Relation between the one-particle spectral function and dynamic spin susceptibility of superconducting ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8-\delta }$

Angle-resolved photoemission spectroscopy (ARPES) provides a detailed view of the renormalized band structure and, consequently, is a key to the self-energy and the single-particle Green’s function. Here, we summarize the ARPES data accumulated over the whole Brillouin zone for the optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8-\delta }$ into a parametric model of the Green’s function, which we use for calculating the itinerant component of the dynamic spin susceptibility in absolute units with many-body effects taken into account. By comparison with inelastic neutron scattering (INS) data, we show that the itinerant component of the spin response can account for the integral intensity of the experimental INS spectrum. Taking into account the bilayer splitting, we explain the magnetic resonances in the acoustic (odd) and optic (even) INS channels.

Figure 1

(Color online) Comparison of the model with experimental ARPES spectra of optimally doped Bi-2212 at $30\mathrm{K}$ taken with [(a) and (b)] $50\mathrm{eV}$ and (c) $38\mathrm{eV}$ photon energies. The model spectra are smoothed with a Gaussian to account for $20\mathrm{meV}$ energy resolution and $0.025{\mathrm{\AA }}^{-1}$ angular resolution. (a) Spectra at the $(\pi ,0)$ point with the corresponding energy distribution curves (below) taken along the dashed lines. (b) Nodal spectra along the $(\pi ,\pi )$ direction. (c) Comparison of the experimental and model spectra taken at an intermediate position in $k$ space to check the validity of the interpolation of the self-energy between the nodal and antinodal directions. (d) The Kramers-Kronig consistent real and imaginary parts of the nodal and antinodal self-energies. (e) Positions of the cuts (a)–(c) in $k$ space.

Figure 2

(Color online) (a) Energy dependence of the real and imaginary parts of the Lindhard function ${\chi }_0$ at the $(\pi ,\pi )$ point for the odd and even channels. (b) Contributions of the normal (thin solid curves) and anomalous (dashed curves) components to the real and imaginary parts of ${\chi }_0^{\mathrm{o}}$ in the SC state. The sum of two components is shown as thicker curves. In our calculations we used the $\Gamma$ value in Eq. (2) of $5\mathrm{meV}$, which could introduce insignificant additional broadening of ${\chi }_0$ as compared to the bare band calculations. The energy integration range in Eq. (2) was chosen to be $\pm 0.25\mathrm{eV}$.

Figure 3

(Color online) (a) $\mathbf{k}$ integrated ${\chi }^{″}=\mathrm{Im}\left(\chi \right)$ in the odd (solid curve) and even (dashed curve) channels. (b) $\mathbf{k}$ dependence of the resonance energies in odd (*) and even $\left(▴\right)$ channels along the high-symmetry directions $(0,0)-(\pi ,\pi )-(\pi ,0)$. The dashed lines mark the onset of the particle-hole continuum (position of the “step” in ${\chi }_0^{″}$). Second row: Constant-energy cuts of ${\chi }^{″}$ in the odd channel (c) below the resonance, (d) at the resonance energy, and (e) above the resonance. The center of each BZ image corresponds to the $(\pi ,\pi )$ point.