table 1 Structure information for triclinic Cu₄(OH)₆Cl₂ at 100 K

Table 2 Bonding lengths, bonding angles and their standard deviations for the Cu²⁺ spins in the (111) Kagome lattice plane in triclinic Cu₄(OH)₆Cl₂ at 100 K

Fig. 1: Crystal structure of pseudo-trigonal Cu₄(OH)₆Cl₂ in comparison with Clinoatacamite.

a Structure of pseudo-trigonal Cu₄(OH)₆Cl₂ viewed along the [111] direction with the revealed magnetic structure at 1 K illustrated in (b). Cu ions are present in blue, O in red, H bonded to O in pearl white, and Cl in green. The (111) lattice plane forms a Kagome layer. The numbers on Cu correspond to the crystallographic sites as indicated in Table 1. The Cu ions in sites of 3A, 3B, 3C, 4A, 4B, and 4C form a Kagome layer with inter-Kagome-layer Cu1A, Cu2A, Cu2B and Cu2C. Only the Cu ions within the Kagome layers are bonded by a single Cu-O-Cu bonding. c Similar Kagome layer in the (101) lattice plane in Clinoatacamite (7), wherein only the atoms near the Kagome layer are shown. All triangles in the Kagome layer consist of one Cu1 and two Cu3 ions, with distances of Cu1-Cu3 = 3.4015(3) Å, 3.4204(3) Å; Cu3-Cu3 = 3.4037(5) Å.

Table 3 Structure information for triclinic Cu₄(OD)₆Cl₂ at 30 K

Fig. 2: Lattice constants between 1 K and 300 K.

The pseudo trigonal nature continued to low temperatures. Source data are provided as a Source Data file.

Unconventional magnetic transition demonstrating bond percolation

The magnetic DC susceptibility in Fig. 3a showed strong magnetic interactions featured by the high Curie-Weiss temperature of θ_CW = −170(1) K, close to that in Clinoatacamite⁷. The χT – T showed an antiferromagnetic transition feature at T_N = 5.5 K, slightly lower than the T_N2 ~ 6.0 K in Clinoatacamite. A distinct difference is that the triclinic Cu₄(OH)₆Cl₂ did not show any anomalous change at higher temperatures. The specific heat in Fig. 3b proved a transition at 5.5 K without high-temperature anomaly. The small total entropy release of ~ 0.2 Rln2 indicated the highly frustrated magnetism in this system. The field-independent ${C}_{{{{\rm{mag}}}}} \sim {T}^{2}$ behavior below T_N = 5.5 K, as shown in Fig. 3c, indicated a two-dimensional (2D) antiferromagnetic ground state, consistent with the theory for quasi-2D antiferromagnetic Heisenberg model²¹. Obviously, this 2D order was established in the (111) Kagome layers illustrated in Fig. 1a. Recently, we found that the intermediate T_N2 < T < T_N1 magnetic order in Clinoatacamite was built in its (101) lattice plane, which then was transformed by the inter-Kagome-layer couplings into a three-dimensional low-temperature spin order²². The 2D order in the present material, as will be shown later, is strikingly like the T_N2 < T < T_N1 order in Clinoatacamite. However, the Kagome order in the pseudo trigonal system is stable down to the lowest temperature, reflecting its much less lattice distortion.

Fig. 3: Magnetic transitions in triclinic Cu₄(OH)₆Cl₂.

a χT vs T curves in comparation with the monoclinic Clinoatacamite (χ: DC susceptibility) showing transition at T_N = 5.5 K. The inset plot is the 1/χ – T with a Curie-Weiss fitting showing nearly equivalent θ_CW and μ_eff for both Clinoatacamite and triclinic Cu₄(OH)₆Cl₂. b Specific heat and magnetic entropy in external fields of B = 0–6 T. c The magnetic specific heat with a C_mag ~ T² fitting, demonstrating a two-dimensional magnetic interaction in triclinic Cu₄(OH)₆Cl₂ below T_N = 5.5 K. In both susceptibility and specific heat, no change can be recognized in the high-temperature range in the triclinic compound as compared to small anomalies at T_N1 ~ 18.1 K in Clinoatacamite. Source data are provided as a Source Data file.

An unconventional magnetic transition below ca. 20 K was revealed by μSR, as presented in Fig. 4. In the whole temperature range, the zero-field μSR asymmetry spectra were well-fitted by

$$A\left(t\right) ={A}_{0}\left\{\left(1-{F}_{{{{\rm{MO}}}}}\right)\left[{F}_{\mu -{{{\rm{H}}}}}{G}_{\mu -{{{\rm{H}}}}}{{{{\rm{e}}}}}^{-\varLambda t}+\left(1-{F}_{\mu -{{{\rm{H}}}}}\right){G}_{{{{\rm{KT}}}}}{{{{\rm{e}}}}}^{-\lambda t}\right] \right. \\ \quad \left.+{F}_{{{{\rm{MO}}}}}\left[\left(1-p\right)\frac{1}{3}+p\left(\frac{1}{3}{{{{\rm{e}}}}}^{-{\lambda }_{{{{\rm{L}}}}}t}+\frac{2}{3}{{{{\rm{e}}}}}^{-{\lambda }_{{{{\rm{T}}}}}t}\right)\right]\right\},$$

(1)

wherein F_MO is the fraction of magnetically ordered phase, F_μ-H and G_μ-H the fraction and depolarization function due to μ-H bonding, G_KT the Kubo-Toyabe function due to nuclear fields, and F_BG = 0.0742(2) the background due to the silver sample holder (The muon depolarization functions G_μ-H and G_KT are explained in detail in “Methods”). As seen in the 29.3 K spectrum in Fig. 4a, 68.7(9) % of the implanted muons (μ⁺) were found to stop nearby OH⁻ acting like H⁺ to form water-like O-μ⁺-H bonding, wherein the nuclear spin of H caused the μ⁺ spin to rotate slowly. The remaining ca. 30% of muons stopped near Cl⁻, which was well fitted by the Kubo-Toyabe function ${G}_{{{{\rm{KT}}}}}\left(t\right)$. The ~7:3 asymmetry ratio of muons stopping at the OH and Cl sites in Cu₄(OH)₆Cl₂ implies that the muons were almost equally implanted near the OH⁻ and Cl⁻ anions. As described in “Methods”, the component ${F}_{{{{\rm{MO}}}}}\left[\left(1-p\right)\frac{1}{3}+p\left(\frac{1}{3}{{{{\rm{e}}}}}^{-{\lambda }_{{{{\rm{L}}}}}t}+\frac{2}{3}{{{{\rm{e}}}}}^{-{\lambda }_{{{{\rm{T}}}}}t}\right)\right]$ in Eq. (1). describes a magnetic order with two kinds of internal fields due to magnetic spins (the faster transverse depolarization term for one was lost at an early time). The absence of muon spin precession oscillations implied a broad distribution in the internal field probed by muons.

Fig. 4: A unique type of magnetic transition revealed by μSR in triclinic Cu₄(OH)₆Cl₂.

a Zero-field μSR spectroscopy. The solid lines are fitted as described in the text. b Unconventional linear growth of magnetic order fraction F_MO below T^* ~ 20 K and transition to full magnetic order at T_N = 5.5 K. c Muon spin depolarization rates for the zero-field μSR spectroscopy. The inset plot is an enlarged view. Source data are provided as a Source Data file.

The most striking result is that a static internal field, i.e., the magnetic order, began to develop below ca. 20 K. The fraction of the ordered fraction F_MO increased in a linear manner below T^* = 20 K, then abruptly jumped from 0.331(7) at 5.8 K to 1 at T_N = 5.5 K (Fig. 4b). The fraction right before 5.5 K can be estimated from the linear slope to be 0.338. The λ_L ~ 0.5 μs⁻¹ in Fig. 4c remained at 1.7 K, indicating persistent spin fluctuations for the magnetic order. From the decoupling behaviors of the longitudinal μSR (LF- μSR) (Supplementary Fig. 7), static internal fields produced by the magnetically ordered spins were estimated to be ca. 290 G and ca. 400 G, respectively, at 10.1 K and 1.7 K. As shown in Fig. 4b, the linear increase of the magnetically ordered fraction F_MO was realized with a simultaneous decrease of an equivalent amount in the F_μ-H. Meanwhile, the F_KT for the Cl-site muons remained unchanged. This means that the magnetic ordering occurred with only a part of the spins near the μ-H site. Referring to the crystal structure in Fig. 1a for the position of OH, we can see that the magnetic order from T^* = 20 K occurred in the Kagome layers, i.e., in the same layer as the fully ordered low-temperature phase below T_N = 5.5 K. Therefore, a fraction of 0.492(8) of the Kagome-layer spins, which was estimated from the ratio of F_MO/F_μ-H at 5.5 K, were magnetically ordered right before the critical point T_N = 5.5 K. It should be noted that since the magnetic order below T_N is a two-dimensional one for spins within the (111) Kagome lattice plane as demonstrated by the magnetic specific heat, the inter-Kagome-layer ${{{{\rm{Cu}}}}}^{2+}$ spins should remain paramagnetic with their thermal fluctuation too fast to fall outside of the μSR time window. A similar case occurred in the intermediate T_N2 < T < T_N1 phase in Clinoatacamite²², wherein the inter-Kagome spins remained paramagnetic and unseen by the μSR, resulting in a seemingly 100% long-range order¹⁴.

The unconventional magnetic order below T^* = 20 K at first sight looked like a long-range order since a static internal magnetic field was clearly observed without even a slight fluctuation (λ_L = 0 as shown in the inset plot in Fig. 4c). Short-range magnetic order usually appeared in μSR as dynamically fluctuating, as exemplified in various magnetic systems (e.g., ^23,24,25,26). This is the magnetic version of the crystallization process during the freezing process of gas-liquid-solid, wherein the liquid has short-range ordered atomic positions. An exceptional case occurred in a spin liquid candidate of triangular-lattice compound NiGa₂S₄²⁷, wherein an inhomogeneous static internal field resulting from short-range order was observed²⁸. As will be discussed, both the present system and NiGa₂S₄ have a common property of order coexisting with spin liquid phases, by which the unusual static short-range order can be consistently explained.

The neutron diffraction patterns demonstrated a long-range order for the low-temperature phase, with magnetic reflections indexed in Fig. 5 at 1 K. However, no trace of any magnetic reflection was recognized at 9 K. Considering the fact that Clinoatacamite showed clear magnetic reflection peaks at 9 K with spin moments of 0.245(15)−0.39(3) μ_B obtained for the Cu1 and Cu3-site spins in its Kagome plane using the same low-background beamline²², the possibility of a long-range order in the $T < {T}^{*}=20\,{{{\rm{K}}}}$ phase is very low, or at least much lower than 0.2 μ_B. In combination with the percolation process as viewed in μSR, the intermediate phase in 5.5 K < T < 20 K should be short-range ordered.

Fig. 5: Long range order observed in triclinic Cu₄(OD)₆Cl₂.

Magnetic reflections, as indexed in the plot, were observed at 1 K. No trace of any magnetic reflection was observed at 9 K. Contrary to previous neutron diffraction study, magnetic reflections have been observed by us in Clinoatacamite at 9 K (22). Considering the similar XRD patterns of Clinoatacamite and triclinic Cu₄(OH)₆Cl₂, which are almost undistinguishable at low diffraction angles, as well as the revealed magnetic properties in this work, we have sufficient reason to suspect that the previous neutron study (15) intended for the intermediate T_N2 < T < T_N1 phase of Clinoatacamite might be actually done on a wrongly identified triclinic Cu₄(OD)₆Cl₂. Source data are provided as a Source Data file.

Therefore, the unconventional magnetic transition below T^* = 20 K can be understood as a linear increase in the volume fraction of short-range-ordered spin clusters within the Kagome layers. When 49.2(8)% of the Kagome-layer spins were ordered short-range, all Kagome spins transitioned to a long-range order at T_N = 5.5 K. This reminds us of the universal percolation theory for all kinds of phase transitions, wherein the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters at a critical fraction of addition. It has been predicted that the critical threshold for 2D square lattice is exactly $\frac{1}{2}$ for bond percolation²⁹. For site percolation, the critical threshold was numerically computed to be 0.59274621(13)³⁰. We can see that our experimentally observed critical threshold of 0.492(8) excellently matches the theoretically predicted bond percolation threshold for the 2D square lattice, which has the same neighboring number 4 as in the Kagome lattice. Therefore, the unconventionally developed magnetic transition in triclinic Cu₄(OH)₆Cl₂ presents a unique case to exemplify a complete percolation process toward the critical transition in magnetic systems.

We would like to compare our results to theoretical studies for magnetic transitions, but we only found references for site percolation. On the quantum phase transition of the randomly diluted Heisenberg antiferromagnet on a square lattice, its ground state was investigated using the quantum Monte Carlo method, and the critical concentration of magnetic sites was found independent of the spin size S and equal to the two-dimensional percolation threshold ~0.593 for site percolation^31,32. In the present spin system, since all ${{{{\rm{Cu}}}}}^{2+}$ spins are magnetically coupled via the Cu-O-Cu super-exchange interaction, this makes a contrasting difference with the magnetic vacancies (nonmagnetic impurities) in refs. ^31,32, bond percolation apparently suits the present system than the site percolation model.

$${{{\rm{The}}}}\; {{{\rm{Coexisting}}}}\; {{{\rm{Disorder}}}}\; {{{\rm{in}}}}\quad{T}_{{\rm{N}}}\, < \, T \, < \, {T}^{*}=20\,{{{\rm{K}}}}$$

We can further explore the coexisting state of the disordered phase in T_N < T < 20 K, as well as the reason for the unconventional static short-range-ordered spin clusters. First, we found that the remanent magnetic moment provided a clue. As shown in Fig. 6, the transition temperature T_N, which is characterized by the peak in dMT/dT in Fig. 6a, was slightly enhanced by increasing the external magnetic field in consistency with the specific data in Fig. 3b. Notable hysteresis loops in M-H were observed for temperatures up to T > T_N (Fig. 6b). Furthermore, time-dependent remanent magnetization appeared, showing a time relaxation behavior obeying ${M}_{r}={M}_{0}(1-\alpha \log \left(t-{t}_{0}\right))$ (Fig. 6c). This kind of slow time relaxation was well known in spin glass systems. However, as demonstrated by the μSR experiments, the correlated spin clusters are far from spin glass. As plotted in Fig. 6d, the remanent magnetic moment persisted up to 30 K, and the time constant α showed a two-peak temperature dependence with the peaks ending coincided with the transition temperatures T^* and T_N. These results suggest that an external field can induce ferromagnetic moments that remain for a long time after removing the field at temperatures up to 30 K, which is well above the T^* = 20 K. The retaining property of the field-induced moments was inferred to be due to a pinning effect by coexisting spin liquid, as will be discussed hereafter.

Fig. 6: Magnetization and relaxation in triclinic Cu₄(OH)₆Cl₂.

a Temperature dependence of the magnetic moments measured in various fields upon cooling. b Magnetization M vs H at typical temperatures showing notable hysteresis loop below 7 K. c An example of remanent magnetization M_r at 12 K after removing a field of 400 Oe. The remanent magnetization showed a slow time relaxation following ${M}_{r}={M}_{0}(1-\alpha \log \left(t-{t}_{0}\right))$, wherein t₀ was used to correct the time. d Change of the initial M_r at t ~ 0 and the constant α. The dashed lines help to see the correlation of the α to the transition temperatures T_N and T^*. Source data are provided as a Source Data file.

Inspection of possible magnetic structures gave further clues to the coexisting disorder. On the condition that the T < T_N = 5.5 K order was formed by the Kagome-layer spins, possible magnetic structures can be examined from the observed magnetic reflections. For the k = (000) magnetic reflections for Kagome-layer spins, there are 32 collinear spin models. We fitted all these models and obtained similar fitting qualities for eight spin structures, as defined in Supplementary Table 1, with spin moments around 0.38μ_B preferentially along the a-axis direction. Basically, they can be grouped into two types of models, as illustrated in Fig. 7. In the type represented by Model 1 in Fig. 7a, there are five other spin structures with similar goodness of fitting, as illustrated in Supplementary Figs. 8 and 9. The only difference is which pair of triangles in the Kagome ring have the ↑↑↑, ↓↓↓ spin arrangement. In the type illustrated as Model 4 in Fig. 7b, there is another similar spin structure (Model 3 in Supplementary Fig. 8), wherein the only difference is that which Cu-O-Cu bond of the ↑↑↓ in each triangle is ferromagnetically coupled as ↑↑. Since the triclinic crystal structure is pseudo-trigonal with a minimal difference in the three crystal directions, it is practically impossible to experimentally determine which one is the correct spin model. However, since theoretical calculations have been proven effective for the present crystal structure, as denoted in “Methods”, we further performed theoretical calculations of the total energy for these possible candidate magnetic structures. As shown in Supplementary Fig. 10, DFT calculation has shown a remarkably stable total energy for Model 4 distinct from all seven other candidates. As a matter of fact, this calculation-supported magnetic structure can be perfectly explained by the Cu-O-Cu super-exchange bonding. As shown in Table 2, the underlined Cu-O-Cu bonding, which has the smallest angle in each triangle, is ferromagnetically coupled in the Model 4 structure, while all other ${{{{\rm{Cu}}}}}^{2+}$ ions are antiferromagnetically coupled. It turned out that the low-temperature magnetic structure is like that in the 6.0 K = T_N2 < T < T_N1 = 18.1 K intermediate phase in Clinoatacamite with similar bigness of spin moments. In the latter, we have found that a unique up-up-down—down-down-up spin order for the neighboring triangles in the Kagome ring with spin moments of 0.245(15) μ_B and 0.39(3) μ_B for the Cu1 and Cu3-site ions, which was formed in the T_N2 < T < T_N1 phase in its (101) lattice plane²². Similarly, in the intermediate phase in Clinoatacamite, two Cu spins in each triangle bonded by the smaller angle parallelly aligned (up-up) while those bonded by larger angles antiparallelly aligned (up-down).

Fig. 7: Candidate magnetic structures in triclinic Cu₄(OD)₆Cl₂ below T_N = 5.5 K.

a Model 1 with χ² = 1.0468, M = 0.378(6) μ_B (M_x: 0.375(4) μ_B, M_y: 0.07(17) μ_B, M_z: 0.1(2) μ_B). Of the six triangles in one Kagome ring, two identical triangles on the opposite sides consisting of the 3B, 3 C, and 4B Cu ions are ordered as ↑↑↑, ↓↓↓, separated by two alternative ↑↑↓, ↓↓↑ triangles on each side. b Model 4 with fitting χ² = 1.0473, M = 0.38(15) μ_B (M_x: 0.37(12) μ_B, M_y: 0.05(24) μ_B, M_z: 0.2(3) μ_B). All six triangles in the Kagome ring are alternatively ordered as ↑↑↓, ↓↓↑. The oval loops indicate the possible formation of spin singlets in the intermediate T_N < T < 20 K phase, as discussed in the text. As shown in Supplementary Fig. 8, there is another Model 3 similar with Model 4, wherein the only difference lies in which Cu-O-Cu bond in each ↑↑↓ triangle is ferromagnetically coupled as ↑↑, and there are five other possible spin structures similar to Model 1, wherein different pairs of triangles have the ↑↑↑ ↓↓↓ spin arrangement. Source data are provided as a Source Data file.

The manner of the growth of short-range-ordered spin clusters and the remanent magnetism can be well explained in relation to the magnetic structure. As shown in Fig. 4b, the T < T_N long-range 2D Kagome order was brought out because of percolation of short-range-ordered spin clusters. The spin clusters grow continuously by decreasing the disordered spins. The T < T_N model in Fig. 7b can better explain the manner of the evolution of the long-range magnetic order from the short-range correlated spin clusters. As indicated by the oval loops in Fig. 7b, spin singlets of VBS (valence-bond-solid) could be considered for the coexisting disordered spins in the intermediate T_N < T < 20 K phase. This picture agrees well with the μSR results. The simultaneous increase of magnetically ordered fraction F_MO with the decrease of the F_μ-H showed that while a part of Kagome spins near the μ-H site were ordered; other disordered Kagome spins did not produce a dynamical field at the μ-H site, implying that their moments were null in consistency to the formation of spin singlets. Therefore, the intermediate phase can be pictured as spin-ordered clusters floating in a spin liquid.

One would notice that the situation in Fig. 7b strikingly resembles the VBS (valence-bond-solid) + Nėel model in Kagome layers initially proposed by Lee et al. for Clinoatacamite¹⁵. They proposed a full VBS spin liquid state for Clinoatacamite’s intermediate T_N2 < T < T_N1 phase in its Kagome layers, followed by coexisting VBS and colinear antiferromagnetic order below T_N2. These states were not realized in clinoatacamite, as was denied by the μSR experiment for the intermediate T_N2 < T < T_N1 phase¹⁴. Our recent neutron diffraction on Clinoatacamite revealed a Kagome magnetic structure in the intermediate phase and investigations using single crystalline Clinoatacamite indicated a three-dimensional spin structure consistent with ref. ¹⁶ for the T < T_N2 phase²². However, the proposed coexisting VBS + Nėel model for the T < T_N2 phase of Clinoatacamite can be used to account for the T < T^* intermediate phase in the present pseudo-trigonal Cu₄(OH)₆Cl₂.

The remanent magnetism and the slow time relaxation can be consistently explained by the coupling between the field-enhanced spin clusters with the surrounding spin liquid, which can be understood through the spin liquid picture illustrated by the oval loops in Fig. 7b. Since the spin clusters are strongly coupled to the coexisting spin liquids, the recovering to the zero-field state would be pinned by the collectively correlated spin singlets. The unconventional static property of the short-range order can be readily explained by a similar pinning effect. Static short-range order was also observed in the triangular-lattice compound NiGa₂S₄²⁸, wherein they tentatively attributed a missing 20% of the full μSR asymmetry to an unlikely texturization of the sample. The static short-range order in NiGa₂S₄ can be more reasonably explained by a pinning effect due to coexisting spin liquid.

It is interesting to compare the unique magnetic transition to the development of disordered static magnetism and spin-liquid-like behaviors in YCu₃(OH)₆O_xCl_3-x³³, wherein no evidence for long-range ordering were observed but distinct ground states were found: disordered static magnetism developed in the x = 0 compound below T < 14 K, and stabilization of a quantum spin liquid in the x = 1/3 one was suggested. We can see that the x = 0 YCu₃(OH)₆Cl₃ is a structurally complete Kagome magnet (without the inter-Kagome-layer Cu) but may contain site disorder of O and Cl. A comparison may raise concerns about the influence of randomness on spin liquid behaviors^12,13.

In summary, we have discovered an unconventional static short-range order stabilized by a pinning effect due to coexisting spin liquids in a rediscovered structure of Cu₄(OH)₆Cl₂. The present work provides a unique real magnetic system for the first time to demonstrate the universal effectiveness of percolation theory. The geometric frustration and the unconventional stable short-range order pinned by the coexisting spin liquids have caused a gradually evolved magnetic transition to perfectly demonstrate a complete percolation process. The uncovered properties and growth of spin order from the spin liquid in this chemically clean quantum magnet would shed new light on the hotly debated spin liquid physics. In particular, the similar and yet strikingly different magnetism in the triclinic and monoclinic Cu₄(OH)₆Cl₂, and their close relation to the trigonal paratacamite, wherein only minimal structural difference can be recognized, provide a rare case to study the critical role of crystal structure on the geometric frustration and spin liquid magnetism.

Methods

Material

Polycrystalline samples of triclinic-structure Cu₄(OH)₆Cl₂ and Cu₄(OD)₆Cl₂ were obtained by heating stoichiometric amounts of high-purity powders of thoroughly dried CuCl₂ and NaHCO₃ in sealed Teflon vessel at 180–200 °C for 3–24 h. Careful attention should be paid to exclude any moisture in the starting materials to avoid partial oxidation. For this synthesis, NaHCO₃ and NaDCO₃ were prepared beforehand by continuously pumping CO₂ gas into Na₂CO₃ dissolved in H₂O and D₂O, respectively. After the synthesis, the impurity NaCl in the product was thoroughly washed out by pure water.

The sample synthesis dated back to our early synthesis of Clinoatacamite in aqueous solution, during which we noted that some samples showed magnetism quite similar to Clinoatacamite but without the slight magnetic susceptibility anomaly at T_N1 = 18.1 K. Those samples also showed contrastingly gradual μSR changes below ca. 20 K. We once attributed them to defective Clinoatacamite. Then we realized that they belonged to a different crystal structure but wrongly suspected it as a metastable structure of rhombohedral Paratacamite³⁴ since its X-ray diffraction pattern showed a seemingly high symmetry. Finally, with successful purification in single-phase synthesis using the solid reaction synthesis as described, we were able to identify this anomalous compound to be a triclinic structure. To our surprise, this crystal structure has already been theoretically predicted in the Material Project (mp-1201073). It is also similar with the crystal structure previously reported for a natural mineral Cu_1.978Ni_0.028(OH)₃Cl_1.015³⁵, which, however, was (wrongly) discredited as twined Clinoatacamite.

Measurements and calculation

Synchrotron radiation X-ray diffraction experiments were carried out at SPring-8 using a wavelength of λ = 0.499843687 Å, Δλ = 0.000000924. Magnetization measurement was performed using the Quantum Design MPMS3 SQUID magnetometer. The remanent field in the equipment was reduced using the reset magnet facility. μSR experiments utilizing polarized positive surface muon beams with conventional He gas flow cryostats were conducted both at the DC muon facility of TRIUMF, Vancouver, and the pulse muon facility of RIKEN-RAL, Oxford. The heat capacity was measured using an adiabatic heat-pulse method with a ³He cryostat and dilution refrigerator using ~0.5 g of the polycrystalline sample mixed with gold powder. Neutron diffraction experiments were carried out at the SuperHRPD beamline, J-PARC, using polycrystalline Cu₄(OD)₆Cl₂. The time-of-flight data from the pulsed beamline were analyzed using the FULLPROF-suite software based on Rietveld refinement³⁶, assisted by the representation analysis program SARAh³⁷.

Theoretical calculations were performed to verify the candidate magnetic structures by using the Vienna ab initio Simulation Package (VASP) to apply Density Functional Theory (DFT) along with the projector-augmented-wave formalism (PAW)^38,39,40,41. The electronic wave functions were expanded as plane waves with an energy cutoff of 550 eV, and the convergence criterion was set to 10⁻⁷eV. The electron exchange and correlation (XC) were used with the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof (PBE). The force convergence criterion for the structural relaxation was set to 0.001 eV/Å. Additionally, a 5 × 5 × 5 Monkhorst–Pack k-point mesh was used for the calculations.

μSR as a probe for magnetic order and spin fluctuations

Since muon has a large gyromagnetic ratio (γ_μ = 2π × 135.5 MHz/T), it is a sensitive magnetic probe. It is also an optimum probe for zero-field magnetism since it does not rely on an external magnetic field. The positive muon beam, polarized to the beam direction, is stopped in the specimen and depolarized by its internal fields produced by magnetic spins or nuclear spins. The time histograms of muon decay positrons are recorded by forward (F) and backward (B) counters as a function of resident time for each μ⁺ within the specimen. A positron is emitted preferentially toward the muon spin direction. Therefore, asymmetry A(t) = [F(t) − B(t)]/[F(t) + B(t)] of the two histograms (after correction for solid-angle effects) reflects the time evolution of muon spin polarization, as well as the time evolution and space distribution of the internal fields in the specimen. In a randomly oriented polycrystal specimen that is magnetically ordered, the longitudinal polarization function P_Z(t) in μSR is expected to be: P_z(t) = $\frac{1}{3}{e}^{-{\lambda }_{{{{\rm{L}}}}}t}+\frac{2}{3}{e}^{-{\lambda }_{{{{\rm{T}}}}}t}\cos ({\gamma }_{\mu } \, < \, {B}_{{loc}}\, > \,t)$, wherein, λ_L and λ_T respectively denote the spin–lattice and spin–spin relaxation rates, γ_μ is the muon gyromagnetic ratio (γ_μ = 2π × 135.5 MHz/T), and < B_loc > stands for the mean value of the local field at the muon site. With a typical time window of 10⁻⁵ to 10⁻¹⁰s, μSR is frequently used to investigate spin fluctuations.

In the present material, some implanted muons (μ⁺) were found to stop nearby OH⁻ acting like H⁺ to form water-like O-μ⁺-H bonding, wherein the nuclear spin of H caused the μ⁺ spin to rotate slowly. This kind of “hydrogen bonding” has been observed in other compounds containing F or OH^42,43,44,45, wherein the μSR polarization function was well described by a “two spin model”, ${G}_{\mu -{{{\rm{H}}}}}\left(t\right)=\frac{1}{6}+\frac{1}{6}\cos (\omega t)+\frac{1}{3}\cos (\frac{1}{2}\omega t)+\frac{1}{3}\cos (\frac{3}{2}\omega t)$, with ω = ћγ_μγ_Ν/r³, γ_μ and γ_Ν being the gyromagnetic ratios of μ⁺ and H⁺ nuclear spins, respectively, and r the distance between μ⁺ and H⁺. A r = 1.538(2) Å was obtained. The remaining muons stopped near Cl⁻, which was well fitted by the Kubo-Toyabe function ${G}_{{{{\rm{KT}}}}}(t)=\frac{1}{3}+\frac{2}{3}(1-{(\sigma t)}^{2}{e}^{-\frac{1}{2}{(\sigma t)}^{2}})$⁴⁶$,$ wherein σ = 0.282(5) μs⁻¹ was obtained, reflecting the average nuclear dipolar field at this muon site.