A quantum spin liquid is a state of matter characterized by quantum entanglement and the absence of any broken symmetry. In condensed matter, the frustrated rare-earth pyrochlore magnets Ho2Ti2O7 and Dy2Ti2O7, so-called spin ices, exhibit a classical spin liquid state with fractionalized thermal excitations (magnetic monopoles). Evidence for a quantum spin ice, in which the magnetic monopoles become long range entangled and an emergent quantum electrodynamics arises, seems within reach. The magnetic properties of the quantum spin ice candidate Yb2Ti2O7 have eluded a global understanding and even the presence or absence of static magnetic order at low temperatures is controversial. Here we show that sensitivity to pressure is the missing key to the low temperature behaviour of Yb2Ti2O7. By combining neutron diffraction and muon spin relaxation on a stoichiometric sample under pressure, we evidence a magnetic transition from a disordered, non-magnetic, ground state to a splayed ferromagnetic ground state.
The pyrochlore lattice, comprised of corner-sharing tetrahedra, is the archetype of magnetic frustration in three dimensions1 (Fig. 1). Since its early study by Anderson in 1956 (ref. 2), frustrated spin Hamiltonians on the pyrochlore lattice have provided a seemingly inexhaustible source for the study of fundamental physics3,4. In particular, spin liquid ground states have been predicted for such a lattice decorated with Heisenberg5,6 or XXZ7 spins. More recently, pyrochlore magnets have been put forward as realistic vehicles for the realization of a quantum spin ice state, using the generic S= nearest-neighbour anisotropic exchange Hamiltonian8,9,10. Yb2Ti2O7 is a promising quantum spin ice candidate as it possesses both an (effective) S= spin, thanks to the well isolated crystal field Kramers doublet ground state appropriate to Yb3+11, and strong quantum fluctuations brought by anisotropic exchange interactions and an XY g-tensor12. Several studies have focused on the nature of the ground state in Yb2Ti2O7, yet a consensus has been elusive to date13,14,15,16,17,18. Early neutron scattering experiments ruled out the presence of conventional static order down to 90 mK in a polycrystalline sample15, whereas other single crystal studies concluded the ground state was ferromagnetic14,16. The results of local probes are even more puzzling. Muon spin relaxation (μSR) measurements evidenced the presence of true static moments on the muon timescale, through the observation of both a drop of asymmetry and a decoupling of the muon spins in longitudinal applied fields18, along with a drastic slowing down of the fluctuation rate below Tc for certain samples13. In contrast, μSR studies by D’Ortenzio et al.17 found a non-magnetic, fluctuating ground state, in both stoichiometric polycrystalline and single crystal samples, despite the presence of pronounced specific heat anomalies at Tc=265 mK and Tc=185 mK, respectively. It is clear that local defects, either oxygen vacancies19 or excess magnetic ions20 (referred to as stuffing), vary significantly between polycrystalline powders and single crystals, and are likely responsible for such sample dependencies.
Here, by applying hydrostatic pressure to well-characterized Yb2+xTi2−xO7+δ samples, with x=0 and x=0.046 (ref. 20), we observe a magnetic transition in the stoichiometric, x=0 sample from a disordered ground state into a splayed ferromagnetic ground state. This result sheds light on the origin of the sample dependence in the ground state selection for Yb2Ti2O7 and is consistent with the recent theoretical proposal that Yb2Ti2O7 lies close to a phase boundary in the generic quantum spin ice Hamiltonian phase diagram21.
Muon spin relaxation
μSR measurements under hydrostatic pressures as high as 25 kbar, and at temperatures as low as 0.245 K, were performed on Yb2+xTi2−xO7+δ samples, with x=0 and x=0.046, at the GPD beamline of PSI. The muons are implanted inside the bulk of the material, and act as local magnetic probes. The signal coming from the muons that stop inside the pressure cell was measured separately and subtracted (see Supplementary Figs 1 and 2) from the overall signal.
Figure 2a shows the temperature dependence of the μSR signal for the stoichiometric, x=0 sample in zero field, Rzf(t), as a function of time t and under an applied pressure P=19.7 kbar. Well above Tc=0.265 K, at T≥0.97 K, the majority of the Yb3+ magnetic moments are paramagnetic and in a fast fluctuating regime, and display single-exponential relaxation. For T≤0.5 K, we observe the development of a small magnetic fraction f of the Yb3+ moments, which grows nonlinearly as the temperature decreases. The absence of oscillations at short time is indicative of a highly disordered magnetic state. The zero-field relaxation is well described by a Gaussian distribution of static internal fields with standard deviation Δ (see Supplementary Note 1), and the following phenomenological function:
In a purely static scenario, the second term (1/3-tail) should be constant. Here, a fluctuating component is nonetheless observed and we modelled this using a relaxation rate λ. The third term accounts for the paramagnetic component, and assumes the same relaxation rate λ, for simplicity. The unconventional shape of the zero-field longitudinal relaxation was discussed in detail in refs 13, 22. In contrast, the evolution of the relaxation in temperature of the x=0 sample under zero applied pressure, shown in Fig. 2b, shows little or no magnetic fraction (f≃6%) at any temperature above our base T=0.245 K, in agreement with D’Ortenzio et al.17 previously reported μSR studies. Using equation (1) we extract the magnetic fraction for each pressure and temperature, and collect the results in Fig. 2c. The development of the magnetic fraction with temperature is clearly pressure dependent, and turns on strongly at low temperatures, below Tc=0.265 K, for our minimum pressure of 1.2 kbar. For each pressure, one can define a critical temperature Tc, such that for T≤Tc, 50% of the magnetic moments are frozen. The corresponding P−T phase diagram is shown in Fig. 3. Clearly, the phase transition extrapolated from finite pressure measurements to zero pressure agrees well with the sharp Cp anomaly at Tc=0.265 K, appropriate to the x=0 sample. However the zero-pressure state for the x=0 sample at 0.245 K, below Tc, is disordered, indicating that the ground state of the stoichiometric, x=0 sample, is a spin liquid.
We now turn to the x=0.046 sample. The zero-field relaxation at T=0.245 K under zero and an applied pressure P=24.1 kbar are shown in Fig. 2d. Strikingly, no frozen magnetic fraction is observed upon the application of a pressure as high as P=24.1 kbar. Instead, we observe an increase of the relaxation for this x=0.046 sample, demonstrating its sensitivity to pressure. The temperature dependence of the relaxation is reported in Fig. 2e,f. One can speculate that a transition to a fully ordered state, as it is observed for the x=0 sample, would require higher pressures or lower temperatures, consistent with the lower Tc=0.185 K of the x=0.046 sample.
μSR studies on other samples have reported a drastic slowing down of spin fluctuations13, or static order18, under zero applied pressure for temperatures below 0.25 K. In the light of our results, even relatively low (applied or chemical) pressure can destroy the disordered spin liquid state and induce magnetic order. A low level of defects in the different samples is a natural explanation to the contradictory μSR results. Such disorder, at the ∼2% level, is difficult to characterize, but it is largely absent in polycrystalline samples, synthesized at lower temperatures by solid state methods.
Armed with the knowledge of the P−T phase diagram in Fig. 3, we sought to determine the nature of the pressure-induced magnetic order in Yb2+xTi2−xO7+δ samples, with x=0, by performing neutron diffraction on the stoichiometric powder sample at the D20 high-flux diffractometer of the ILL. The detection of small magnetic moments under pressure using neutron diffraction is challenging due to the significant background signal of the pressure cell itself. Figure 4a shows the neutron diffraction data for the maximum hydrostatic pressure of the cell, P=11(2) kbar, and temperatures from 400 to 100 mK, from which a background measured at 800 mK was subtracted. We clearly observe the development of magnetic Bragg intensities at the (111), (311), (222) and (004) positions upon cooling below 400 mK. This is firm evidence for the existence of long-range magnetic order in Yb2+xTi2−xO7+δ samples, with x=0, under an applied pressure P=11(2) kbar. The refinement of the neutron diffraction data gives us the temperature dependence of the ordered moment, shown in Fig. 4b. The contrast with previous experiments under zero pressure is striking. First, the saturated moment μ=0.33(5) μB is much smaller than that μ∼1 μB reported previously for different Yb2Ti2O7 samples16, although similar to the ordered moment in the Γ5 ordered state of Yb2Ge2O7 (ref. 23). Second, the ordered moment vanishes cleanly above Tc∼0.4 K, with no anomalous magnetic Bragg intensity well above Tc (refs 24, 25). The previously reported order parameter at P=0 of our x=0 polycrystalline sample is anomalous24; it shows no change across Tc and only falls off at much higher temperatures. Consistency with our P=0 μSR results on the same sample requires that this Bragg-like scattering is dynamic on slow time scales. That notwithstanding, the magnetic structure previously refined on the basis of a very high temperature (∼8 K) background subtraction gave a splayed ice-like ferromagnetic structure24, with the moments on a tetrahedron lying mainly in the  direction with a positive splay angle θ=14(5)°, such that the moments tilt towards the local  direction (Fig. 4c). The components perpendicular to the local  axis obey the 2-in/2-out ice rule on a single tetrahedron. A different type of splayed ferromagnet, with the perpendicular components satisfying the all-in/all-out structure, has also been reported recently26, in addition to a nearly collinear ferromagnet (θ∼0°)16, for other samples. The magnetic structure associated with the true Bragg scattering we refine here in the stoichiometric x=0 sample under P=11(2) kbar is also a splayed ice-like structure, but with a much reduced splay angle θ=5(4)°, such that it is close to a collinear  ferromagnet (Fig. 4d).
These results bring a fresh perspective on the long standing debate about the presence or absence of static magnetic order in the quantum pyrochlore magnet Yb2+xTi2−xO7+δ. The acute sensitivity to local (through the Yb3+ stuffing) or applied pressure is surprising. However, a corollary of our new P−T phase diagram is that non-stoichiometric samples with non-zero chemical pressure can easily display an ambient applied pressure phase transition to a splayed ferromagnetic state at Tc. Yet, this interpretation remains challenged by the fact that our x=0.046 sample does not show evidence for magnetic order at ambient pressure, and by previous reported observations of a magnetic transition in polycrystalline, likely x∼0, sample even under zero pressure13,18. This may indicate that the non-magnetic low-temperature region of the phase diagram is extremely narrow, existing only for a certain range of x, whose absolute values are still to be determined. This would actually be reminiscent of the recent findings on the Tb2+xTi2−xO7+δ pyrochlore magnet, which has been shown to display an ordered phase that is extremely sensitive to disorder, appearing only for 0<x<0.01 (refs 27, 28, 29).
Furthermore, the present work illustrates the relevance of applying hydrostatic pressure to tune the magnetic properties of frustrated pyrochlore compounds, a path that was followed by pioneering work on the other spin liquid candidate Tb2Ti2O7 (ref. 30). In case of Yb2Ti2O7, we found that the pressure tunes the delicate balance between the anisotropic exchanges of the quantum spin ice Hamiltonian, and selects a splayed ferromagnetic ground state away from the degenerate antiferromagnetic ground states manifold. This scenario confirms recent theoretical proposals that Yb2Ti2O7 lies close to phase boundaries derived from the generic Seff= quantum spin ice Hamiltonian21, and provides the missing key to understand its exotic magnetic properties. Particularly appealing is the prediction that accidental degeneracies in the vicinity of these phase boundaries can lead to the emergence of a quantum spin liquid31. This would offer a natural explanation for a non-magnetic, disordered state under zero pressure in stoichiometric Yb2Ti2O7 and recent observations of a continuum of gapless quantum excitations24,32 at low temperatures.
The Yb2+xTi2−xO7+δ samples with x=0 and x=0.046 were prepared at the Brockhouse Institute for Materials Research, McMaster University. The x=0 powder sample was obtained through conventional solid-state reaction between pressed powders of Yb2O3 and TiO2 sintered at 1,200 °C in air. The x=0.046 powder sample was obtained by crushing a single crystal grown by the floating zone method in 4 atm of O2 with a growth rate of 5 mm h−1. More details on the details of the synthesis and the characterization can be found in ref. 25.
Muon spin relaxation
μSR measurements were carried out at the GPD instrument of the Paul Scherrer Institut, Switzerland. About 1 g of each powder sample was mixed with ∼3 mm3 of a pressure medium (Daphne 7373 oil) and placed inside the sample channel of a double-wall pressure cell. Two different cells were used, labelled as (1) and (2) (see Supplementary Note 1), and are described in more details in ref. 33. The muon momentum was adjusted in order to obtain an optimal fraction of the muons stopping in the sample, with optimal values found at P=106 and P=107 MeV c−1. The relaxation of both cells were measured without any sample down to 0.245 K. The applied pressure was determined by measuring the superconducting transition temperature of a small piece of pure indium inserted in the sample channel33.
The neutron diffraction experiment was conducted at the D20 beamline, a high intensity two axis diffractometer, at the Institut Laue-Langevin, using a neutron wavelength λ=2.4 Å. A mass of 1.5 g of Yb2Ti2O7 powder sample and a small amount of NaCl powder, which serves as a pressure calibration, were both mounted in a high pressure clamp cell and inserted in a 3He–4He dilution fridge. Fluorinert was used as a pressure transmitter. A minimum of 12 h of data was collected at each temperature. The diffraction pattern obtained for T=800 mK is shown in Supplementary Fig. 3. Structural refinements for both NaCl and Yb2Ti2O7 and magnetic refinements for Yb2Ti2O7 have been performed using the Fullprof program suite34.
The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
How to cite this article: Kermarrec, E. et al. Ground state selection under pressure in the quantum pyrochlore magnet Yb2Ti2O7. Nat. Commun. 8, 14810 doi: 10.1038/ncomms14810 (2017).
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Work at McMaster University was supported by NSERC of Canada. This work is based on experiments performed at SμS, Paul Scherrer Institute, Villigen, Switzerland and at the Institut Laue-Langevin, Grenoble, France. This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under the NMI3-II Grant number 283883. E.K. acknowledges useful discussions with P. Mendels, F. Bert, S. Petit, L. D. C. Jaubert and C. Decorse.
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