Spin susceptibility in bilayered cuprates: Resonant magnetic excitations

We study the momentum and frequency dependences of the dynamical spin susceptibility in the superconducting state of bilayer cuprate superconductors. We show that there exists a resonance mode in the odd as well as the even channel of the spin susceptibility, with the even mode being located at higher energies than the odd mode. We demonstrate that this energy splitting between the two modes arises not only from a difference in the interaction, but also from a difference in the free-fermion susceptibilities of the even and odd channels. Moreover, we show that the even resonance mode disperses downward at deviations from $\mathbf{Q}=(\pi ,\pi )$. In addition, we demonstrate that there exists a second branch of the even resonance, similar to the recently observed second branch (the $Q^{*}$ mode) of the odd resonance. Finally, we identify the origin of the qualitatively different doping dependence of the even and odd resonances. Our results suggest further experimental tests that may finally resolve the longstanding question regarding the origin of the resonance peak.

Figure 1

(Color online) Calculated Fermi surface for the bilayered cuprates as obtained from Eq. (2). The arrows indicate the transition between bonding-bonding $\left(bb\right)$, antibonding-antibonding $\left(aa\right)$, and antibonding-bonding $(ab,ba)$ states for antiferromagnetic wave vector $Q=(\pi ,\pi )$.

Figure 2

(Color online) (a) $\mathrm{Im}{\chi }_0^{e,o}$, (b) $\mathrm{Re}{\chi }_0^{e,o}$, and (c) $\mathrm{Im}{\chi }_{\mathrm{RPA}}^{e,o}$ as a function of frequency at the antiferromagnetic wave vector $\mathbf{Q}=(\pi ,\pi )$ at optimal doping. Here, we use $g_0=0.55\mathrm{eV}$.

Figure 3

(Color online) RPA results for magnetic excitations in a bilayered $d_{x^2-y^2}$ superconductor at optimal doping. Calculated $\mathrm{Im}{\chi }^o$ and $\mathrm{Im}{\chi }^e$ obtained from Eq. (8) for $g_0=0.55\mathrm{eV}$ as a function of momentum along the diagonal $[\mathbf{q}=\eta (\pi ,\pi )]$ and bond $\left[q_x(q_y=\pi )\right]$ direction and frequency in the SC state.

Figure 4

(Color online) (a) Doping dependence of (a) the resonance frequency at $\mathbf{Q}$ in the odd and even channels, and (b) the superconducting gap ${\Delta }_0$ and $g_0$. [(c)–(f)] Dispersion of the even and odd modes for various doping concentrations in the (c) underdoped, (d) optimally doped, and [(e) and (f)] overdoped regime.

Figure 5

(Color online) $\mathrm{Re}{\chi }_0^{e,o}(\mathbf{Q},0)$ as a function of doping concentration in the normal state.