doi:10.1038/nature09910 Supplementary Online Material METHODS Single crystals were made at Kyoto University by the electrooxidation of BEDT-TTF in an 1,1,2- tetrachloroethylene solution of KCN, CuCN, and 18-crown-6 ether. The polycrystalline sample used for the muon measurements had a mass of 70 mg and was made up of a mosaic many small crystals. For the SR 12 experiments the crystal mosaic was wrapped in a thin packet of 12.5 m Ag foil and mounted on a Ag sample plate in the sample cryostat that was placed in a beam of spin-polarized positive surface muons (with energy 4 MeV and momentum 28 MeV/c). Further Ag foils were used to degrade the muon beam so that the stopping profile matched the sample. Thus the background signal from muons missing the sample was mainly from Ag, which gives insignificant relaxation compared to the sample signal. For the measurements in transverse fields up to 3 T the LTF instrument of the S S Facility at the Paul Scherrer Institut in Switzerland was used. The applied field direction was perpendicular to the plane of the mosaic. Although the crystals were not deliberately aligned, some degree of alignment is expected due to the crystal shape and the field direction is therefore expected to lie primarily in the interlayer direction. Data analysis was carried out using the WiMDA program 30. The precession signal count rate N(d,t) measured as a function of time t in each detector d from the muon decay positrons was fitted to the form N d, t N BG d = N 0 d e t τ μ 1 + A d cos [γ μ B + φ d t ) e (γ μ B rms t ) 2 /2 = N d, t where is the muon lifetime and is the muon gyromagnetic ratio, d the detector (L=left, R=right) and for each detector N BG (d) is the background count rate, N 0 (d) the signal count rate, A(d) the asymmetry, (d) the detector phase and t = t-t 0 (d) is the time corrected for the time offset of the detector channel. The global parameters for the average and width of the internal field distribution B and B rms are derived from simultaneous fitting of the two detector channels. Detector pair asymmetry for detectors d 1 and d 2 is derived from the background corrected signal N (d,t) as A d1 d 2 t = N d 1, t N d 2, t N d 1, t + N d 2, t Where is a parameter used to correct any imbalance of the detector sensitivities. Example raw data for A LR are shown in Figure S1. The Poisson counting statistics of the muon decay positron events at the detectors determines the error of the raw count signal. Errors are propagated through the analysis procedure under WWW.NATURE.COM/NATURE 1
the assumption of a normal distribution, which is a good approximation for the count levels that were obtained within the time region dominating the data analysis. Parameters P and corresponding errors E derived from the data fitting are expressed as P(E) in the text, where P- E to P+E represents a 68% confidence interval on the fitted parameter (standard error). All error bars on the plots reflect this standard error. Zero field and low transverse field measurements were measured at the ISIS/RIKEN-RAL facility in the UK using a range of spectrometers: ARGUS and EMU for measurements down to 300 mk and MUSR and HIFI for measurements down to 40 mk. For these measurements the pair asymmetry A FB from detectors forward and backward to the initial muon spin is studied. The zero field relaxation of A FB has contributions from both electronic and nuclear moments within the system that are comparable in size and the relaxation of the forward-backward asymmetry A FB (t) was fitted to the form A FB t A BG = A FB 0 A BG KTZ Δ, t e λt where A BG is a non-relaxing component primarily due to the Ag, KTZ is the zero-field Kubo- Toyabe function reflecting the nuclear contribution to the relaxation in the sample with characteristic field width parameter KTZ Δ, t = 1 3 + 2 3 1 γ μ 2 Δ 2 t 2 exp 1 2 γ μ 2 Δ 2 t 2 The nuclear width was obtained from fitting the low temperature data and kept fixed thereafter. The relaxation contribution originating from fluctuating electronic spins in the sample is reflected by the parameter. Studies of the dependence of on longitudinal field confirm that the muon probe is in a fast fluctuation regime with respect to electronic spin fluctuations. is therefore proportional to the spin correlation time and the electronic spin fluctuation rate is inversely proportional to. WWW.NATURE.COM/NATURE 2
Supplementary Figure S1: Transverse field data Muon spin precession data measured at 120 mk are shown for fields both above and below the critical field 0 H c = 14 mt. The increased width of the internal field distribution at higher fields is reflected by the faster damping rate of the precession envelope (the envelope for H<H c is shown on all plots to aid comparison of the damping rates). WWW.NATURE.COM/NATURE 3
BACKGROUND Supplementary Figure S2: Lattice structure and suggested spin excitations (a) 2D lattice structure of the dimer layers in -(BEDT-TTF) 2 Cu 2 (CN) 3. The molecule abbreviated by BEDT- TTF is bis(ethylenedithio)- tetrathiafulvalene. The ratio of exchange couplings J /J is estimated 5 to be close to 0.9. (b) The Brillouin zone for spin excitations in the ideal triangular lattice. The calculated 3 spin excitation energy goes linearly to zero at the points K i (solid circles). An underlying spinon dispersion with Dirac minima at (K i -K j )/2 (open circles) has been suggested to explain roton-like deviations of the numerically calculated dispersion from that of linear spin wave theory 3. The primitive unit cell of the Brillouin zone (dashed line) contains four such points. WWW.NATURE.COM/NATURE 4
Supplementary Figure S3: Xu and Sachdev global phase diagram (a) The global phase diagram for the S=1/2 frustrated triangular lattice quantum antiferromagnet model of Xu and Sachdev 18, expressed in terms of coupling parameters s and v associated with bosonic spinon and vison excitations. Negative s leads to weakly antiferromagnetic (WAF) states, positive s leads to weakly gapped spinon (GS) states: a Z 2 spin-liquid (Z 2 -SL) for positive v (with vison gap v ) and valence bond solid (VBS) for negative v (with VBS transition at T v ). The blue dot illustrates the most likely location for -(BEDT-TTF) 2 Cu 2 (CN) 3 within this model on the basis of our data, the circle illustrates the less likely scenario of a VBS ground state. The predicted orders of the non-magnetic to magnetic transitions are indicated 18 : O(4) for Z 2 -SL to WAF and CP 1 for VBS to WAF (b) The s dependence of characteristic energy scales for H=0; the spinon gap on the GS side and the spin stiffness on the WAF side (dashed lines). The small estimated spinon gap s for -(BEDT-TTF) 2 Cu 2 (CN) 3 (~ 4 mk) reflects its closeness to the QCP at s=0. In the fan-shaped higher T region purely quantum critical (QC) behaviour is expected. (c) For T=0 an applied magnetic field H induces BEC of the spinons with the small critical field H 0 ( 0 H 0 ~ 5 mt). WWW.NATURE.COM/NATURE 5
MODELS AND CRITICAL EXPONENTS The reported critical exponents computed for various models using field-theory or Monte Carlo simulation are listed in Table 1 alongside our measured exponents. The exponent is linked to and by the scaling relation 2 = (1+ ). The exponents for the the CP 1 class that is predicted to govern the VBS-WAF transition 18 are not yet accurately established and an average of several studies has been taken 31-34. For the model of Kaul and Sachdev (KS) the following 1/N expansion formulae were given for the critical exponents 17 ν N b, N f = 1 16 3π 2 N b + 4 π 2 7N f 9N b N b + N f 2 20 3π 2 N b + N f η N b, N f = 1 + 32 128 3π 2 N b 3π 2 N b + N f A more detailed exploration of the exponents of the KS model in relation to the experimentally measured values is given in Figure S4, along with a possible interpretation in terms of spinon Fermi surface pockets (see Figure S2b). Supplementary Table 1 Model Notes 0.39(2) L from experiment 0.83(4) H from experiment KS(4,2) 0.51 0.55 0.39 Ref. [17] O(2) 0.67 0.04 0.35 Ref. [22] O(3) 0.71 0.04 0.37 Ref. [22] CP 1 0.72 0.31 0.47 Average of Refs. [31-34] O(4) 0.75 0.04 0.39 Regular (vector order parameter) Ref. [22] 0.75 1.38 0.89 Complex (tensor order parameter) Refs. [22,23] KS(4,8) 0.87 0.91 0.83 Ref. [17] WWW.NATURE.COM/NATURE 6
Supplementary Figure S4: Critical exponents in the Kaul Sachdev model (a,b) Map of the computed values of the critical exponent for the KS(N b,n f ) model 17, in which the quantum critical fluctuations reflect N b species of bosons and N f species of fermions. Experimental values of H and L and corresponding error bounds are shown by the contour lines. Circles indicate some possible values that lie within the error bounds and the red circles indicate the closest value in each case, N f is assumed to be even to reflect two-fold spin degeneracy. (c) A possible interpretation of the optimum N f value that gives the H observed in the WAF H ( deconfined ) phase: four spinon pockets with two-fold spin degeneracy give N f =8. (d) In the WAF L ( confined ) phase the reduced value of L is reflected in this model by a single pocket with N f =2. WWW.NATURE.COM/NATURE 7
SUPPLEMENTARY REFERENCES 31. Motrunich, O.I. and Vishwanath, A. Comparative study of Higgs transition in onecomponent and two-component lattice superconductor models, arxiv:0805.1494v1 (2008). 32. Motrunich, O.I. and Viswanath, A., Emergent photons and transitions in the O(3) sigma model with hedgehog suppression, Phys. Rev. B 70, 075104 (2004). 33. Sandvik, A.W., Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin Interactions, Phys. Rev. Lett. 98, 227202 (2007). 34. Melko, R.G. and Kaul, R.K., Scaling in the fan of an unconventional quantum critical point, Phys. Rev. Lett. 100, 017203 (2008). WWW.NATURE.COM/NATURE 8