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In most La-based cuprate superconductors, static order of both spin and charge (so-called stripe order) has been unambiguously identified by spectroscopic and thermodynamic probes1,2. At low temperature, magnetic fields weaken superconductivity and at the same time reinforce the magnitude of such orders8,9,10. As the superconducting transition temperature (Tc) in the La-based materials is substantially lower than in other cuprate materials, it has been argued that stripe order is detrimental to high-temperature superconductivity. In underdoped YBa2Cu3Oy (YBCO), there is now compelling evidence of competing order even though Tc = 94 K at optimal doping. The discovery of quantum oscillations11 combined with the negative Hall12 and Seebeck13 coefficients at low temperature has demonstrated that the Fermi surface of underdoped YBCO undergoes a reconstruction at low temperature and consists of at least one electron pocket. A comparative study of thermoelectric transport in underdoped YBCO and in La1.8−xEu0.2SrxCuO4 (a cuprate in which stripe order is well established) has been interpreted as charge stripe order causing reconstruction of the Fermi surface at low temperature14. High-field nuclear magnetic resonance (NMR) measurements have revealed that the translational symmetry of the CuO2 planes in YBCO is broken by the emergence of a modulation of the charge density at low temperature4. In addition, NMR measurements show that the modulation is observed above a threshold magnetic field and suggest that charge order is most likely uniaxial4. In zero field, long-range charge fluctuations in YBCO were recently observed with resonant soft X-ray scattering (RSXS) up to 150 K and 160 K for p = 0.11 (ref. 5) and p = 0.133 (ref. 6), respectively, whereas hard X-ray scattering experiments suggest that a charge order develops below 135 K for p = 0.12 (ref. 7). All measurements identify charge fluctuations at two wave vectors corresponding to an incommensurate periodicity of approximately 3.2 lattice units. The identification of a thermodynamic phase transition is thus important to determine where long-range charge order exists in the phase diagram and particularly whether static order occurs only in high magnetic fields.

Here we report sound velocity measurements, a thermodynamic probe, in magnetic fields large enough to suppress superconductivity. The sound velocity is defined as , where ρ is the density of the material, ci j = ∂2F/∂ ɛi∂ ɛj (ref. 15), F is the free energy and ɛi is the strain along direction i (in the contracted Voigt notation). Changes in the elastic constants ci j are expected whenever a strain-dependent phase transition occurs. Owing to their high sensitivity, sound velocity measurements are a powerful probe for detecting such phase transitions, in particular charge ordering in strongly correlated electron systems16.

We have measured several elastic constants (see Supplementary Table S1 for the description of the elastic modes) in high magnetic fields in an underdoped YBCO6.55 sample with Tc = 60.7 K corresponding to a hole doping p = 0.108 (ref. 17). Figure 1a,b shows the field dependence of the relative variation of the sound velocity Δvs/vs corresponding to the c11 mode, at different temperatures. At T = 4.2 K, the softening of the elastic constant at the vortex lattice melting field Bm≈20 T corresponds to the first-order melting transition from a vortex lattice to a vortex liquid18,19 (see Supplementary Information for more details). At T = 29.5 K, this anomaly shifts to lower field (Bm≈5 T) and because the pinning potential becomes less effective, the magnitude of the change of c11 at the melting transition becomes smaller20. At T = 29.5 K and above Bm, a sudden increase of the elastic constant can clearly be resolved at Bco = 18 T, which corresponds to a thermodynamic signature of a phase transition. Whereas Bco is almost temperature independent at low temperature, it increases rapidly between 35 and 50 K (see red arrows in Fig. 1b). For T≥50 K, no change of c11 can be resolved up to the highest field. Owing to the difference in the temperature dependence of Bm and Bco, the phase transition at Bco cannot originate from vortices. Figure 2 shows the phase diagram in which both Bm and Bco deduced from sound velocity measurements are plotted as a function of temperature. The identification of this phase stabilized by the magnetic field above Bco is straightforward. High-field NMR measurements in YBCO at similar doping have shown that charge order develops above a threshold field Bco>15 T and below TcoRMN = 50±10 K (ref. 4). Given the similar field and temperature scales, it is natural to attribute the anomaly seen in the elastic constant at Bco to the thermodynamic transition towards the static charge order.

Figure 1: Field dependence of the sound velocity in underdoped YBa2Cu3Oy.
figure 1

a,b, Field dependence of the longitudinal mode c11 (propagation q and polarization u of the sound wave along a axis) in underdoped YBCO (p = 0.108) at different temperatures from T = 4.2 K to T = 24.9 K (a), and from T = 29.5 K to T = 50 K (b). The curves are shifted for clarity. The measurements were performed in static magnetic field up to 28 T. Black arrows indicate the field Bm corresponding to the vortex lattice melting. At low temperature, the loss of the vortex lattice compression modulus can be estimated and is in agreement with previous studies (see Supplementary Information). For T>40 K, Bm cannot be resolved. Red arrows indicate the field Bco where the charge-order phase transition occurs. This transition is not related to vortex physics because it is also seen in acoustic modes c44 and c55 (Fig. 3 and Supplementary Fig. S3), which are insensitive to the flux line lattice because those modes involve atomic motions parallel to the vortex flux lines (u∥H∥c).

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Figure 2: Thermodynamic phase diagram.
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Magnetic field–temperature phase diagram of underdoped YBCO (p = 0.108) obtained from the anomalies seen in the elastic constant c11 (Fig. 1). Black squares indicate the transition from a vortex lattice to a vortex liquid at Bm, which cannot be resolved above 40 K. Red circles correspond to the phase transition towards static charge order at Bco, as observed in c11. The error bars on the field scale Bm (Bco) correspond to the width of the transition in the derivative (raw data) of c11(B). The charge-order transition is almost temperature independent up to ≈40 K. Above 40 K the field scale Bco at which charge order sets in rises. In the Supplementary Information, we argue that the overall behaviour of the charge-order phase boundary in this B–T diagram is consistent with a theoretical model of superconductivity in competition with a density-wave state21. The green diamond is the temperature TcoNMR = 50±10 K at which NMR experiments detect the onset of a charge modulation at a field B = 28.5 T in YBCO at doping p = 0.11 (ref. 4). Within the error bars, this onset temperature agrees with our findings. Dashed lines are guides to the eye.

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The phase diagram in Fig. 2 shares common features with the theoretical phase diagram of superconductivity in competition with a density-wave order21 (see discussion in the Supplementary Information). For T below 40 K or so, static charge order sets in only above a threshold field of 18 T, akin to the situation in La2−xSrxCuO4 (x = 0.145) in which a magnetic field is necessary to destabilize superconductivity and to drive the system to a magnetically ordered state9. Close to the onset temperature of static charge order, Tco, the threshold field Bco sharply increases and the phase boundary tends to become vertical. This is in agreement with the theoretical phase of competing order with superconductivity that predicts that superconducting fluctuations have no significant effect on charge order in this part of the phase diagram.

We now turn to the analysis of the symmetry of the charge modulation. In the framework of the Landau theory of phase transitions, an anomaly in the elastic constant occurs at a phase transition only if a coupling in the free energy Fc = gm nQm ɛn (where m and n are integers and gm n is a coupling constant) between the order parameter Q and the strain ɛ is symmetry allowed, that is, only if Qm and ɛn transform according to the same irreducible representation22. In Fig. 3 we compare the field dependence at T = 20 K of four different modes c11, c44, c55 and c66 that display an anomaly at Bco. To explain the presence of such coupling for all these modes, we rely on group theory arguments. YBCO is an orthorhombic system (point group D2h), and given the even character of the strains we have only to consider the character table of point group D2 shown in Table 1.

Figure 3: Charge-order transition seen in different modes.
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Field dependence of different elastic constants of YBCO (p = 0.108) at T = 20 K. a, Longitudinal mode c11. b, c44 (q∥b, u∥c) in green and c55 (q∥a, u∥c) in magenta. c, c66 (q∥b, u∥a). c44 and c55 have similar amplitude and field dependence at the transition Bco towards the charge order, indicating the biaxial character of the charge-density modulation. The anomalies seen in c66 and c11 are larger than in c44 and c55. This is probably due to a greater dependence of charge order with respect to in-plane strains than out-of-plane strains.

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Table 1 Character table of point group D2.
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To represent the different symmetric charge modulations that transform according to each irreducible representation of the point group D2 and to determine to which acoustic mode they couple, we will follow the procedure of ref. 16 and use the projection operator in the basis defined by the four Cu atoms at the corner of the CuO2 plane (see Supplementary Information). This simplification is justified because the charge order is an intrinsic property of the CuO2 plane4,6. The four kinds of symmetric charge distribution allowed in this basis are shown in Supplementary Fig. S5. In Fig. 4, we illustrate them with concrete charge modulation patterns in the CuO2 plane with an arbitrary periodicity. A uniaxial charge modulation along the a axis should induce an anomaly in c44 but not in c55. The very similar field dependence of c44 and c55 around Bco (Fig. 3) indicates a biaxial charge modulation both along the a axis (Qa) and along the b axis (Qb). As the irreducible representation and ɛ6 transforms according to B1, a further coupling βa bQaQbɛ6 explains the anomaly seen in c66. A similar coupling exists for c11 because  (i = 1,2 or 3).

Figure 4: Charge distribution in the CuO2 plane.
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Sketches of the different charge modulation patterns in the CuO2 plane coupled with the measured elastic constants. Blue and green spheres represent Cu atoms with different charge density. Grey spheres represent O atoms. The periodicity of the modulation is arbitrary. We assume that charge modulation takes place on the Cu atom, but including modulation of the charge on the O atoms does not change the conclusion. If the charge distribution is uniform, anomalies would be seen only in c11. A uniaxial charge order with a modulation along the a(b) axis couples linearly with c44 (c55). The fact that both c44 and c55 show an anomaly at Bco indicates that the charge modulation is biaxial.

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As a thermodynamic bulk probe with a typical energy scale below 1 μeV, sound velocity measurements establish that static charge order sets in below Tco≈45 K in magnetic fields above Bco = 18 T in YBCO at a doping level p = 0.108. For a nearby doping p = 0.11 (non-ortho-II ordered sample), zero-field RSXS measurements show that charge fluctuations appear at T = 150 K (ref. 5). The fluctuating character of the charge order at high temperature is consistent with recent resonant ultrasonic measurements in YBCO at similar doping level. Whereas anomalies in the sound velocity are detected at the pseudogap temperature, T*≈240 K, and at the superconducting transition temperature Tc, no further phase transition is observed in zero field23. An important aspect of these fluctuations is their biaxial character inferred from both RSXS (refs 5, 6) and X-ray7 measurements. However, those techniques cannot determine whether there is a single charge density wave (CDW) with biaxial modulation or whether there are domains of two uniaxial CDWs. From sound velocity measurements, the latter scenario seems unlikely because for the anomaly in c66 to exist, charge modulations along the two directions should in principle coexist within the same sample volume. The biaxial character of the CDW observed in our study and in X-ray measurements supports the scenario in which the static charge ordering occurring at low temperature and finite magnetic field is a consequence of the freezing of the charge fluctuations observed in zero field at high temperature. A biaxial CDW seems to differ from the conclusion drawn from NMR measurements, which have been interpreted as a uniaxial modulation of the charge density along the a axis4. However, an extra charge modulation along the b axis, as reported here, is not excluded from the NMR spectra. The biaxial nature of the CDW differs from the phenomenology of stripe order, which corresponds to a uniaxial charge order, as observed in La-based cuprates1,2. This is further supported by the absence of spin order down to the lowest temperature at this doping level as inferred from NMR measurements4. It raises the questions of the origin of the Tc anomaly close to p = 0.12 in underdoped YBCO and of the peculiar behaviour of the transport properties at high fields around this doping level24. These questions and the interplay between biaxial CDW and high-temperature superconductivity call for further investigations. The biaxial character of the charge order has another important consequence, namely on the topology of the reconstructed Fermi surface. In the absence of spin order, as inferred from NMR (ref. 4), uniaxial order by itself cannot produce electron pockets in the reconstructed Fermi surface25, unless one assumes that the Fermi surface at high temperature already breaks C4 symmetry26. On the other hand, no particular assumption is needed to produce electron pockets when the Fermi surface is folded along two perpendicular directions27.