Magnetotransport signatures of antiferromagnetism coexisting with charge order in the trilayer cuprate HgBa2Ca2Cu3O8+δ

Multilayered cuprates possess not only the highest superconducting temperature transition but also offer a unique platform to study disorder-free CuO2 planes and the interplay between competing orders with superconductivity. Here, we study the underdoped trilayer cuprate HgBa2Ca2Cu3O8+δ and we report quantum oscillation and Hall effect measurements in magnetic field up to 88 T. A careful analysis of the complex spectra of quantum oscillations strongly supports the coexistence of an antiferromagnetic order in the inner plane and a charge order in the outer planes. The presence of an ordered antiferromagnetic metallic state that extends deep in the superconducting phase is a key ingredient that supports magnetically mediated pairing interaction in cuprates.


Supplementary Note 2. Quantum oscillation data
The complete data of quantum oscillations in two samples of Hg1223 is displayed in Fig. 2.  Table 1 shows the effective masses of the different frequencies obtained from the temperature dependence of the amplitude of the oscillation given by the Fourier transform. It is worth to note that the background of the TDO signal is strongly field dependent. Since the amplitude of the QOs is small compared to this background, its subtraction is not straightforward, in particular in presence of several QO frequencies. The peaks at low frequency in the Fourier transform are artefacts coming from the fact that the oscillatory part of the TDO signal is not perfectly symmetric about the x-axis. Since we have a limited field range to analyse the QOs, a field window H =35 T -85.5 T corresponds to less than 2 oscillations for a frequency of F = 110 T. We cannot exclude the presence of such low frequency but it is impossible to reveal it by Fourier transform. A strong argument against the presence of such low QO frequency is the temperature dependence of the peak in the Fourier transform, that is not in agreement with the behaviour expected for QO.

Supplementary Note 3. Fitting quantum oscillations
In order to confirm the value of the frequencies obtained by Fourier transform, some fits of the QOs were performed. This is an additional check of our analysis leading to our interpretation. We fit the field dependence of the TDO signal to f T DO = (a 0 + a 1 H + a 2 H 2 + ...) + ∆f osc , where the first term is a polynomial representing the non-oscillating background and ∆f osc is given by the Lifshitz-Kosevich (LK) theory, where A 0i are prefactors, F i are the oscillation frequencies and γ i are the phase factors. We neglect any contribution from magnetic breakdown and spin damping.  [1]. Some of the scattering going into T D probably comes from scattering due to vortices, but we leave this as a field-independent scattering contribution for simplicity. For each frequency, there are five free parameters. In order to constrain the parameters, we perform simultaneous fits with equation (1) to the total data set at different temperatures, where all parameters are temperature independent.
We use a gradient based search algorithm where the optimal solution is the set of parameters that gives the smallest least squares value. The gradient based search algorithm we used converges only in the temperature range from T = 1.4 K to T = 4.2 K at p = 8 % and from T = 0.6 K to T = 2.9 K at p = 8.7 %. Due to local minima, there exist a large number of solutions and we have constrained a few parameters like the effective masses (to agree with the Fourier transform analysis) and the Dingle temperature (to take into account the disorder protected nature of the inner plane). Fig. 2 of the main text shows the raw data (symbols) for the p = 8 % sample along with the results of the fitting procedure (black lines) at different temperatures from T =1.4 K to T =4.2 K and in the field range H = 40 -83 T. Table 2 shows the relevant parameters deduced from the fit, e.g. frequencies, effective masses and Dingle temperatures. The contribution of each frequency to the total signal is seen in the DFT depicted in the inset of Fig. 2. In addition to the fundamental frequencies, .   2) and the p = 8.7 % sample (see Fig. 3).
T =0.6 K to T =2.9 K and in the field range H = 40 -85 T (see Fig. 3).

Supplementary Note 4. Hall effect data
The transverse Hall resistance R xy of our Hg1223 p = 8.8 % sample was measured in pulsed fields. Fig. 4 shows the isotherms of the Hall coefficient, where t is the thickness of the sample. Data in the temperature range T = 10 -150 K was obtained in a 68 T pulsed magnet, where the signal to noise ratio was good. However, at T = 10 K, 68 T was not enough to reach the normal state value of the Hall coefficient. We have thus performed additional measurements in a dual coil magnet to produce non-destructive magnetic fields up to 88 T at lower temperature, T = 4.2 K and T = 1.5 K. Due to larger dH/dt, the signal to noise ratio is worse but we can clearly state that R H remains positive down to the lowest temperature. This means that the Fermi surface of Hg1223 contains at least one mobile hole pocket at this doping level (see next section).

Supplementary Note 5. Two-band model
In the main text, we argue that one scenario compatible with the QO spectrum consists of antiferromagnetism (hole pocket) in the inner plane and charge order (electron pocket) in the outer plane (see Fig. 4b of the main text). Since the inner plane is protected from out-ofplane disorder, we assume that the hole-like carriers are more mobile than the electron-like carriers.
Using a simple Drude model in the low field limit, the Hall coefficient is given by: where σ and µ are the conductivities and mobilities, respectively. To calculate the carrier density, we rely on the measured QO frequencies. According to Onsager's relation, the size of the Fermi surface pocket A FS in reciprocal space is : where F is the SdH oscillation frequency. The number of carriers per layer is given by: where A FBZ is the size of the first Brillouin zone in the reciprocal space. In a unit cell there are three layers with one Cu atom per layer. We suppose that the Fermi surface of the outer planes contains a single electron pocket due Fermi surface reconstruction by charge order. The Fermi surface of the inner layer contains two hole pockets due to Fermi surface reconstruction by antiferromagnetism. Then the carrier density n h(el) of holes(electrons) is calculated according to the following formula: where V = 236.9Å 3 is the unit cell volume of Hg1223. As stated in the main text, a Hall coefficient R H ≈ 6.5 mm 3 /C (see Fig. 3) corresponds to a ratio of mobilities µ h /µ e =2.85, a reasonable value owing to the disorder-protected nature of inner plane compare to outer plane. Now we have to justify the low-field limit. This is in fact indicated in the data of Fig. 4 by the slight field dependence of R H . In the field-dependent two-band model, R H is given by: where σ h(e) and R h(e) are the conductivity and Hall number of the h(e) carriers, respectively: σ h(e) = en h(e) µ h(e) and R h(e) = 1 en h(e) .  Table 1) and the order of magnitude of the Dingle temperature for the p = 8.7 % sample (see Table 2). We use the parameters of the p = 8.7 % sample since Hall effect measurements have been performed in a sample with a similar doping level, namely p = 8.8 %. As seen in Fig. 5, we expect a slight departure from the low field limit since the Hall coefficient is not absolutely constant over our field range. Due to the signal-to-noise ratio at low temperature and the limited field range, it is hard to tell from the experimental data if such departure of the low field limit exists. Note that this model is very simple since for instance, it does not take into account the anisotropy of the scattering rate along the Fermi surface, which can be substantial in the cuprates. But one should keep in mind that the main target of the analysis is to get reliable value of the QO frequencies and explain the positive Hall number at low temperature.

Supplementary Note 6. Sketch of Fermi surface
The sketches presented in Fig. 4b of the main text and Fig. 6 are based on calculations of Fermi surface reconstruction due to AFM order and/or CDW order. For the sake of simplicity, we treat the inner and the outer planes as independent even if they are coupled by an interlayer tunneling t ⊥ . For the Fermi surface reconstruction, we consider only one band within each CuO 2 plane. The starting point of the calculations is the unreconstructed Fermi surface of the single-layer Hg1201 compound, parametrized by the tight-binding equation: where (t 1 , t 2 , t 3 , t 4 ) = (0.48, −0.105, 0.08, −0.02) eV, k x and k y are the in-plane wavevectors, a and b the in-plane lattice constants. The tight-binding parameters used in the calculations are chosen in order to reproduce the photoemission data reported in Hg1201 [2]. The chemical potential (µ) is adjusted to yield a doping level p = 8.5% (for µ = -0.355 eV) so that the total carrier density is equal to 1 + p holes, according to the Luttinger sum rule.
For the calculation of Fermi surface reconstruction due to AFM order, we used the following analytical formula for the electron (E e (k)) and hole (E h (k)) band structure: where V AF is the AFM potential, Q AF = (π, π) is the antiferromagnetic wavevector. The hole pocket (purple FS in Fig. 4) whose area corresponds to a frequency of 500 T is obtained using V AF = 0.445 eV.
For the calculation of Fermi surface reconstruction due to bi-axial charge order, only the first-order translations of the band-structure due to charge-ordering with a characteristic wavevector Q CDW are considered [3]. The resulting 4 × 4 Hamiltonian is then diagonalized numerically, with a typical mesh of 10 6 points. We choose Q CDW = 0.275 r.l.u and  obtained by local density approximation calculations [4]. It consists of three large hole-like tubular CuO 2 sheets centered on the corner of the Brillouin zone plus a small electron-like Fermi surface located at the anti-node. The latter disappears with doping [4]. The size of the large orbits translates into a QO frequency of about 15 kT, a value much larger than the observed frequencies.   Fig. 6c corresponding to a quantum oscillation frequency of F h = 850 T (F e = 500T) are obtained using V AF = 0.25 eV. They are compatible with the observed QO frequencies F 2 and F 3 . The third observed frequency (F 1 = 350 T) could then be due to magnetic breakdown between these two orbits. However, the magnetic breakdown field B 0 required to enable magnetic breakdown orbits is B 0 ∝ ∆k 2 where ∆k is the separation between the two orbits. This large gap precludes the observation of magnetic breakdown in the QOs. A simple estimate of the magnetic breakdown field can be achieved using the Blount criterion: where E F = e m * × F QO is the Fermi energy, E g is the energy gap due to AFM order, m * the effective mass and F QO the QO frequency. Using E g = 0.25 eV, m * = 1.6m 0 (1.2m 0 ) and F QO = 850 T (500 T) yields an estimated magnetic breakdown field B 0 ≈ 13000 T (14000T), more than two orders of magnitude larger than the magnetic field used in the present study.