Field-induced spin exciton doublet splitting in $d_{x^2-y^2}$-wave $\mathbf{Ce}M{\mathbf{In}}_5(M=\mathbf{Rh},\mathbf{Ir},\mathbf{Co})$ heavy-electron superconductors

We investigate the spin exciton modes in the superconducting $d_{x^2-y^2}$ state of $\mathrm{Ce}M{\mathrm{In}}_5$ heavy-fermion compounds found at the antiferromagnetic wave vector by inelastic neutron scattering. We present a theoretical model that explains the field dependence for both field directions. We show that the recently observed splitting of the spin exciton doublet in ${\mathrm{CeCoIn}}_5$ into two nondegenerate modes for an in-plane field appears naturally in this model. This is due to the spin anisotropy of $g$ factors and quasiparticle interactions, which lead to different resonant conditions for the dynamic susceptibility components. We predict that the splitting of the spin-resonance doublet becomes strongly nonlinear for larger fields when the energy of both split components decreases. For a field along the tetragonal axis, no splitting but only a broadening of the resonance is found in agreement with the experiment.

Figure 1

(a) Fermi surface and (b) quasiparticle density of states for a zero magnetic field, here, $2{\Delta }_1=0.56{\Delta }_0$ and $2{\Delta }_2=1.5{\Delta }_0$. (c) and (d) show Fermi surfaces for spin up and spin down, respectively, in the presence of the magnetic field $B_0\widehat{z}$, corresponding to Zeeman energy splitting $h_{B_0}^f=g_{\parallel }^f\tilde{\chi }{\mu }_B{B}_0=0.3{\Delta }_0$, which is also used in subsequent figures.

Figure 2

Individual components [${\chi }_0^{l{l}^{\prime }}(\mathbf{Q},\omega )$] and total [${\chi }_0^{\mathrm{tot}}(\mathbf{Q},\omega )$] of the unperturbed pseudospin susceptibility in (a) the absence, (b) the presence of the magnetic field along the $\widehat{z}$ direction, and for (c) the magnetic field along the $\widehat{x}$ direction. ($B_0$ is the same as Fig. 1.)

Figure 3

The total unperturbed physical susceptibility (including matrix elements $m_{\parallel },m_{\perp };{\chi }^{\mathrm{tot}}(\mathbf{Q},\omega )={\chi }^{zz}+\frac{1}{2}[{\chi }^{\pm }+{\chi }^{\mp }]$) for the zero field and finite field along the $\widehat{z}$ and $\widehat{x}$ directions. ($B_0$ is the same as in Fig. 1.)

Figure 4

RPA susceptibility in the absence and presence of the magnetic field: (a) along the $\widehat{z}$ direction and (b) along the $\widehat{x}$ direction. Parameters corresponding to the behavior as observed in CeCoIn${}_5$ with $g_f^{\perp }/g_f^{\parallel }=2.3$ and $J_{\mathbf{Q}}^{\perp }=2.6{\Delta }_0;J_{\mathbf{Q}}^{\parallel }=13.5{\Delta }_0$. ($B_0$ is the same as in Fig. 1.)

Figure 5

RPA susceptibility peak positions corresponding to spin-resonance energies as function of field strength: (a) along the $\widehat{z}$ direction and (b) along the $\widehat{x}$ direction. Anisotropic $g$ factors with $g_f^{\perp }/g_f^{\parallel }=2.3$ and quasiparticle interaction parameters $J_{\mathbf{Q}}^{\perp }=2.6{\Delta }_0;J_{\mathbf{Q}}^{\parallel }=13.5{\Delta }_0$, leading to similar behavior as in CeCoIn${}_5$ for both field directions. Here, the maximum $h_B^f=0.7{\Delta }_0$ corresponds to a field $B=39.7/\left(g_f^l\tilde{\chi }\right)$ in Teslas for direction $l=\perp ,\parallel$.