Main

In the field of quantum magnetism, geometrically frustrated lattices generally imply major difficulties in analytical and numerical studies. For very few particular topologies, however, it has been shown that the ground state, at least, can be calculated exactly as for the Majumdar–Ghosh model15 that solves the J1 − J2 zigzag chain when J1 = 2J2. In two dimensions, the Shastry–Sutherland model2 consisting of an orthogonal dimer network of spin S = 1/2 was developed to be exactly solvable. For an inter-dimer J′ to intra-dimer J exchange ratio α ≡ J′/J ≤ 0.5 the ground state is a product of singlets on the strong bond J. Numerical calculations have further shown that this remains valid up to α ≤ ∼0.7 and for small values of three-dimensional (3D) couplings J′′ between dimer layers. At the other end, for ∼0.9 ≤ α ≤ ∞ the system approaches the well-known 2D square lattice, which is antiferromagnetically (AFM) ordered, albeit with significant quantum fluctuations that are believed to include resonating singlet correlations resulting in fractional excitations16. The phase diagram of the Shastry–Sutherland model, both with and without applied magnetic field, has been intensively studied by numerous theoretical and numerical approaches4. In the presence of magnetic field, magnetization plateaus at fractional values of the saturation magnetization corresponding to Mott insulator phases of dimer states, as well as possible superfluid and supersolid phases have been extensively studied7,17,18,19. At zero field, the main unsolved issue is the existence and nature of an intermediate phase for ∼0.7 ≤ α ≤ ∼0.9. A variety of quantum phases and transitions between them have been predicted depending on the theoretical technique used: a direct transition from dimer singlet phase to AFM order2,6,7, or an intermediate phase with helical order5, columnar dimers11, valence bond crystal12 or resonating valence bond plaquettes9,10. Recent results indicate that a plaquette singlet phase is favoured4,20. From such a phase, which would have an additional Ising-type order parameter, a subsequent transition to AFM order could provide a realization of the so far elusive deconfined quantum critical point21.

The compound strontium copper borate SrCu2(BO3)2 is the only known realization of the Shastry–Sutherland model with S = 1/2 spins4 and has thus triggered considerable attention in the field of quantum magnetism. The spectrum of SrCu2(BO3)2 exhibits an almost dispersionless Δ = 3 meV gap, and a bound state of two triplets (BT) forms at EBT ≃ 5 meV. The unusual size and dispersionless nature of the gap is an effect of the frustration that prevents triplets from hopping up to sixth order4. The estimated exchange parameters in the material J ∼ 85 K and α = 0.635 (ref. 4) or J ∼ 71 K and α = 0.603 (ref. 8) place the compound close to an interesting regime α ∼ 0.7 where correlations may change dramatically at a critical point.

A precious means to tune a quantum magnet across a quantum phase transition is the application of hydrostatic pressure as it directly modifies the atomic distances and bridging angles, such as Cu–O–Cu and thus the magnetic exchange integrals. Quantum phase transitions were successfully discovered in dimer magnets following application of pressure22. However high-pressure measurements remain technically challenging. In the case of SrCu2(BO3)2, magnetic susceptibility23 and electron spin resonance24 to moderate pressures (p ≤ 12 kbar) indicate a softening of the gap, while the combined effect of pressure and field was measured by susceptibility and NMR25. In the latter case, magnetic order occurring at 24 kbar and 7 T on a fraction of the dimers was proposed. In an X-ray diffraction investigation, the temperature dependence of the lattice parameters was analysed as an indirect proxy for the singlet–triplet gap leading to the suggestion that it closes at 20 kbar26. At even higher pressures, neutron and X-ray diffraction experiments observed a transition above 45 kbar from the ambient I2M tetragonal space group to monoclinic27,28,29,30.

Here we present neutron spectroscopy results, which directly determine the pressure dependence of the gap and through the dynamic structure factor allow us to address the nature of the correlations. Figure 1 summarizes the phase diagram of SrCu2(BO3)2, which we determined in this study. The exact dimer phase survives up to 16 kbar. The gap decreases from 3 meV to 2 meV, but does not vanish. At 21.5 kbar, we discover experimentally a new, intermediate phase. We identify it by its inelastic neutron scattering spectrum as the formation of 4-spin plaquette singlets. Above 40 kbar and below 117 K we find by neutron diffraction that AFM order appears (Supplementary Fig. 6) while the compound probably still has tetragonal symmetry with orthogonal dimers. Above ∼45 kbar, a structural distortion takes place and the symmetry becomes monoclinic, implying non-orthogonal dimers28,29. SrCu2(BO3)2 is magnetically ordered after the distortion, but can no longer be described appropriately by the original Shastry–Sutherland model. The transition from 2-spin dimer to 4-spin plaquette singlets appears to be of first order, whereas the transition from the plaquette to the AFM phase could be of second order and concomitant with the continuous closure of the plaquette gap as sketched in Fig. 1 or of first order9,20.

Figure 1: Phase diagram of SrCu2(BO3)2 as a function of pressure and temperature, including excitation energies.
figure 1

The blue region is the dimer phase, the red region the newly identified plaquette phase, and the green region is the antiferromagnetic phase where Q = (1, 0, 0) magnetic Bragg peaks, indicated by green squares, are observed only above 40 kbar. Circles are the triplet gap energy Δ at Q = (2, 0, L), diamonds are the corresponding two-triplet bound state (BT) energy EBT and the star is a new low-energy excitation (LE) observed at Q = (1,0,1). The magenta line shows the tetragonal to monoclinic structural transition. The procedure used to obtain it and its error bars is described in ref. 28. The corresponding monoclinic space groups are indicated29,30. The dashed line in the plaquette phase is the extrapolated energy gap using ref. 9. The insets depict the corresponding ground states. All of the experimental points are from this study.

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To allow a quantitative comparison to theoretical predictions, we establish the pressure dependence of the exchange parameters Jχ(p), Jχ′(p) and α(p) by measuring magnetic susceptibility χ(p, T) and fitting it using 20-site exact diagonalization. The peak in susceptibility shifts to lower temperature as pressure increases up to 10 kbar (Fig. 2a). This suggests a reduction of the spin gap. We parametrize the pressure dependence of J and J′ by linear fits (Fig. 2b). J has the larger slope so that α increases with pressure. Having established α(p) we see that the critical pressure lying between 16 kbar and 21.5 kbar corresponds to 0.66 < αc < 0.68, in good agreement with theoretical predictions4,12,20.

Figure 2: Pressure dependence of the magnetic susceptibility and of the exchange parameters in SrCu2(BO3)2.
figure 2

a, Magnetic susceptibility at three pressures below 10 kbar with fits to calculations by exact diagonalization (solid lines), H = 0.5 T. b, Extracted exchange parameters Jχ(p) and Jχ′(p) with linear fits and their ratio α(p) . The error bars in b represent standard deviation of the fit.

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A selection from the neutron spectra leading to the phase diagram is presented in Fig. 3; additional spectra at various momenta transfer Q are shown in the Supplementary Information. Up to 16 kbar an essentially Q-independent linear decrease of the gap energy is observed (Figs 1 and 3a). The measurement of the dispersion and of the structure factor in that pressure range shows that the spin system is still in its original ‘exact dimer’ phase. The gap value and the integrated intensity decrease linearly with pressure. The dispersion increases slightly with pressure, which can be understood by the increase of α (ref. 6). Interestingly, the bound triplet energy EBT softens twice as fast, implying that the triplet binding energy, δ = 2Δ − EBT = 1.19(2) meV, remains quasi pressure independent. This results in the unusual situation that extrapolating the softenings, the bound triplet would reach zero energy before the single triplet, and hence that, before that point, exciting a bound state of two triplets would cost less energy than exciting one triplet.

Figure 3: Inelastic neutron scattering measurements of SrCu2(BO3)2 under hydrostatic pressure.
figure 3

a, Energy spectra with triplet gap Δ and two-triplet bound-state BT energies softening in the dimer phase (Set-up 1). b, Discontinuity in the gap softening between 16 and 21.5 kbar (Set-up 2). c, New low-energy excitation LE at Q = (1, 0, 1) (Set-up 3). d, Pressure dependence of the gap integrated intensity (Set-ups 1–3). e, Momentum dependence of the intensity at the gap energy Δ, background subtracted (Set-up 2). The black line is the isolated dimer structure factor. f, Intensities of Δ and LE at 21.5 kbar. The red (blue) line is the full plaquette T1 (T2) structure factor (see Fig. 4). g, Dispersion of Δ and LE at 21.5 kbar. The black line is a scaled ambient pressure dispersion and the red dashed line is a guide to the eye. Error bars for inelastic neutron scattering intensities (a,b,c,e) are obtained from the square root of the number of counts assuming a Poisson distribution. The error bars in d,f,g are standard deviations of the fit.

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SrCu2(BO3)2 enters a new quantum phase between 16 and 21.5 kbar, where a discontinuity in the gap softening occurs. The inelastic neutron scattering peaks corresponding to the gap energy, Δ ≃ 2 meV, at these two pressures remain unchanged (Fig. 3b). The discontinuity is also visible in the intensities (Fig. 3d), where the linear decrease with pressure exhibits an abrupt halt above 16 kbar.

The transition to a new quantum phase is further asserted by a new type of excitation suddenly appearing at the higher pressure (Fig. 3b, c). We label this new low-energy excitation LE. LE is clearly visible around 1 meV for Q = (1,0,1), (−1,0,1) and (1,0,1.5) at 0.5 K and is not observed at 15 K, which proves its magnetic origin.

At 21.5 kbar, beyond the discontinuity, the 2 meV excitation displays a remarkable similarity to the ambient-pressure Δ as shown by constant energy scans along Q = (H, 0,1) in Fig. 3e. Both qualitatively follow the isolated dimer structure factor. This is further confirmed by extracting the structure factors from energy scans (Fig. 3f) and by comparing the dispersion to the ambient pressure dispersion (Fig. 3g). We therefore keep labelling this excitation Δ. LE on the other hand is more dispersive, ∼0.4 meV in the measured momentum range, and has a different structure factor strongly peaking between Q = (1,0,1) and Q = (1.25,0,1). This behaviour is reminiscent of a 4-spin plaquette structure factor (red line in Fig. 3f) that is further discussed in Fig. 4.

Figure 4: Plaquette phase in SrCu2(BO3)2.
figure 4

a, Full plaquettes containing a diagonal bound. b,c, Full plaquette structure factors for excitations T1 (b) and T2 (c); c is identical to the structure factor of two orthogonal isolated dimers. d, Void plaquettes. e,f, Void plaquette structure factors for excitations T1 (e) and T2 (f). In each case the structure factors Sxx + Syy + Szz are calculated as the sum over the two possible full (a) or two possible void (d) plaquette orientations in the SrCu2(BO3)2 geometry. g, Structure factor along Qhk = (Qh, Qk) = (H, 0) for T1; void plaquette in green and full plaquette in blue are identical. h, Structure factor along Qhk = (Qh, Qk) = (H, 0) for T2; void plaquette in green and full plaquette in blue. The blue line is also the isolated dimer structure factor. i, Excitation energies as a function of α(p) extrapolated from Fig. 2b. Comparison between experiment (same points as in Fig. 1) and theoretical predictions: dimer gap energy adapted from ref. 11 (solid line), low- and high-energy triplet excitations in the plaquette phase from ref. 10 (dotted lines), and for columnar plaquette block energies ref. 13 (black squares).

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To interpret the new excitation and the observed momentum dependence of the dynamical structure factors, it is illustrative to consider the simplified case of an isolated 4-spin plaquette, described in the Methods, which has a singlet ground state and shows two low-lying excitations T1 and T2. The structure factors of these excitations, summed over the two possible ‘full’ plaquette orientations (Fig. 4a), are shown in Fig. 4b, c together with those of a ‘void’ plaquette (Fig. 4d–f) containing no diagonal bond. T1 has a structure factor peaking between Qhk = (Qh, Qk) = (1,0) and Qhk = (1.25,0) in the 2D geometry of SrCu2(BO3)2 for both full and void plaquettes (Fig. 4g). T2, however, has a structure factor identical to that of an isolated dimer on the diagonal bond only for the full plaquette (Fig. 4c, f, h). While an extended many-body calculation would be needed for a fully quantitative comparison, the isolated plaquette considered here displays the main characteristics of the new intermediate pressure phase: a non-magnetic gapped ground state; a low-energy triplet (LE) with structure factor peaking above Qhk = (1,0); and another low-energy excitation (Δ) with structure factor identical to the singlet–triplet transition in the exact dimer phase. We thus identify the discovered phase as composed of 4-spin plaquette singlets, with excitation LE corresponding to T1 and excitation Δ corresponding to T2. Comparing the experimental intensities to this simple calculation favours the singlets sitting on ‘full’ plaquettes containing diagonal bonds, but calculations of the structure factor for the extended model are required for verification of this point.

To analyse further the interacting plaquette system, we plot in Fig. 4i the measured energies E/J versus α, which enables a direct comparison between our results and the calculations for the low- and high-energy resonating valence bond-like plaquette excitations10, and columnar plaquette block energy13. Experimental and calculated6 gap energies in the dimer phase are in excellent agreement. Beyond the transition, there is qualitative agreement for the energy scales; in particular, the observed energies of LE and of Δ for 21.5 kbar are close to the expected low- and high-energy plaquette excitations of ref. 10 for α = 0.68.

Our results can also explain the occurrence of magnetic ordering proposed by NMR measurements at 24 kbar and 7 T (ref. 25): the new spin S = 1 excitation LE being low in energy (0.8 meV), a 7 T field is sufficient to close the related gap and to obtain a magnetic ground state. This field-induced quantum critical point and resulting phase will be related to the field-induced BEC physics observed in dimer singlet systems31, but could reveal new phenomena due to the strong frustration in the Shastry–Sutherland model. In particular, the evolution of the magnetization plateaus in SrCu2(BO3)2 with pressure remains to be studied. On the basis of the results presented here, we can predict that, in particular, the pressure range between 15 and 25 kbar will be of high interest.

In conclusion, we have performed high-pressure experiments on SrCu2(BO3)2 and tuned the compound to experimentally identify a novel singlet phase consistent with the conjectured plaquette state at intermediate exchange ratio in the Shastry–Sutherland lattice. We observed a first-order transition taking place between two magnetically disordered states: the exact 2-spin dimer singlet and the 4-spin plaquette singlet phase. The dominant correlations in the plaquette phase involve a four-spin unit and are characterized by a low-lying triplet excitation that is not present in the dimer phase and that gives access to new types of field- and pressure-induced quantum critical points. The plaquette phase itself is suppressed at higher pressures where classical Néel order is found. Particularly exciting is the fact that the existence of two possible plaquette singlet coverings offers an Ising-type order parameter. This may turn the transition from plaquette to Néel phase into a deconfined quantum critical point at 40 kbar.

During the review of this manuscript, a new publication32 came to our attention, where high field magnetization measurements confirm the existence of a novel phase at 22 kbar, and its implication on the magnetization plateaus of SrCu2(BO3)2 is discussed.

Methods

Experiments.

Inelastic neutron scattering data were collected on three instruments: IN14 at ILL, TASP at SINQ-PSI and PANDA at FRM-2 (ref. 33). Piston–cylinder pressure cells based on hard Al alloy and hard steel allowed for a single-crystal sample mass of 3 g below 16 kbar (ref. 34). The 16 and 21.5 kbar pressures were reached with a McWhan34 pressure cell and a sample mass of 0.2 g. At 21.5 kbar, the sample was cooled down to both 2 K and 0.5 K to account for a possible unusual finite temperature damping35. AFM ordering was investigated by neutron diffraction on IN8 at ILL, with an opposed anvils Paris–Edinburgh press34,36 and a sample mass of about 0.1 g. Synchrotron X-ray diffraction was performed on ID9a at ESRF with a diamond anvil cell and microgram samples28. The details of the set-ups used with corresponding crystal orientations are given in Supplementary Table 1.

The pressure dependence of magnetic susceptibility was measured on a MPMS SQUID magnetometer (Quantum Design) using non-magnetic CuBe clamp pressure cells (CamCell) and pressure was calibrated by the superconducting transition of Pb.

Data analysis.

The pressure-dependent gap Δ(Jχ(p), Jχ′(p)) obtained through the Q = 0 expansion of ref. 11 with exchange parameters from fits to susceptibility data is in good agreement with the direct inelastic neutron scattering gap measurement ΔQ(p). To take into account the small Q-dependence of ΔQ, due to Dzyaloshinskii–Moriya interactions37, we additionally used ΔQ(p) = Δ(Jχ(p), Jχ′(p)) + DQ(p), where the dispersion of DQ is of the order of 0.2 meV.

The 4-spin plaquette is described by the Hamiltonian:

where the last term represents a diagonal bond between sites 1 and 3 (a ‘full’ plaquette), and should be removed for a ‘void’ plaquette without such a diagonal bond. The eigenstates of can be separated over two sectors depending on the value of the quantum number S1,3 for the spins S1 + S3 on the diagonal bond and S2,4 for the spins S2 + S4 on the outer sites30,38. A study of the excitation spectrum of such a plaquette shows that for α ≥ 0.5 the ground state is an S=0 singlet of four spins. Two low-lying excitations T1 and T2 are present. For α ≥ 1, T1 has the lower energy, while for 0.5 ≥ α ≥ 1 T2 does. T1 corresponds to a triplet excitation with both S1,3 and S2,4 equal to 1. In the full plaquette, T2 is four-fold degenerate and corresponds to a singlet on the diagonal S1,3 = 0 plus two free spins, S2,4 = 0 or 1. The corresponding structure factor is identical to that of the singlet–triplet excitation on the isolated diagonal bond. For the void plaquette, T2 is seven-fold degenerate and the structure factor does not match the isolated dimer. We note that, in general, the maxima and minima of the isolated models structure factors are not commensurate with the reciprocal lattice.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.

Additional Information

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