Introduction

One of the surprising features of cuprate superconductors is the ubiquity of the interplay between antiferromagnetic (AFM) order, charge density waves (CDWs), and superconductivity1. In general, these orders compete, and each occupies its own piece of the phase diagram; however, significant coexistence regimes are observed, and these can lead to novel cooperative behavior. For example, incommensurate CDW and spin orders coexist as magnetic stripes in the La-based cuprates2,3. In other cuprates, CDWs are nonmagnetic, and there have been ongoing efforts to understand their connection to the strong electron correlations, low dimensionality, and AFM fluctuations that are hallmarks of the cuprates1. The Peierls paradigm that requires extended sections of Fermi surface to be nested by some wavevector q, leading to a peak in the non-interacting charge susceptibility, is certainly not at the origin of the cuprate CDWs. There are other CDW materials without obvious Fermi surface nesting—for example, layered dichalcogenides such as NbSe24 and nonmagnetic pnictides in the BaNi2As2 family5—for which there is compelling evidence that the CDWs are phonon-driven6,7,8,9. The corresponding evidence in the cuprates is weak10.

As an alternative to phonons, spin fluctuations provide a natural mechanism for CDW formation in cuprates. The mechanism for cuprate superconductivity is widely accepted to be of magnetic origin11, and several analytical calculations predict that spin-fluctuation-mediated CDWs degenerate with superconductivity at the AFM quantum critical point (where the Néel temperature vanishes)12,13,14,15. This is important because it provides a route to charge order at high temperatures without Fermi surface nesting. Instead, CDW wavevectors connect Fermi surface “hotspots” that are determined by the q-dependence of the spin susceptibility. Early versions of the theory incorrectly predicted that the CDW q-vectors lie along the Brillouin zone diagonals, with q = (q, ± q); however, quantitatively correct axial CDW wavevectors, q = (q, 0), (0, q), are obtained when the dominant diagonal-q instability is suppressed. This occurs close to the AFM quantum critical point16,17,18, within AFM19,20 or loop-current21 phases, or when strong correlations are accounted for (see refs. 22,23,24,25). Note, however, that realistic band structures produce weak peaks in the bare charge susceptibility with similar q-vectors26 despite the lack of proper Fermi surface nesting. Thus, while there is experimental support for Fermi surface hotspots27, it is unresolved whether they are selected by the Fermi surface or the interaction.

CDWs have clear signatures in the transport properties of YBa2Cu3O6+x (YBCO) and single-layer HgBa2CuO4+δ (Hg1201). At low temperatures, the observation of small-frequency quantum oscillations28,29 combined with negative Hall30,31,32 and Seebeck32,33,34 coefficients indicate the presence of a closed electron pocket in the reconstructed Fermi surface. Details of the reconstruction are still debated35, but a plausible scenario is a biaxial charge order that creates a small electron-like pocket located at the nodes. Quantum oscillations with small frequencies have also been observed in the underdoped trilayer cuprate HgBa2Ca2Cu3O8+δ (Hg1223) at hole doping p = 0.08–0.09 but without a sign change in the Hall coefficient down to the lowest temperatures36. A plausible interpretation is the coexistence of an AFM order in the inner plane and CDW in the outer planes. Multilayer cuprates, with three or more CuO2 layers per unit cell, offer a new twist on the story because a single material can host distinct phases in different layers simultaneously37. This raises intriguing questions about the coexistence and competition between these phases, and the possible emergence of novel cooperative phases. In particular, symmetry-inequivalent CuO2 planes differ both in doping and disorder levels37,38,39,40. In addition, AFM order is more robust in multilayer cuprates and persists up to higher doping as the number of CuO2 planes increases37.

Results

Hall effect measurements

Figure 1a shows the Hall coefficient plotted as a function of magnetic field H for Hg1223 p = 0.118 at different temperatures from T = 20 K down to T = 4.2 K. While RH is positive above T = 20 K, it becomes negative at T = 4.2 K and flattens at fields above H ≈70 T, which indicates that the normal state is reached. The temperature-dependence of the normal-state Hall coefficient measured at the highest fields (see Supplementary Note 5) between T = 1.5 K and 100 K is shown in Fig. 1b at three doping levels, p = 0.118 and p = 0.101 (this work) and p = 0.08936. For p = 0.118 and p = 0.101, there is a sudden sign change of the Hall coefficient below T ≈10 K that is not observed at lower doping. The sign change is accompanied by a maximum of the resistance at T ≈10 K (see Fig. 1c for p = 0.118). In other cuprates, notably underdoped YBCO41,42,43 and Hg120144,45, this behavior is attributed to the emergence of charge order. Here, both the abruptness of the transition and its low temperature, are surprising.

Fig. 1: Transport measurements in Hg1223.
figure 1

a Field dependence of the Hall coefficient RH in Hg1223 (p = 0.118) at various fixed temperatures, as indicated. b Temperature dependence of the normal-state Hall coefficient RH, measured at high fields (see Supplementary Note 5), in Hg1223 p = 0.089 (red circles from ref. 36), p = 0.101 (black squares, this work), and p = 0.118 (blue triangles, this work). At p = 0.101 and p = 0.118, RH changes sign abruptly below T = 10 K while it remains positive down to the lowest temperature for p = 0.089. c Temperature dependence of the normal-state resistance in Hg1223 (p = 0.118). The sign change of the Hall coefficient is accompanied by a maximum in the resistance, that strongly suggests the occurrence of a phase transition.

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Quantum oscillation measurements

Because of the low temperatures, we are in the rare situation of being able to track the Fermi surface morphology through the CDW transition using quantum oscillation measurements. Figure 2a shows the variation of the tunnel diode oscillator (TDO) circuit frequency as a function of the magnetic field at different temperatures for the p = 0.112 sample (see Methods). A smooth background subtraction leads to the oscillatory part of the signal depicted in Fig. 2b. At T = 4.2 K, there is a strong low-frequency oscillation whose amplitude decreases with decreasing temperature. By T = 1.75 K, the low-frequency oscillations are weak, and small amplitude oscillations at higher frequencies have emerged. This is inconsistent with the Lifshitz–Kosevich theory for a temperature-independent Fermi surface46, but rather signals a Fermi surface reconstruction. The temperature evolution of the oscillation spectrum is clearly seen in the Fourier analysis of the oscillatory part of the data (Fig. 2c). At T = 4.2 K, the spectrum is dominated by a low-frequency peak (F = 680 T, black arrow) and its first harmonic. The low-frequency peak weakens and shifts to a higher frequency (F = 830 T) as the temperature decreases. Two peaks at high frequency, F = 2100 T and F = 2800 T (red arrow), emerge at the lowest temperatures. While the high frequencies could result from a large effective mass (a Lifshitz–Kosevich analysis for F = 2800 T would have given m* = 6.3 ± 0.5 me, see Supplementary Fig. 6), the Fermi surface reconstruction scenario is required by the T-dependence of the low-frequency oscillation. Note that the evolution of the quantum oscillation spectrum versus temperature has been reproduced in another sample at doping level p = 0.102 (see Supplementary Fig. 7 and Notes 6). The comparison of quantum oscillations in Hg1223 at different doping levels from p = 0.080 to p = 0.112 is shown in Supplementary Fig. 8.

Fig. 2: Quantum oscillation measurements in Hg1223.
figure 2

a Field dependence of the TDO frequency after the heterodyne circuit in Hg1223 (p = 0.112) at various fixed temperatures, as indicated. b Oscillatory part of the TDO signal after removing a smooth background (spline) from the data shown in panel (a). −Δf is plotted and the curves have been offset for clarity. At T = 4.2 K, only two oscillations are detected due to the small field window above the superconducting transition. Their amplitude does not increase significantly with the magnetic field, which is likely due to the presence of closely spaced frequencies and a beating-like effect. c Discrete Fourier analysis of the oscillatory part of the TDO signal shown in panel (b) in the field range 70 T ≤H≤ 87 T. The black (red) arrow marks the low (high) frequency observed at T = 4.2 K (T = 1.8 K). The broadening of the Fourier transform at low frequency comes from the small number of oscillations in the field range. The inset shows a zoom of the Fourier transform between F = 2 kT and F = 4 kT, where one can clearly see the emergence of two peaks at F = 2100 T and F = 2800 T (red arrow).

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Theory

Our results demonstrate that a crossover takes place at low temperatures for p = 0.10–0.12 when superconductivity is quenched by a magnetic field. To gain insight, we numerically simulated quantum oscillations of the density of states for a single CuO2 trilayer. Since the detailed band structure of underdoped Hg1223 is unknown, the trilayer model was adapted from existing tight-binding models for related multilayer compounds (see Supplementary Note 1). A layer-dependent quasiparticle scattering rate, which modifies QO amplitudes but not frequencies, was incorporated. We then introduced, by hand, commensurate AFM and CDW order parameters. The AFM and CDW magnitudes, and the CDW periodicity, are the main tuning parameters in our model, and they were adjusted to give plausible oscillation spectra at each temperature.

We found that CDW order alone is insufficient, but that the addition of AFM order in the inner layer explains the measured oscillations. The required AFM order parameter, M ≲ 50 meV, is small compared to the values M ≳ 100 meV inferred from experiments at slightly smaller doping36,39. Consistent with this substantial change of energy scales, NMR experiments on related multilayer cuprates37 find that the Néel temperature collapses rapidly near the end of the metallic AFM phase. Taken together, these suggest that the current doping is close to the AFM quantum critical point. We introduce CDW order parameters Δi and Δo for the inner and outer layers, respectively. The CDWs are biaxial, with wavevectors of magnitude q = 2π/4.5a0, and with a bond-centered d-wave form factor. The true form factor is more complex47, but would not change our results qualitatively. Similarly, the calculated QOs are not strongly dependent on the value of Δo, which we cannot therefore establish reliably. Conversely, our results are highly sensitive to both q and Δi. We include scattering rates Γo,k and Γi,k for the outer and inner layers, respectively. We assume that Γo,k is impurity-dominated, and, therefore, isotropic and T-independent, while Γi,k is anisotropic—either due to pseudogap physics or to order parameter fluctuations—and decreases with decreasing T.

The density of states, N0(H), is obtained at the Fermi energy as a function of magnetic field using a numerical recursion method for large supercells (see Supplementary Note 2), for the parameters in Table 1. Figure 3a shows that the oscillations at 4.2 K are dominated by a single frequency, while multiple frequencies are evident at 1.8 K. Fourier transforming with respect to 1/H reveals a strong low-frequency peak at F ≈600 T at 4.2 K, along with harmonics at 1200 and 1800 T [Fig. 3b]. The peak weakens and shifts to a higher frequency, F ≈800 T, at 1.8 K. Additional peaks, most notably at 1100 and 2700 T, emerge at low T. These frequencies agree surprisingly well with the experiments.

Table 1 T-dependent order parameters and scattering rates used to simulate quantum oscillations at T = 4.2 K and T = 1.8 K
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Fig. 3: Simulation of the density of states in Hg1223.
figure 3

a Calculated density of states N0(H) at the Fermi energy as a function of magnetic field H for T = 4.2 K and T = 1.8 K and b its Fourier transform, using the parameters in Table 1. The inset shows an expanded view of the high-frequency region. Arrows indicate the low- (black) and high-frequency (red) peaks and their harmonics (gray). c, d Show simplified Fermi surfaces for the 4.2 and 1.8 K, respectively. Quantum oscillation frequencies calculated from the Fermi surface areas are listed next to each panel.

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The reconstructed Fermi surfaces are too complicated to provide a semiclassical explanation of the peaks (see Fig. 1 of the Supplementary Information). Simplified Fermi surfaces [Fig. 3c, d)] show that the main peak at 4.2 K comes from hole pockets (blue) generated by AFM order; the other Fermi surfaces (solid red and green) are unchanged by AFM because they have primarily outer-layer character, but are reconstructed by the outer-layer CDW to form electron pockets (dashed red and green). By assumption, the inner-layer order parameters M and Δi depend on T. Consequently, the main peak’s frequency increases between 4.2 K and 1.8 K because the hole pocket area grows as M decreases; the peak’s height decreases because of increased quasiparticle scattering by the CDW (Δi grows as T decreases) and magnetic breakdown when M is small. Ultimately, Δi reconstructs the hole pocket at low T to form a diamond-shaped electron pocket [dashed blue in Fig. 3d] with frequency, F ~2700 T. This provides an explanation for the high-frequency peak seen experimentally [Fig. 2c]. Other low-T peaks in Fig. 3b, at 1100 and 2100 T, are clearly tied to the growth of Δi, but are difficult to ascribe to a single semiclassical orbit.

Discussion

Our calculations support the scenario shown in Fig. 4. Above 10 K, inner-layer AFM order coexists with short-range outer-layer CDW order. Below 10 K, the CDW is sufficiently long-range that the outer-layer Fermi surfaces are reconstructed and generate an electron-like Hall coefficient; however, the relatively large quasiparticle scattering rate in the outer layers damps the corresponding quantum oscillations. (In our simulations, the outer layers account for less than 10% of oscillation amplitude.) Though centered on different layers, the AFM and CDW orders compete because the layers are coupled, and the CDW order gradually replaces the AFM order on the inner layer. In our simulations, the shifts of the main peak height and frequency, and the appearance of a high-frequency peak are intimately connected to this crossover. While zero-field charge order was detected in Hg122348, we cannot exclude that it is enhanced by the magnetic field, as in YBCO49.

Fig. 4: Sketch of the Fermi surface.
figure 4

a Sketch of the trilayer structure of Hg1223 in different temperature ranges. SR(LR)-CDW stands for short-range (long-range) CDW and AF corresponds to antiferromagnetism. b Sketch of the Fermi surface corresponding to the sequence of crossover as the temperature is reduced (see text). qa = (q, 0) and qb = (0, q) are the wavevector of the biaxial CDW. Q is the AFM wavevector.

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The results reported here have a remarkable implication. In our simulations, the CDW wavevector q was chosen to explain the low-T oscillation peak at ~2800 T. The CDW hotspots implied by this q-value are, within the resolution of our simulations, coincident with the AFM hotspots, i.e., points on the Fermi surface connected by the AFM wavevector \({\bf{Q}}=(\frac{\pi }{{a}_{0}},\frac{\pi }{{a}_{0}})\) (Fig. 4 bottom). This coincidence is a key feature of models in which CDW order is mediated by critical spin fluctuations carrying momentum Q. In contrast, the CDW order tends to connect points away from the AFM hotspots when it emerges on top of a pre-existing AFM state20,50. As discussed above, we infer from our measurements that the AFM quantum critical point is nearby. Our measurements are thus suggestive that charge order at this doping level is mediated by critical spin fluctuations.

One possible counter-argument is that the superconducting and long-range CDW transition temperatures differ by an order of magnitude, contrary to early calculations suggesting they should be the same at the AFM quantum critical point13,14; however, more recent work showed that the predicted degeneracy breaks down when the band structure lacks particle-hole symmetry19, as in Hg1223. Another possible counter-argument is that CDW order persists in YBCO51, in La2−xSrxCuO452, and in overdoped Tl2Ba2CuO6+δ53 to doping levels far from the AFM quantum critical point. It is unlikely that the AFM hotspot picture makes sense in these doping regimes, and other (noncritical) interactions are likely important. The question posed by our work is whether the hotspot picture makes sense close to the AFM quantum critical point. In this regard, it would be interesting to determine the evolution of the CDW wavevectors with doping because the CDW hotspots should be pinned to the AFM Brillouin zone boundary wherever critical spin fluctuations are the dominant interaction.

The fact that the CDW and AFM Fermi surface hotspots coincide in our simulations provides a natural explanation for the competition between the two phases: both phases derive part of their condensation energy from the gap they open at the hotspot and the phase with the largest condensation energy will form preferentially. Our model suggests that these energies must be roughly equal at 4.2 K, and that there is growth of the outer-layer CDW below 4.2 K that tips the balance in the inner layer toward the CDW phase. Since the order parameters reported in Table 1 are directly related to the hotspot gaps, it is tempting to make naive statements about the energetics of the competition between the CDW and AFM phases. Noting that the d-wave character of the CDW means that the CDW gap near the antinode is roughly twice the value shown in the table, and that Δo is not well-established by our simulations and could be significantly larger, it is plausible that the AFM and CDW phases have similar gap magnitudes. However, these arguments miss the important point that the CDW and AFM condensation energies have distinct dependencies on the Fermi surface geometry (unlike BCS superconductors, which depend only on the density of states). Simple arguments about the energetics of the two phases are, therefore, not possible, and a proper microscopic calculation is required.

In conclusion, we have measured the evolution of the Fermi surface at low temperatures in Hg1223 samples with p = 0.10–0.12, and demonstrated that there is a Fermi surface reconstruction at low temperature. Through numerical simulations, we attribute this to competition between CDW and AFM order. The CDW wavevectors required to reproduce the experimental oscillation spectrum suggest that the CDW and AFM phases share Fermi surface hotspots. This naturally explains the competition between the two phases, but is also a hallmark of CDWs that are mediated by critical spin fluctuations.

Methods

Samples

Single crystals of the trilayer cuprate HgBa2Ca2Cu3O8+δ have been synthesized using a self-flux growth technique as described in ref. 54. Using adequate heat treatment, Hg1223 can be largely underdoped and its doping level controlled. The doping p has been deduced from the empirical relation 1 − Tc/Tc,max = 82.6(p−0.16)2, where Tc is the onset superconducting transition measured by SQUID (see Supplementary Fig. 3) and Tc,max = 133 K. The list of samples studied by transport or TDO are shown in Table 2 with their Tc and corresponding hole doping. The results for the lowest doping p = 0.089 are from ref. 36.

Table 2 Tc, hole doping and probe measurement for the five Hg1223 samples measured in this study
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TDO measurements

Quantum oscillation measurements using the tunnel diode oscillator technique55,56 were performed at the pulsed-field facility in Toulouse (LNCMI-T) up to 88 T in two samples of Hg1223 at doping levels p = 0.102 (Tc = 96 K) and p = 0.112 (Tc = 108 K). Typical sample dimensions are 500 × 500 × 100 μm3. The experimental setup consists of an LC-tank circuit powered by a tunneling diode oscillator biased in the negative resistance region of the current-voltage characteristic. The sample is placed in a compensated 8-shape coil (the diameter and length of the coil are adapted for each sample to optimize the filling factor). The fundamental resonant frequency f0 of the whole circuit is about 25 MHz. The RF signal is amplified and demodulated down to a frequency of about 1 MHz using a heterodyne circuit. A high-speed acquisition system is used to digitize the signal. The data are post-analysed using a software to extract the field dependence of the resonance frequency fTDO, which is sensitive to the resistivity through the change in skin depth.

Hall effect measurements

The Hall effect was measured at the LNCMI-T in two distinct samples of Hg1223 at similar doping levels, p = 0.101 (Tc = 95 K) and p = 0.118 (Tc = 114 K). The magnetic field H was applied along the c-axis of the tetragonal structure, perpendicular to the CuO2 planes for both field polarities. Data for the sample p = 0.089 is from ref. 36. Typical sample dimensions are 700 × 400 × 90 μm3. Gold contacts were sputtered onto the surface of the sample before a heat treatment leading to contact resistances of a few ohms at room temperature and below 1 Ω at low temperature. The measurements were performed up to 86 T in a dual coil magnet using a conventional four-point configuration with a current excitation of ≈5 mA at a frequency of ≈60 kHz. A high-speed acquisition system was used to digitize the reference signal (current) and the voltage drop across the sample at a frequency of 500 kHz. The data were post-analyzed with software to perform the phase comparison.

Calculations

We numerically simulated quantum oscillations of the density of states for a tight-binding model of a single CuO2 trilayer using a numerical recursion method for large supercells (see Supplementary Notes 1, 2 for a detailed description).