Commensurate antiferromagnetic excitations as a signature of the pseudogap in the tetragonal high-Tc cuprate HgBa2CuO4+δ

Antiferromagnetic correlations have been argued to be the cause of the d-wave superconductivity and the pseudogap phenomena exhibited by the cuprates. Although the antiferromagnetic response in the pseudogap state has been reported for a number of compounds, there exists no information for structurally simple HgBa2CuO4+δ. Here we report neutron-scattering results for HgBa2CuO4+δ (superconducting transition temperature Tc≈71 K, pseudogap temperature T*≈305 K) that demonstrate the absence of the two most prominent features of the magnetic excitation spectrum of the cuprates: the X-shaped ‘hourglass' response and the resonance mode in the superconducting state. Instead, the response is Y-shaped, gapped and significantly enhanced below T*, and hence a prominent signature of the pseudogap state.

where (gr e ) 2 = 0.2905 barn sr -1 , k f and k i are the final and incident neutron wave-vectors, and ⎜F(Q)⎜ 2 is the magnetic form factor. Supplementary Fig. 7c shows a constant energy image with peaks of intensity at q AF at ω = 53 meV. The scattering intensity is larger at Q = (0.5, 0.5, 4.55) than in higher (two-dimensional) Brillouin zones (Q = (0.5, 1.5, 6.34) and (1.5, 0.5, 6.34)), consistent with magnetic scattering, for which the form-factor is smaller at larger Q. After accounting for the anisotropic Cu 2+ d x 2 -y 2 form factor 2 , χ″ becomes equivalent in all zones ( Supplementary Fig. 7d).

Supplementary Note 3
Limits on the low-energy response. Supplementary Fig. 6 shows the temperature dependence of response at ω = 15 ± 5 meV and ω = 36 ± 3 meV. There is no discernible magnetic scattering above the concave background at ω = 15 meV, and we estimate conservative upper bounds of 15% and 8%, respectively, compared to the signal at ω = 36 meV and 51 meV. We note that χ″ 0 (ω) for ω > Δ AF extrapolates to zero at ω ~ 25 meV ( Fig. 3a; 5 K data), consistent with Δ AF = 27 meV, which is defined as the energy below which we can no longer observe a peak at Q AF .

Supplementary Note 4
Polarized neutron scattering. Spin-polarized measurements ( Supplementary Fig. 7a,b) were carried out on the triple-axis spectrometer IN20 at the Institute Laue Langevin.
Heusler alloy crystals were used as monochromator and analyzer to select the initial and final neutron energies and spin polarizations. The polarization of the neutron beam in the vicinity of the sample was maintained by CryoPAD, which provides high stability and reproducibility of the neutron spin polarization 4 .
For longitudinal polarization analysis, it is convenient to define the coordinate system based on the relative orientations of the neutron spin polarization (P) at the sample, the scattering wave-vector Q, and the scattering plane that contains Q: the three orthogonal axes are defined by P||Q and P⊥Q in the scattering plane, and P||z, where z is the direction perpendicular to the scattering plane. In the absence of chiral magnetic correlations, the measured spin-flip (SF) and non-spin-flip (NSF) scattering intensities in the three principal spin-polarization geometries are given by the following relations 1 : where N inc,isotope and N inc,spin are the nuclear isotope and spin incoherent cross sections, respectively, N coh is the coherent nuclear cross section, M is the magnetic cross section, and BG is the background contribution. Supplementary Fig. 7a shows excess scattering in !∥! !" above the background level at the two-dimensional AF wave vector at ω = 48 meV. We attribute this to scattering from magnetic fluctuations. To further confirm this, we measure all six SF and NSF neutron polarization configurations for select wave vectors and extract the pure magnetic intensity from both SF and NSF channels: Supplementary Fig. 7b shows unambiguous evidence of magnetic scattering at q AF from both SF and NSF scattering.

Supplementary Note 6
Temperature dependence of the commensurate response. Supplementary Fig. 10 shows the evolution of the low-energy magnetic response across the temperatures T c = 71 K and T CDW = 200 K (ref. 7) for HgUD71. In order to reduce background contributions, 18 the scattering at 350 K is first subtracted from that at 5 K, 85 K and 220 K ( Supplementary Figs. 10a,b,c). The gapped commensurate q AF response is already observed in the (H,1/2) vs. energy slices. The background subtraction procedure described in Supplementary Note 1 is then applied at all energies to more clearly isolate the AF fluctuations in Supplementary Figs. 10e,f,h. Besides the slightly broader response at 5 K described in the main text, the overall commensurate spectrum remains largely impervious to the onset of superconductivity and CDW order. This is also apparent from the line cuts of the data in Supplementary Figs. 10i. Supplementary Figs. 10d,h shows the enhancement of the response between T CDW and T c . The lack of significant changes in the Q-ω dependence or in the scattering intensity (Fig. 1a) across T CDW and T c highlights that the commensurate low-energy spectrum is a signature of the PG state which has higher characteristic temperature T* (T c < T CDW < T*).