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A quantum magnetic analogue to the critical point of water


At the liquid–gas phase transition in water, the density has a discontinuity at atmospheric pressure; however, the line of these first-order transitions defined by increasing the applied pressure terminates at the critical point1, a concept ubiquitous in statistical thermodynamics2. In correlated quantum materials, it was predicted3 and then confirmed experimentally4,5 that a critical point terminates the line of Mott metal–insulator transitions, which are also first-order with a discontinuous charge carrier density. In quantum spin systems, continuous quantum phase transitions6 have been controlled by pressure7,8, applied magnetic field9,10 and disorder11, but discontinuous quantum phase transitions have received less attention. The geometrically frustrated quantum antiferromagnet SrCu2(BO3)2 constitutes a near-exact realization of the paradigmatic Shastry–Sutherland model12,13,14 and displays exotic phenomena including magnetization plateaus15, low-lying bound-state excitations16, anomalous thermodynamics17 and discontinuous quantum phase transitions18,19. Here we control both the pressure and the magnetic field applied to SrCu2(BO3)2 to provide evidence of critical-point physics in a pure spin system. We use high-precision specific-heat measurements to demonstrate that, as in water, the pressure–temperature phase diagram has a first-order transition line that separates phases with different local magnetic energy densities, and that terminates at an Ising critical point. We provide a quantitative explanation of our data using recently developed finite-temperature tensor-network methods17,20,21,22. These results further our understanding of first-order quantum phase transitions in quantum magnetism, with potential applications in materials where anisotropic spin interactions produce the topological properties23,24 that are useful for spintronic applications.

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Fig. 1: Specific heat of water and of SrCu2(BO3)2, shown together with calculated critical properties of the Shastry–Sutherland model.
Fig. 2: Specific heat and critical scaling at zero magnetic field.
Fig. 3: Evolution of the specific heat with magnetic field around the QPT.
Fig. 4: Phase diagram of SrCu2(BO3)2 in pressure, field and temperature.

Data availability

The data that support the findings of this study are available at and from the corresponding author upon reasonable request.

Code availability

The code that supports the findings of this study is available from the corresponding author upon reasonable request.


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We are grateful to R. Gaal, J. Piatek and M. de Vries for technical assistance. We acknowledge discussions with D. Badrtdinov, C. Boos, T. Fennell, A. Sandvik, A.-M. Tremblay, A. Turrini, A. Wietek and A. Zheludev. We thank the São Paulo Research Foundation (FAPESP) for financial support under grant no. 2018/08845-3, the Qatar Foundation for support through Carnegie Mellon University in Qatar’s Seed Research programme, the Swiss National Science Foundation (SNSF) for support under grant no. 188648 and the European Research Council (ERC) for support under the EU Horizon 2020 research and innovation programme (grant no. 677061), as well as from the ERC Synergy Grant HERO. We are grateful to the Deutsche Forschungsgemeinschaft for the support of RTG 1995 and to the IT Center at RWTH Aachen University and the JSC Jülich for access to computing time through JARA-HPC. The statements made herein are not the responsibility of the Qatar Foundation.

Author information




The experimental project was conceived by H.M.R. and Ch.R. and the theoretical framework was put forward by F.M. The crystals were grown by E.P. and K.C. Specific-heat measurements were performed by J.L.J. with assistance from M.E.Z., R.L. and H.M.R. S.P.G.C. and P.C. performed iPEPS calculations. A.M.L. performed complementary exact diagonalization calculations. L.W. and S.W. performed quantum Monte Carlo simulations on the fully frustrated bilayer model. Data analysis and figure preparation were performed by J.L.J., E.F., S.P.G.C., L.W., S.W., P.C. and H.M.R. The detailed theoretical analysis was provided by P.C., S.P.G.C., F.M., A.H., B.N., L.W. and S.W. The manuscript was written by B.N. and F.M. with assistance from all the authors.

Corresponding author

Correspondence to H. M. Rønnow.

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The authors declare no competing interests.

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Peer review information Nature thanks Jong Yeon Lee and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 a.c. calorimetry on SrCu2(BO3)2.

a, The a.c. calorimeter was prepared by depositing two Pt thin films (the shinier surfaces) over both halves of the sample. One film was used as the heater and the other for optimal thermal contact and measurement. The heating current (Iex at frequency f) was supplied through the pair of Constantan wires labelled H1a and H1b, and H2a and H2b were used to measure the electrical resistance, RPt, of the Pt film. TC1 and TC2 are thermocouples and K represents the thermal contact between the sample and the cryostat (through the pressure cell). b, Isothermal (T = 4.5 K) and isobaric (P = 20 kbar) f dependence of the modulated pick-up voltage, Vac, which is directly proportional to the temperature differential, ΔTac, measured by the thermocouples at two different positions. c, Isobaric (P = 18.2 kbar) f-dependence measurements of Vac at different temperatures with I0 = 1.6 mA at T ≥ 3.9 K, I0 = 0.8 mA at T = 2 K and I0 = 0.4 mA at T < 2 K. d, Sample heat capacity normalized to the input heating power (\({P}_{0}={I}_{0}^{2}{R}_{{\rm{Pt}}}\)), comparing the fit of Vac(f) obtained from the steady-state equation (‘f scan’)29 with values obtained directly from a variable-temperature measurement performed at the fixed working frequency fC = 1.5 Hz (‘T scan’). Solid and dashed lines in b and c represent fits using the steady-state equation29,44.

Extended Data Fig. 2 Correlation length.

ξ/a for the Shastry–Sutherland model, calculated by iPEPS with D = 8 as a function of the coupling ratio, J/JD, at a fixed temperature Tc(D = 8) = 0.0522JD/kB. The three panels show increasing magnification of the J/JD axis from the equivalent of Fig. 1 (upper right) through the step sizes of Fig. 2c, d (10−3J/JD, centre) to 10−5J/JD (lower left).

Extended Data Fig. 3 Critical point in the presence of Dzyaloshinskii–Moriya interactions.

Thermodynamic data obtained from iPEPS calculations with D = 10 performed for the Shastry–Sutherland model in the presence of Dzyaloshinskii–Moriya interactions. These interactions, of strength DD, are placed on the dimer (JD) bonds and have the magnitude known for SrCu2(BO3)2. They create an entangled ground state in the dimer phase, which resolves the numerical instabilities observed for the pure Shastry–Sutherland model at low temperatures, although the reduced symmetry limits the maximum D to 10. a, Specific heat, C( J/JDT)/T, shown in the same format as for Fig. 1b, c. b, Correlation length, ξ/a, showing clearly the region of ‘pressure’ and temperature over which Ising correlations develop. c, Dimer spin–spin correlation function, Si · Sj, emphasizing the abrupt onset with decreasing temperature of a sharp discontinuity as a function of J/JD. It is clear that these Dzyaloshinskii–Moriya interactions have no qualitative effect whatsoever on the physics of the critical point.

Extended Data Fig. 4 Ising critical points in different lattice models.

Specific heat, C/T, for a number of 2D models, illustrating its universal behaviour around the Ising critical point. a, Ising model on the square lattice in a longitudinal magnetic field, h, obtained by contracting the exact D = 2 tensor-network representation of the partition function using the corner-transfer-matrix method with a boundary bond dimension χ = 24 (ref. 55). b, Fully frustrated bilayer model, obtained by using the stochastic series expansion quantum Monte Carlo approach developed in refs. 27,45 to perform simulations on systems of sizes up to 2 × 32 × 32 as a function of J/J. c, Shastry–Sutherland model, obtained by iPEPS calculations with D = 20 as in Fig. 1c. The dashed lines show the positions of the local maxima of the specific heat, C(J/JD), which we label by their temperatures, Tmax. These two lines reach an absolute minimum, Tmax = Tc, where they meet at the Ising critical point, with Tmax increasing as the control parameter is changed away from the QPT. Thus the specific heat defines two characteristic lines in the phase diagram of the Ising critical point, instead of the single line given by the correlation length (Fig. 1d) and the critical isochore (Fig. 1e). This contrasting behaviour has been demonstrated in models where the critical pressure is temperature-independent57 and the issue of characteristic lines has also been discussed in the Mott metal–insulator phase diagram38,40. We stress that such behaviour is a fundamental property of the Ising model, and hence of all models sharing its physics. For the Shastry–Sutherland model (c), the two lines of maxima can be taken to provide a qualitative definition of regimes dominated by ‘dimer-like’ spin correlations (lower left) and by ‘plaquette-like’ correlations (lower right, but above the plaquette-ordered phase), accompanied by a third regime (above both lines) bearing no clear hallmarks of either T = 0 phase. We remark that the values of Tc in units of the relevant energy scale, Tc/J ≈ 2.3 (a), Tc/J ≈ 0.53 (b) and Tc/J ≈ 0.04 (c), vary widely among the three models. This can be traced to the change in slope of the ground-state energy at the transition, the compensation of which by entropy effects restores a derivable free energy at Tc.

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Jiménez, J.L., Crone, S.P.G., Fogh, E. et al. A quantum magnetic analogue to the critical point of water. Nature 592, 370–375 (2021).

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