Abstract
The complex antiferromagnetic orders observed in the honeycomb iridates are a doubleedged sword in the search for a quantum spinliquid: both attesting that the magnetic interactions provide many of the necessary ingredients, while simultaneously impeding access. Focus has naturally been drawn to the unusual magnetic orders that hint at the underlying spin correlations. However, the study of any particular broken symmetry state generally provides little clue about the possibility of other nearby ground states. Here we use magnetic fields approaching 100 Tesla to reveal the extent of the spin correlations in γlithium iridate. We find that a small component of field along the magnetic easyaxis melts longrange order, revealing a bistable, strongly correlated spin state. Far from the usual destruction of antiferromagnetism via spin polarization, the highfield state possesses only a small fraction of the total iridium moment, without evidence for longrange order up to the highest attainable magnetic fields.
Introduction
Spin systems with highly anisotropic exchange interactions have recently been proposed to host quantum spinliquid states^{1}. It has been suggested that the extreme exchange anisotropy required to achieve such states can be mediated by the strong spin–orbit interactions of transitionmetal ions situated in undistorted and edgesharing oxygen octahedra^{2}. Both the twodimensional (α) and threedimensional (3D) (β and γ) polymorphs of the honeycomb iridates (A_{2}IrO_{3}, A = Na, Li) closely fulfill these geometric requirements and display intriguing lowfield magnetic properties that indirectly indicate the presence of anisotropic exchange interactions between the iridium spins^{3,4,5}. Kitaev’s spinliquid ground state, however, results from the presence of exclusively bondspecific Ising exchange interactions^{1}, which have not yet been verified experimentally. To date, all honeycomb iridates deviate sufficiently from this ideal, such that they transition to longrange magnetic order at finite temperature^{6, 7} drawing attention to the complex magnetic structures of their ground states^{8,9,10,11,12,13}.
γlithium iridate features an incommensurate magnetic structure with noncoplanar and counterrotating moments below T _{N} = 38 K^{6}. The magnetic anisotropy within the ordered phase was extensively characterized in our previous study^{4}. With increasing magnetic field below T _{N}, the magnetic torque τ, divided by the applied field H increases linearly up to an angledependent field H ^{*}, defining the phase boundary of the lowfield ordered state^{4}. Importantly, the sharp feature at H ^{*} is not accompanied by full saturation of the iridium moment^{4}. Instead, the magnetic moment at H ^{*} is only ~0.1 μ _{B} ^{4}. The lack of a fully saturated moment at H ^{*} implies either the onset of another magnetically ordered phase above H ^{*}, or alternatively a transition into a paramagnetic state lacking longrange order. The latter implies that the spin correlations are controlled by exchange interactions much stronger than the applied magnetic field. Just as the strange metallic state near the quantum critical point in the holedoped cuprates is revealed once superconductivity is suppressed with magnetic field^{14}, we use high magnetic fields to destroy the antiferromagnetic order and expose the spin correlations in γlithium iridate.
In the following, we show that above H ^{*} γLi_{2}IrO_{3} is characterized by a highly anisotropic nonlinear magnetic response with a small net magnetic moment up to the highest attainable magnetic fields. The absence of a sharp boundary with the hightemperature paramagnetic phase suggests that the highfield state does not break additional symmetries and does not exhibit longrange order. Furthermore, a hysteresisfree magnetic anisotropy that can be tied to the crystallographic directions evolves continuously with field out of the ordered state, uncharacteristic of glassy behavior. With these features in mind, and in light of recent studies that find dynamic evidence for liquid behavior in related transitionmetal honeycomb structures^{15, 16}, we will refer to the observed highfield state as a spinfluid.
Results
Experimental technique
We use torque magnetometry to study the highfield state of single crystal γlithium iridate. Specifically, we examine the nonlinear response at high magnetic fields. The anisotropy of the magnetic susceptibility α _{ ij } = χ _{ i } − χ _{ j } in the linear response regime leads to a smooth sin 2θ angle dependence with the torque vanishing when field is applied along the high symmetry directions (Supplementary Information of ref. ^{4}). Deviations from this smooth angle dependence provide a direct probe of magnetic correlations. The extreme magnetic anisotropy of γlithium iridate necessitated that both smaller volume samples and stiffer cantilevers were employed for the pulsed highfield measurements compared to our prior lowfield (DC magnet) measurements^{4}. To this end, we utilized focused ion beam lithography to cut and precisely align samples on Seiko PRC120 piezoresistive levers. This had the added benefit of reducing the lever deflection with field and systematic angle offsets, particularly when field is aligned close to the magnetically hardaxis.
Magnetic anisotropy
Figure 1a, b show τ/H for field rotation in two planes that include the caxis: the bcplane \(\left( {{\it{\varphi }}{{ = 0}^ \circ }} \right)\) and a plane that is perpendicular to one of the honeycomb planes, referred to as the hcplane \(\left( {{\it{\varphi }}{{ = 55}^ \circ }} \right)\) (Fig. 1c). We find that H ^{*} closely follows a \(1{\rm{/}}\left {{\rm{cos}}\,\theta \,{\rm{cos}}\,{{\varphi }}} \right\) angle dependence (Fig. 2a, b), where θ denotes the angle between the abplane and the applied field in both rotation sets. The collapse of 1/H ^{*} vs. the normalized bcomponent of magnetic field \(\left {{H_b}} \right{\rm{/}}\left H \right = \left {{\rm{cos}}\,\theta ,{\rm{cos}}\,{{\varphi }}} \right\) onto a straight line in both rotation planes (Fig. 2c) indicates that the torque at H ^{*} is dominated by the bcomponent of magnetization M _{ b } and that the absolute value of magnetization M(H ^{*}) is nearly independent of field orientation. Moreover, the absolute value in the denominator of the angle dependence of H ^{*} indicates bistable behavior, \({M_b} = M_b^*{\rm{sign}}\left( {{H_b}} \right)\). To determine whether the bistable behavior of M _{ b } persists to fields above H ^{*}, we turn to the angle dependence of τ/H across the entire field range.
Below H ^{*}, τ/H follows a sin 2θ dependence (Fig. 3a) that is characteristic of the linear response regime, where \({M_i} = {\chi _{ij}}{H_j}\) (Supplementary Information of ref. ^{4}). The sin 2θ dependence below H ^{*} is in stark contrast with the angle dependence at high fields, where torque exhibits a sharp discontinuity as the magnetic field crosses the caxis (Fig. 3b). We observe the nonlinear susceptibility at fields above H ^{*} as a sin θ component, accompanied by a sign(cos θ) factor that captures the observed discontinuity at 90°. The sin θ angle dependence indicates that the torque \(\tau = {M_b}{H_c}  {M_c}{H_b}\) in this highfield regime is dominated by the first term, where H _{ c } = H sin θ. The negligible contribution of the second term M _{ c } H _{ b } is confirmed by the observed torque in the hcplane: if the discontinuity in τ/H vs. θ is driven exclusively by the saturated bcomponent of magnetization, then one would expect the amplitude of sin θ in the hcplane to be reduced by a factor cos 55° ≈ 0.577 compared to the bcplane. As expected, the reduced amplitude factor is 0.573. Thus, M _{ b } dominates the total magnetization at very high fields. Futhermore, the highfield magnetic response indicates that the Isinglike discontinuity \({M_b} = M_b^*{\rm{sign}}\left( {{H_b}} \right)\) across the caxis extends well beyond H ^{*}.
In the ultra highfield limit, when all spins are nearly aligned with the applied magnetic field, the effective local anisotropy energy (per formula unit) is \(E \approx \left( {\mu _{{\rm{Ir}}}^2\beta {\rm{/}}2} \right){\rm{cos}}\,2\theta \), where μ _{Ir} is the magnetic moment of the iridium ion and β is the anisotropic spin stiffness. Therefore when all correlations are overcome by a very large applied magnetic field, the torque \(\tau = {\rm d}E{\rm{/}}{\rm d}\theta \approx \mu _{{\rm{ir}}}^2\,\beta \,{\rm{sin}}\,2\theta \) is expected to recover the sin 2θ angle dependence observed at low fields. By contrast, in γlithium iridate, the sin θ sign(cos θ) angle dependence of the torque persists up to the highest applied magnetic fields (Fig. 3c), indicating the presence of robust spin correlations throughout the entire field range and providing a lower bound for the magnitude of the exchange interactions.
To assess the extent of this correlated spinfluid, we examine the thermal evolution of the nonlinear angle dependence of the torque at fields above H ^{*} (vertical line in Fig. 4b). Figure 4a shows the angle dependence of τ/H at 15 T for a range of temperatures, revealing a gradual decrease of the nonlinear response as temperature is raised. The nonlinearity becomes undetectable at this magnetic field above 70 K, the same temperature where paramagnetic behavior onsets in magnetic susceptibility measurements^{4}, indicating a continuous crossover to a strongly correlated spinfluid at low temperatures. This crossover is similarly captured by a smoothly evolving torque signal upon cooling through the transition temperature at fields above H ^{*} (Fig. 1d). We note that specific heat measurements also observe a broad peak upon cooling at high fields^{17}, consistent with a decrease in entropy due to the onset of spin correlations^{18}
Discussion
The orientation of the spinanisotropic exchange interactions with respect to the crystal directions gives rise to a strong magnetic anisotropy in the ordered state^{4}. The extreme softening of χ _{ b } occurs because the baxis is the only direction coaligned either parallel or perpendicular to all IrO_{2}Ir planes. None of the IrO_{2}Ir planes are parallel or perpendicular to either the a or caxis (Fig. 4c). In this study, we find that the bcomponent of magnetization continues to dominate the magnetic response beyond the ordered state to the highest measured fields (Fig. 4d). The highfield anisotropy bears resemblence to the broken symmetry state observed at low temperatures and low fields. This may suggest that the correlated object is a subunit of the complex magnetic structure observed at zero field: H ^{*} signifies the destruction of longrange order while leaving most of the local magnetic correlations intact. Perhaps the counterpropagating spin spirals seen by Xray scattering in βlithium iridate^{6}, are decoupled by a small component of magnetic field along the bdirection, leading to a finite correlation length with only minimal polarization of the individual spins.
The unusual behavior of the spinfluid directly indicates that the observed magnetic anisotropy is driven by exchange interactions, rather than gfactor anisotropy that is tied to the honeycomb planes (Supplementary Information of ref. ^{4}). This observation, coupled with the anomalously small magnetic moment induced for all field orientations, identifies spin correlations that persist over a broad field and temperature range. In this context, we note that many conventional, as well as correlated, metals are unstable at low temperatures and undergo symmetry breaking that gaps their lowenergy excitations^{19, 20}. In frustrated magnets, longrange order can be stabilized by lattice distortions, crystalfield effects, and alternative exchange pathways. In the specific case of embedding the Kitaev model onto a 3D honeycomb lattice, it appears that longrange order is stabilized by intrinsic symmetry breaking—whereby only one component of the spinanisotropic exchange can be coaligned with a crystal direction. Recent dynamic studies in related compounds^{15, 16} have found exotic gapless excitations persisting after magnetic order is suppressed with relatively small magnetic fields. Although the ordered state at low temperatures in γlithium iridate has quelled the possibility of a Kitaev spinliquid ground state, we have shown that this system hosts an exotic spin state that is otherwise masked by the zerofield antiferromagnetic order. Other studies, such as nuclear magnetic resonance^{16}, inelastic neutron scattering^{15} or linear thermal transport in the zerotemperature limit^{21} are necessary to determine whether the spinfluid in γlithium iridate inherits any properties of a gapless spinliquid.
Data availability
All relevant data is available from the authors.
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Acknowledgements
This work was performed at the National High Magnetic Field Laboratory, and was supported by the US Department of Energy BES ‘Science at 100 T’, the National Science Foundation DMR1157490, and the State of Florida. R.M. acknowledges support from LANL LDRD DR20160085 ‘Topology and Strong Correlations’. J.A. and N.B. acknowledge support by the Department of Energy Early Career program, Office of Basic Energy Sciences under Contract No. DEAC0205CH11231. J.A. and N.B. also acknowledge support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4374.
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K.A.M., B.J.R., J.B.B., R.D.M. and A.S. performed the experiments at the National High Magnetic Field LaboratoryPulsed Field Facility. K.A.M., B.J.R., R.D.M. and A.S. analyzed the data and wrote the manuscript with input from all authors. N.P.B. and J.G.A. synthesized and characterized highquality single crystals of γlithium iridate at the University of California, Berkeley.
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Modic, K.A., Ramshaw, B.J., Betts, J.B. et al. Robust spin correlations at high magnetic fields in the harmonic honeycomb iridates. Nat Commun 8, 180 (2017). https://doi.org/10.1038/s41467017002646
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DOI: https://doi.org/10.1038/s41467017002646
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