Spin-orbital magnetic response of relativistic fermions with band hybridization

Spins of relativistic fermions are related to their orbital degrees of freedom. In order to quantify the effect of hybridization between relativistic and nonrelativistic degrees of freedom on spin-orbit coupling, we focus on the spin-orbital (SO) crossed susceptibility arising from spin-orbit coupling. The SO crossed susceptibility is defined as the response function of their spin polarization to the “orbital” magnetic field, namely, the effect of magnetic field on the orbital motion of particles as the vector potential. Once relativistic and nonrelativistic fermions are hybridized, their SO crossed susceptibility gets modified at the Fermi energy around the band hybridization point, leading to spin polarization of nonrelativistic fermions as well. These effects are enhanced under a dynamical magnetic field that violates thermal equilibrium, arising from the interband process permitted by the band hybridization. Its experimental realization is discussed for Dirac electrons in solids with slight breaking of crystalline symmetry or doping, and also for quark matter including dilute heavy quarks strongly hybridized with light quarks, arising in a relativistic heavy-ion collision process.

Figure 1

The loop diagram corresponding to the response function ${\mathrm{\Pi }}_{A_l}^{S_i}$ given by Eqs. (9) and (11), with the Matsubara formalism. The solid lines represent the fermion propagators, and the wavy line corresponds to the external field.

Figure 2

Band structure of the minimal hybridized model defined by Eqs. (53) and (54), for the energy offset (a) ${\epsilon }_0>0$ and (b) ${\epsilon }_0<0$. The parameters are taken as $m=3|{\epsilon }_0|$ and $h=0.2|{\epsilon }_0|$. The dashed lines show the bands without the hybridization $h$. ${\epsilon }_1$ and ${\epsilon }_2$ in (a) are the energies of the band crossing points in the absence of hybridization, which we shall use in later calculations.

Figure 3

Behavior of the SO crossed susceptibility $\mathrm{\Delta }{\chi }^{\mathrm{so}}\left(\mu \right)$, as functions of the Fermi energy $\mu$. The solid and dashed lines are the static and dynamical susceptibility, respectively, with the hybridization parameter $h$ varied as shown in the inset table. The vertical dashed lines $\left({\epsilon }_{1,2}\right)$ correspond to the band hybridization points, which are identical to those shown in Fig. 2. The parameters are taken as ${\epsilon }_0>0$ and $m=3{\epsilon }_0$.

Figure 4

The SO crossed susceptibilities separated into the Dirac sector ${\chi }_{\mathrm{Dirac}}^{\mathrm{so}}$ and the nonrelativistic (NR) sector ${\chi }_{\mathrm{NR}}^{\mathrm{so}}$, and their sum (Total, ${\chi }^{\mathrm{so}}$). Panel (a) shows the static susceptibilities, while panel (b) shows the dynamical susceptibilities. Both results are obtained with the offset of the NR band ${\epsilon }_0>0$, the effective mass for the NR band $m=3{\epsilon }_0$, and the hybridization parameter $h=0.2{\epsilon }_0$.