Abstract
In the quest for materials with unconventional quantum phases, the organic triangularlattice antiferromagnet κ(ET)_{2}Cu_{2}(CN)_{3} has been extensively discussed as a quantum spin liquid (QSL) candidate. The description of its low temperature properties has become, however, a particularly challenging task. Recently, an intriguing quantum critical behaviour was suggested from lowtemperature magnetic torque experiments. Here we highlight significant deviations of the experimental observations from a quantum critical scenario by performing a microscopic analysis of all anisotropic contributions, including Dzyaloshinskii–Moriya and multispin scalar chiral interactions. Instead, we show that disorderinduced spin defects provide a comprehensive explanation of the lowtemperature properties. These spins are attributed to valence bond defects that emerge spontaneously as the QSL enters a valencebond glass phase at low temperature. This theoretical treatment is applicable to a general class of frustrated magnetic systems and has important implications for the interpretation of magnetic torque, nuclear magnetic resonance, thermal transport and thermodynamic experiments.
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Introduction
Despite intensive efforts devoted to uncover the nature of the lowtemperature properties of the quantum spin liquid (QSL) candidate κ(ET)_{2}Cu_{2}(CN)_{3} (ET = bisethyelenedithioltetrathiafulvalene)^{1,2,3}, a few puzzles remain unresolved, such as the presence of an anomaly near a temperature T* = 6 K in a wide variety of experiments, including ^{13}C nuclear magnetic resonance (NMR)^{4}, muon spin resonance (μSR)^{5,6}, electron spin resonance (ESR)^{7}, specific heat^{8,9}, ultrasound attenuation^{10}, and thermal expansion^{9}. Various scenarios have been suggested to explain this anomaly such as spinchirality ordering^{11}, a spinonpairing transition^{12,13,14,15}, or formation of an exciton condensate^{16}. However, a comprehensive explanation of all aspects of the anomaly and its possible relation to a spin liquid phase is currently lacking.
Motivated by the unconventional scaling observed in magnetic torque measurements, Isono et al.^{17} suggested a lowtemperature quantum critical scenario for κ(ET)_{2}Cu_{2}(CN)_{3} (κCu). A finite torque magnitude τ reflects a variation of the energy E as a function of the orientation of the external magnetic field H (Fig. 1a)
where θ is the angle between H and a reference axis. Conventionally, τ is considered to probe the uniform magnetization M, through τ ∝ M × H. However, the torque is sensitive to any angular variation of the energy, which allows more general instabilities to be probed.
The experimentally observed torque response of κCu^{17} displays several characteristic features. As H is rotated in the crystallographic ac*plane, τ follows a sinusoidal angle dependence τ ∝ sin2(θ − θ_{0}), with an angle shift θ_{0} that increases substantially at low T and H, suggesting the emergence of a new contribution below T*. Indeed, the torque susceptibility χ_{τ} = τ/H^{2} diverges as a power law with the temperature exponent ω at low fields, χ_{τ} ~ T^{−ω}, and with the field exponent ζ at low temperatures, χ_{τ} ~ H^{−ζ}. In experiment, these exponents are approximately the same^{17} ω ≈ ζ ≈ 0.8. Consequently, the torque displays an apparent critical H/T scaling over several orders of magnitude. In this context, the remarkable behavior of κCu has been interpreted^{17} in terms of a fieldinduced quantum critical point with a diverging uniform spin susceptibility.
However, the presence of magnetic interactions beyond the conventional Heisenberg couplings allows more general instabilities of the spin liquid to be directly probed by torque measurements. For example, in κCu, spin–orbit coupling (SOC) leads to a staggered gtensor and a finite Dzyaloshinskii–Moriya (DM) interaction^{18}. These terms ensure torque contributions from the staggered magnetic susceptibility, which may diverge near an instability toward a canted Néel antiferromagnetic order^{19}. Furthermore, higher order multispin interactions^{20} couple the field to the scalar spin chirality, allowing τ to probe transitions to chiral phases^{11}.
In the following we show that through a microscopic calculation of τ, the quantum critical scenario cannot account for the experimental findings, even when all such instabilities are considered. Instead, motivated by the observation of an inhomogeneous NMR response below T*, we consider disorderinduced effects^{21,22}. In particular, we show that the NMR and τ experiments are consistent with proximity to a finite randomness largespin fixed point (LSFP)^{23,24}, where the diverging χ_{τ} reflects a small density of quasifree defect spins. By considering the specific temperature dependence of the response, the anomaly at the temperature T* can be interpreted as the onset of a valence bond glass (VBG), at which the resonating valence bonds of the QSL become randomly pinned. Finally, we discuss how this interpretation is consistent with a remarkable range of experiments.
Results
Effective Hamiltonian
We first consider possible anisotropic terms in the spin Hamiltonian that contribute to τ. These include anisotropic exchange interactions and gtensor anisotropy that arise from SOC, as well as higher order spinchiral terms. Ab initio estimates of each contribution are detailed in Supplementary Note 1.
The bilinear exchange interactions can be written in terms of the Heisenberg exchange J_{ij}, the Dzyaloshinskii–Moriya (DM) vector D_{ij}, and the pseudodipolar tensor Γ_{ij}
where S_{i} denotes the spin at site i. For κCu, the sites consist of molecular dimers, which are arranged on an anisotropic triangular lattice with two dimer sublattices, labeled A and B in Fig. 1b. We have previously estimated the bilinear interactions including SOC for κCu in ref. ^{18}, and found that the DMvectors are nearly parallel, but staggered (D_{ij} ≈ ±D), with magnitude D/J ~5%. It is therefore useful to transform the Hamiltonian^{25} by applying local staggered rotations of the spins \({\bf{S}}_i \to {\tilde{\bf{S}}}_i\) around D by a canting angle ±ϕ_{i}. For a purely staggered Dvector, this eliminates the DM interaction and also, in leading orders, the pseudodipolar tensor. As a result, the transformed bilinear interactions are
with \(\tilde J_{ij} \approx J_{ij}\) and J_{ij}/k_{B} ~ 230–270 K. Since the transformed bilinear interactions are rotationally invariant, they do not explicitly contribute to τ. Instead, the anisotropic effects are shifted to the transformed Zeeman term:
where \(\tilde {\Bbb G}_u\) and \(\tilde {\Bbb G}_{\mathrm{s}}\) are the uniform and staggered components of the gtensor in the rotated frame, and η_{i} = +1(−1) for i ∈ A(B) sublattice. Microscopic expressions for \(\tilde {\Bbb G}_{\mathrm{u}}\) and \(\tilde {\Bbb G}_{\mathrm{s}}\) are detailed in Supplementary Note 1.
For materials close to the Mott transition (such as κCu), it has been suggested that higher order ringexchange terms such as \({\cal{H}}_{(4)} = 1/S^2\mathop {\sum}\nolimits_{\langle ijkl\rangle } {\tilde K_{ijkl}\left( {{\tilde{\bf{S}}}_i \cdot {\tilde{\bf{S}}}_j} \right)\left( {{\tilde{\bf{S}}}_k \cdot {\tilde{\bf{S}}}_l} \right)}\), where \(\tilde K_{ijkl}\) is the ring exchange parameter in the rotated coordinates, may also play a significant role in stabilizing QSL states^{20,26,27}. As detailed in Supplementary Note 1, our ab initio estimates find \(\tilde K/\tilde J\sim 0.1\), which represents a significant contribution^{20,26}. Including SOC leads to anisotropic ringexchange terms, in principle. Fortunately, the leading order contributions to these terms are also eliminated by the \({\bf{S}} \to {\tilde{\bf{S}}}\) transformation, such that the transformed 4spin ring exchange terms do not explicitly contribute to τ at lowest order.
Finally, additional 3spin scalar chiral interactions^{20} may also arise when a finite magnetic flux penetrates the 2D organic layers. In the rotated coordinates, this provides the interaction
where n is the outofplane unit vector and the parameter j_{Φ} is proportional to the magnetic flux \(\frac{\hbar }{q}{\mathrm{\Phi }}\) enclosed by the triangular plaquette 〈ijk〉. Since only the outofplane component of the magnetic field couples to the scalar spin chirality, this term contributes explicitly to τ. We estimate μ_{B}j_{Φ}/k_{B} ~0.03K/T.
Bulk torque expressions near a quantum critical point
In the rotated frame, the bulk contribution to τ depends only on two terms in the Hamiltonian
For notational convenience, we introduce effective fields, \({\bf{H}}_{{\mathrm{eff}},{\mathrm{u}}/{\mathrm{s}}} = \tilde {\Bbb G}_{{\mathrm{u}}/{\mathrm{s}}}^{\mathrm{T}} \cdot {\bf{H}}\) and H_{eff,Φ} = j_{Φ}(n ⋅ H)n, which couple directly to the rotated spin variables
Since the transformed Hamiltonian of Eq. (3) is isotropic, the energy is minimized when the spin expectation values \(\left\langle {\mathop {\sum}\limits_i {{\tilde{\mathbf{S}}}_i} } \right\rangle\) and \(\left\langle {\mathop {\sum}\limits_i {\eta _i{\tilde{\mathbf{S}}}_i} } \right\rangle\) are parallel to the corresponding effective fields H_{eff,u} and H_{eff,s}, respectively. However, in order to compare with experimental results, it is more convenient to express τ in terms of the magnitude H and orientation θ of the original laboratory field H. Evaluating Eq. (6) shows that the H and θ dependences are separable (see Supplementary Note 2 for detailed derivation). The total bulk magnetic torque is the sum of the uniform, the staggered, and the chiral contributions τ_{B} = τ_{u} + τ_{s} + τ_{Φ} with
The fielddependence of each contribution is described by the susceptibilities
in terms of the constants \(\tilde \chi _{0,x}\) and the scaling exponents ζ_{x}, with x = {u, s, Φ}. Since the effective uniform and staggered fields are orthogonal (see Supplementary Note 1), ζ_{u} and ζ_{s} are generally different. Constant susceptibility (i.e. m ∝ H) corresponds to the limit ζ → 0. The angledependence of the torque results entirely from the anisotropy of the effective fields, and is described by
where h_{eff,x} = H_{eff,x}/H. These expressions define the torque up to the constants \(\tilde \chi _{0,x}\) and the scaling exponents ζ_{x}.
Bulk torque response of κCu
To compare with κCu, we computed τ for fields rotated in the ac*plane, and a variety of possible \(\tilde \chi _{0,x}\) and ζ_{x} using Eq. (9) together with ab initio values for the gtensors and DMinteractions (see Supplementary Table 1). Coordinates were defined similarly to the experiment of Isono et al.^{17}, in which θ is the angle between H and the long axis of an ET molecule, approximated by the principal axis of \(({\Bbb G}_u \cdot {\Bbb G}_u^{\mathrm{T}})\) close to the aaxis (see Fig. 1a).
The experimental response^{17} for temperatures T > T* is consistent with a material deep in a vanilla QSL or paramagnetic state that corresponds to ζ_{u} = 0, ζ_{s} = 0 and \(\tilde \chi _{\mathrm{\Phi }} = 0\), i.e., the spin susceptibilities are fieldindependent and there is no chiral response (Fig. 2a). In this case, the uniform contribution τ_{u} dominates, and the torque arises primarily from weak anisotropy of the uniform gtensor \(\tilde {\Bbb G}_{\mathrm{u}}\), providing a simple sin2(θ − θ_{0}) dependence with a fixed θ_{0} ≈ 90°, as defined in Fig. 2a.
For temperatures T < T*, the experiments of Isono et al.^{17} instead revealed a field dependent torque susceptibility that diverges at low temperatures as τ/H^{2} ∝ H^{−ζ}sin2(θ − θ_{0}), with ζ = 0.76–0.83, together with a notable angle shift of θ_{0}(H, T). Importantly, we find that these observations cannot be reconciled with divergence of any bulk susceptibility. For example, the case of a diverging uniform susceptibility \(\tilde \chi _{\mathrm{u}}(H)\) (Fig. 2b, ζ_{u} = 0.8), then τ_{u} would dominate at all temperatures, thus providing a nearly identical angle dependence at all T, with no shifting θ_{0}. This scenario would also imply proximity to a ferromagnetic instability, and therefore appears unlikely given the strong antiferromagnetic interactions^{20,28,29,30,31,32,33}.
In this context, divergence of the staggered susceptibility \(\tilde \chi _{\mathrm{s}}(H)\) appears more likely, which may occur near a critical point between a QSL and a Néel phase. We have previously highlighted this scenario^{18} as a possible explanation of the μSR response of κCu. However, we find this scenario is also inconsistent with the experimental torque. Figure 2c, d depicts the torque for an exponent ζ_{s} = 0.8, for different values of \(\tilde \chi _{\mathrm{s}}/\tilde \chi _{\mathrm{u}}\). For a diverging staggered susceptibility, τ_{s} shows a sawtooth like angle dependence instead of the experimentally observed sin2(θ − θ_{0}). This behavior stems from the strong angle dependence of the effective staggered field H_{eff,s}, which contributes to the denominator of Eq. (11). The strongly anisotropic staggered gtensor \(\tilde {\Bbb G}_{\mathrm{s}}\) leads to field orientations where H_{eff,s} vanishes linearly as H_{eff,s} ~ (θ − θ_{c}). For such orientations, a diverging susceptibility d^{2}E/dH_{eff,s}^{2} implies divergence of dτ/dθ ~ d^{2}E/dθ^{2}, thus giving a sawtooth torque.
Similar considerations apply to the case of a diverging chiral susceptibility, shown in Fig. 2e, f (with na and ζ_{Φ} = 0.8). In this case, the sawtooth form reflects the vanishing of \(\langle {\cal{H}}_{\chi ,{\mathrm{eff}}}\rangle\) for inplane fields H ⊥ n, which leads to similar divergences in dτ/dθ for ζ_{Φ} > 0. Taken together, the observed field and temperature dependent angle shift θ_{0} and absence of sawtooth features^{17} argue against any diverging bulk susceptibility in κCu.
Local spin defects in κCu
In order to account for the experimental torque observations at temperatures T < T*, we considered additional contributions from rare local spin moments induced by disorder, which may also give rise to diverging susceptibilities^{34,35,36,37}. We identify two relevant types of disorder for κCu.
The first type produces a random modulation of the magnetic interactions between dimers, and includes disorder in the conformations of the terminal ethylene groups of the ET molecules^{38,39}, the orientations of the cyanide anions located at crystallographic inversion centers^{40}, and/or the local charge distributions within each dimer^{41,42,43}. In onedimensional spin chains, introducing a finite randomness induces a random singlet ground state^{44,45}, in which the fluctuating singlets of the QSL are randomly pinned to form quasistatic singlets with varying length scales. At any given energy scale, a small fraction of spins remain unpaired, which leads to a diverging susceptibility. As recently emphasized by Kimchi et al.^{21,22}, similar behavior may occur in higher dimensional analogues of the random singlet state, which we refer to as VBG^{46,47} states. A possible scenario^{48} occurs in proximity to a valence bond solid order. In this case, random interactions induce complex patterns of domain walls that can host quasifree orphan spins at their intersections, as depicted in Fig. 3b.
The second type of disorder may result from defects in the anion layer^{7,49}, which may slightly dope the organic layer, producing nonmagnetic spin vacancies. As illustrated in Fig. 3a, these vacancies break singlet bonds, which may produce local moments if the host system is in a confined state (e.g., a valence bond solid)^{50,51,52,53,54,55}. Interestingly, Furukawa et al.^{56} recently showed that the introduction of anion defects via irradiation could suppress magnetic order in the less magnetically frustrated κ(ET)_{2}Cu[N(CN)_{2}]Cl salt.
Regardless of their origin, the lowenergy response of disorderinduced orphan spins has been the subject of many recent works^{23,24,34,36,48,57}, including those with specific reference to κCu^{58,59}. In the following, we start from the assumption that such local moments exist at low energies, and are randomly distributed in the material. We assume that the orphan spins interact via random longrange interactions that arise from bulk fluctuations, and consider their contribution to the magnetic torque.
Defect torque expressions
For simplicity, we consider the case of S = 1/2 impurity spins embedded in a nonordered background, and assume constant bulk susceptibilities \(\tilde \chi _{\mathrm{u}}\) and \(\tilde \chi _{\mathrm{s}}\). Due to fluctuations of the valence bonds around each impurity, the associated magnetic moment will be delocalized over some characteristic localization length. An important distinction from the bulk case is that all Fourier components of the induced defect magnetization density are parallel or antiparallel at lowest order^{52}. Therefore, the impurity induced magnetism can be described by a single effective impurity spin variable \({\tilde{\bf{S}}}_{\mathrm{I}}\), which represents both the impurity and the broad screening cloud surrounding the impurity. An external field couples to \({\tilde{\bf{S}}}_{\mathrm{I}}\) through the uniform and staggered moments induced near the impurity. These are given by \(\mathop {\sum}\limits_{i\prime \sim m} {\langle {\tilde{\bf{S}}}_{i\prime }\rangle = c_{\mathrm{u}}\langle {\tilde{\bf{S}}}_{{\mathrm{I}},m}\rangle }\) and \(\mathop {\sum}\limits_{i\prime \sim m} {\eta _{i\prime }\langle {\tilde{\bf{S}}}_{i\prime }\rangle = c_{\mathrm{s}}\langle {\tilde{\bf{S}}}_{{\mathrm{I}},m}\rangle }\) with \(\eta _{i\prime } = \pm 1\), respectively. Here, the summation runs over dimer sites i′ near the impurity m, and c_{u} and c_{s} are constants related to the Fourier transform of the induced spin density at k = 0 and k = (π, π), respectively. At lowest order, the scalar spin chirality is not coupled to the impurity effects. Therefore only the staggered and uniform moments respond to an external field through
where the effective impurity gtensor is \(\tilde {\Bbb G}_{\mathrm{I}} \approx (c_{\mathrm{u}}\tilde {\Bbb G}_{\mathrm{u}} + c_{\mathrm{s}}\tilde {\Bbb G}_{\mathrm{s}})\), and therefore differs from any of the bulk gtensors. The effective field felt by each impurity is given by
with the corresponding reduced field h_{eff,I} = H_{eff,I}/H. For simplicity, we take \(\tilde {\Bbb G}_{\mathrm{I}}\) to be the same for each impurity.
In the absence of residual interactions between S = 1/2 impurities, they would act as independent Curie spins, with the impurity contribution to the torque τ_{I}/H^{2} diverging as T^{−1} and H^{−1} at lowfield and temperature. However, the overlap of the screening clouds leads to random residual interactions \(\{ J_{mn}^{{\mathrm{e}}ff}\}\) between the randomly distributed impurity moments m and n. Following previous works^{23,24,34,36}, we expect the interacting impurities to become successively coupled into clusters as the energy is lowered below the interaction energy (e.g., \(k_{\mathrm{B}}T \lesssim {\mathrm{max}}\left {J_{mn}^{{\mathrm{e}}ff}} \right\)). The initial distribution of interactions may include both ferromagnetic (\(J_{mn}^{{\mathrm{e}}ff} < 0\)) and antiferromagnetic (\(J_{mn}^{{\mathrm{e}}ff} > 0\)) terms, as the sign of the coupling depends on the relative positions of the impurities. Provided this distribution is not too singular^{24}, the lowenergy response of the impurities is therefore expected to be described by a finite randomness largespin fixed point (LSFP)^{23,24} or the spinglass fixed point (SGFP)^{36}. In the vicinity of such a fixed point, we find that the total impurity torque can be approximated by a modified Brillouin function \({\cal{B}}(S,x)\) (see Supplementary Note 3)
where N_{C} = N_{0}Ω^{2κ} is the number of clusters, and \(S_{{\mathrm{eff}}}^{{\mathrm{avg}}} = S_0{\mathrm{\Omega }}^{  \kappa }\) is the average moment per cluster. Both depend on an effective energy scale defined by
The constants S_{0} and N_{0} define the average cluster spin and density at fixed T and H. The nonuniversal exponent κ is related to the fixed point distribution of energy couplings, and is sampledependent but constrained by \(1 \lesssim (2\kappa )^{  1} \le \infty\) (see Supplementary Note 3).
Evaluating the torque in the highfield limit \(k_{\mathrm{B}}T \ll \mu _{\mathrm{B}}S_{{\mathrm{eff}}}^{{\mathrm{avg}}}\left {{\bf{H}}_{{\mathrm{eff}},{\mathrm{I}}}} \right\), yields that τ_{I} is temperature independent
where \(\tilde \chi _{0,{\mathrm{I}}}^H\) is a constant that depends on the parameters {N_{0}, S_{0}, ζ_{I}} (see Supplementary Note 3). The angle dependence is given by
The susceptibility diverges as \(\tilde \chi _{\mathrm{I}}^H \propto H^{  \zeta _{\mathrm{I}}}\), with the exponent ζ_{I} = (1 + κ)^{−1} being restricted to the narrow range \(2/3 \lesssim \zeta _{\mathrm{I}} \le 1\) due to the constraints for the nonuniversal exponent κ. As a result, the low temperature divergence of τ_{I}/H^{2} with respect to field is always marginally weaker than free Curie spins.
In the lowfield limit \(\mu _{\mathrm{B}}S_{{\mathrm{eff}}}^{{\mathrm{avg}}}\left {{\bf{H}}_{{\mathrm{eff}},{\mathrm{I}}}} \right \ll k_{\mathrm{B}}T\), the torque susceptibility evaluates to
which is independent of field. The derivation of the temperatureindependent contribution \(\tilde \chi _{0,{\mathrm{I}}}^T\) is given in Supplementary Note 3. The angle dependence \(f_{\mathrm{I}}^T(\theta )\) is described by
As discussed in the next section, despite not following a perfect power law, the lowfield divergence of τ_{I}/H^{2} with respect to temperature may appear to follow a power law \(\sim T^{  \omega _{\mathrm{I}}}\) with \(2  \zeta _{\mathrm{I}}^{  1} \le \omega _{\mathrm{I}} \le 1\) at intermediate temperatures.
Total torque response of κCu
Here, we show that impurityrelated orphan spin contributions can reproduce all observed features of the torque response.
In Fig. 4a, b we show the low temperature angledependence of the torque for H rotated in the ac*plane. Including impurity contributions, the total experimental torque is τ = τ_{B} + τ_{I} according to Eqs. (9) and (17), respectively. We employed ab initio values (given in Supplementary Table 1), and an impurity exponent ζ_{I} = 0.8. We assume that the bulk susceptibility is not diverging (i.e., ζ_{u} = ζ_{s} = ζ_{Φ} = 0), and assume the same order of magnitude \(\tilde \chi _{{\mathrm{I}},0}^H = \tilde \chi _{0,{\mathrm{u}}} = \tilde \chi _{0,{\mathrm{s}}}\) for simplicity. In agreement with the measurements for κCu, the total torque has a sinusoidal sin2(θ − θ_{0}) shape. Owing to a nearly isotropic impurity gtensor \(\tilde {\Bbb G}_{\mathrm{I}}\), a diverging \(\tau _{\mathrm{I}}/H^2\) does not lead to a sawtooth appearance. However, provided the existence of a finite DM interaction, then \(\tilde {\Bbb G}_{\mathrm{I}}\) will generally differ from the bulk \(\tilde {\Bbb G}_{\mathrm{u}}\), leading to a shifted angle dependence of the impurity contribution. This explains the experimentally observed^{17} field and temperature dependent angle shift θ_{0}(H,T). As illustrated in Fig. 4c, the value of θ_{0} measures the ratio of impurity and bulk susceptibilities \(\tilde \chi _{\mathrm{I}}/\tilde \chi _{\mathrm{u}}\). The asymptotic value is further controlled by the ratio of the constants c_{s}/c_{u}, corresponding to the staggered (k = (π, π)) and uniform (k = 0) contributions of the induced spin density. For c_{s}/c_{u} = 1 (Fig. 4a) the impurity and bulk contributions have nearly identical angle dependence, since \(\tilde {\Bbb G}_{\mathrm{I}} \approx \tilde {\Bbb G}_{\mathrm{u}}\). This leads to θ_{0} ~−90°. In contrast, for c_{s}/c_{u} > 1 (Fig. 4b), the impurity contribution is shifted with respect to the bulk contribution (Fig. 4b), which leads to an overall shift of the total torque curve. The experimentally observed angle shift^{17} corresponds to c_{s}/c_{u} ≈ 5–10, implying the staggered moment induced around each defect exceeds the uniform defect moment.
To produce Fig. 4d–f, we performed a global fit of the divergent experimental^{17} torque at various T, H, yielding values for N_{0},S_{0} and ζ_{I} in Eq. (14). The fitted exponent (ζ_{I} = 0.79 ± 0.03) falls in the middle of the suggested range \(2/3 \ \lesssim \ \zeta _{\mathrm{I}} \le 1\), and is therefore consistent with a scenario with orphan spins. Figure 4d shows the resulting temperature dependence at various fields. Consistent with the experiment, the theoretical \(\tau _{\mathrm{I}}/H^2\sim T^{  \omega _{\mathrm{I}}}\) appears to follow a power law behavior. From Eq. (21) it follows that the exponent should fall in the narrow range \(2  \zeta _{\mathrm{I}}^{  1} \le \omega _{\mathrm{I}} \le 1\), illustrated by the blue region in Fig. 4d. For this reason, ω_{I} ≈ ζ_{I} (dashed line) at lowfield. Interestingly, the similar values of ζ_{I} and ω_{I} mean that τ_{I} will accidentally appear to display H/T scaling described by \(\tau _{\mathrm{I}}/H^2 \approx T^{  \zeta _{\mathrm{I}}}F[H/T]\), with
This behavior is displayed in Fig. 4f, where the theoretical \((\tau _{\mathrm{I}}/H^2)T^{\zeta _{\mathrm{I}}}\) is plotted against (H/T) for different temperatures. Deviations from scaling appear as a very small separation of the curves at lowfields. The apparent data collapse on a general scaling function is due to the disorder induced mechanism discussed in ref. ^{21}. Finally, in Fig. 4e we show the predicted evolution of the angle shift θ_{0}, for a fixed ratio of induced staggered to uniform defect moments c_{s}/c_{u} = 6. At high temperature, θ_{0} asymptotically approaches 90°, as thermal fluctuations suppress the diverging impurity contribution.
Taken together, the impurity contributions appear to explain all essential features of the experimental torque response below T*, including the reported range of exponents ζ_{exp} = 0.76 − 0.83, the sin2(θ − θ_{0}) dependence, the evolution of the angle shift θ_{0}, and the apparent H/T scaling of the magnitude of the torque.
Inhomogeneous NMR response
Within the same framework, we have also considered the inhomogeneous broadening of the NMR lines in κCu, which has previously been attributed to impurity spins^{4,60}. As the external field aligns the total spin of each cluster, the local defect moments \({\tilde{\bf{S}}}_{{\mathrm{I}},m}\) contributing to the cluster become static on the NMR timescale. The resulting static spin density around each defect is staggered, and decays with distance, leading to an inhomogeneous distribution of staggered Knight shifts within the sample. At first approximation, we assume that the resulting contribution to the NMR linewidth ν_{I} scales as the root mean squared average of \({\tilde{\bf{S}}}_{{\mathrm{I}},m}\), leading to (see Supplementary Note 3)
For the highfield limit \(k_{\mathrm{B}}T \ll \mu _{\mathrm{B}}S_{{\mathrm{eff}}}^{{\mathrm{avg}}}\left {{\bf{H}}_{{\mathrm{eff}},{\mathrm{I}}}} \right\), the local impurities are completely static, leading to constant linewidth (ν_{I} = ν_{0}). In the opposite limit \(k_{\mathrm{B}}T \gg \mu _{\mathrm{B}}S_{{\mathrm{eff}}}^{{\mathrm{avg}}}\left {{\bf{H}}_{{\mathrm{eff}},{\mathrm{I}}}} \right\), expanding the Brillouin function yields
suggesting ν_{I} is linear with respect to field.
In Fig. 5, we show the predicted relative linewidth ν_{I}/ν_{0} as a function of temperature and field employing S_{0}, N_{0} and ζ_{I} fit from the experimental τ. Comparing Eq. (25) to the experimental behavior, we find reasonable agreement. The experimental linewidth saturates (ref. ^{4}, Fig. 2) in precisely the same temperature range where τ/H^{2} also becomes temperature independent (ref. ^{17}, Fig. 3). For T > 1 K, the experimental NMR linewidth also grows approximately linear with H (ref. ^{4}, Fig. 4). This correspondence suggests that the NMR linewidth and torque response have a common impurity origin.
Discussion
In the presence of anisotropic interactions beyond the conventional Heisenberg terms, measurements of magnetic torque τ may provide significant insights into quantum magnets. We have shown that the effects of SOC allow the torque to probe uniform and staggered components of spin susceptibilities. In addition, higher order ringexchange processes, resulting from proximity to a metallic state, directly couple the magnetic field to the scalar spin chirality. For this reason, the torque provides a direct probe for a number of possible instabilities of the spinliquid state in κCu, including proximity to magnetically ordered (π, π) Néel or chiral ordered phases. However, we showed that the experimental response at low temperature is incompatible with the divergence of any bulk susceptibility, arguing against proximity to any such instability.
We therefore considered contributions to τ/H^{2} from rare impurity spins. After deriving approximate expressions for such contributions, we find that effects of local orphan spins can explain essentially all features of the torque experiments in κCu, including the specific angle dependence, observed exponents, and the apparent H/T scaling noted by Isono et al.^{17}. Related expressions also consistently describe the inhomogeneous NMR linewidth^{4} with the same parameters. These independent experiments both point to the presence of local moments in κCu, which we attribute to a disorderinduced VBG^{46,47} ground state analogous to the randomsinglet phase of 1D spin chains^{44,45}. While the small fraction of orphan spins result in a diverging torque response, the majority of lowenergy excitations will consist of local domain wall fluctuations, i.e., shifts of the frozen valence bond pattern^{47}. Such fluctuations can give rise to a linear specific heat^{58,61} C_{v} ~ T, in analogy with structural glasses^{62}. Since most excitations of the VBG are ultimately localized, they will contribute negligibly to thermal conductivity κ_{T} in the T → 0 limit. This feature may therefore explain the linear C_{v} but suppressed κ_{T} observed in κCu^{63,64}. Since disorder is expected to be small compared to the scale of interactions, the appearance of local moments also places some constraints on the ground state of the hypothetical disorderfree sample^{50,51,52,53,54,55}. In particular, proximity to valence bond solid order may aid the formation of a VBG^{48}.
We note, however, that an important discrepancy between the scaling expressions and the experimental response occurs at the T* = 6 K anomaly. At T*, both the experimental torque angle shift θ_{0} and the NMR linewidth ν increase more sharply than suggested by the scaling ansätze, so that strong contributions from orphan spins seem to emerge rapidly at T*. We, therefore, speculate that the anomalies observed in a wide range of experiments are connected to the freezing of valence bonds into the random VBG configuration. In principle, this may be driven by a number of effects. For example, it may reflect a thermal valence bond solid ordering transition in hypothetical disorder free samples, which evolves into a VBG transition with finite disorder. Alternately, external factors such as freezing of local charges or anion orientations may lead to a rapid enhancement of the effective disorder, driving the rapid formation of the VBG. In either case, the singlets would be free to fluctuate for T ≫ T*, thus forming an essentially homogeneous QSL at high temperatures. Strong inhomogeneity in the NMR relaxation (and μSR response^{6}) would only be expected below T*, which is consistent with ref. ^{4}. Moreover, the temperature T* itself should not be strongly sensitive to external fields, which is consistent with negligible field dependence^{9} of the thermal expansion near T*.
On this basis, we conclude that the torque investigations of κCu provide unique insight into the low temperature phase. The analysis presented in this work provide direct evidence of disorderrelated effects and may provide a roadmap for finally revealing the nature of the enigmatic T* anomaly as the formation of a VBG. This framework may also be generalized to other prominent organic QSL candidates like κH_{3}(CatEDTTTF)_{2}^{65} or β′EtMe_{3}Sb[Pd(dmit)_{2}]_{2}^{66}. It is also noteworthy that similar scaling of the susceptibility with exponent ζ = 2/3 is observed in the frustrated Kagome system Herbertsmithite^{67}, where disorder is thought to play a role. This highlights the rich interplay between disorder and frustration in quantum spin materials.
Data availability
Data is available from the corresponding author upon reasonable request.
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Acknowledgements
The authors acknowledge fruitful discussions with S. Brown, V. Dobrosavljevic, S. Hartmann, K. Kanoda, I. Kimchi, M. Lang, and P. Szirmai. The work was supported by the Deutsche Forschungsgemeinschaft (DFG) through project SFB/TRR49.
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Riedl, K., Valentí, R. & Winter, S.M. Critical spin liquid versus valencebond glass in a triangularlattice organic antiferromagnet. Nat Commun 10, 2561 (2019). https://doi.org/10.1038/s41467019106043
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DOI: https://doi.org/10.1038/s41467019106043
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