Orbital magnetization and magnetic susceptibility of interacting electrons

Abstract

We present a rigorous derivation of the orbital magnetization for interacting electrons within the self-consistent Hartree-Fock approximation. Our method also allows us to derive formulas for the orbital magnetic susceptibility. The results are expressed entirely in terms of the self-consistent wavefunctions and the Hartree-Fock energy spectrum at zero magnetic field. We find that the formula for the orbital magnetization is the same as in the non-interacting case, provided that the Hamiltonian and Bloch wave functions are replaced by their Hartree-Fock counterparts. By contrast, the orbital magnetic susceptibility contains an additional interaction-induced contribution that cannot be obtained by such a replacement. We test the formulas on an interacting Rashba model, finding an agreement with calculations performed at a small but non-zero external magnetic field.

Figures (2)

• Figure 1: Energy spectrum of Rashba-like continuum model. Colors represent eigenstate polarization $\langle \psi^{\dagger} \sigma_z \psi\rangle / \langle \psi^{\dagger} \psi\rangle$. (a) Non-interacting energy spectrum vs. momentum at $B=0$. (b) Non-interacting Landau level spectrum vs. $B$ field. (c) Finite temperature ($T$) HF energy spectrum at $B=0$ and fixed chemical potential $\mu$. The black dashed line marks $\mu$. (d) HF Landau level energy spectrum versus $B$ field at same $T$ and $\mu$. The oscillations within the spectrum are not numerical artifacts and come from the oscillatory solution to HF self-consistency equations. $T$ is chosen as low as $T = b_1^2/(380 k_B b_2)$. Parameters: $\{\Delta, U, \mu\} = \{1/4, 10 \hbar^2 b_2, 1.6816 b_1^2/b_2\}$.
• Figure 2: HF grand potential $\Omega$ per unit area $A_0 = \hbar^2 b_2^2/b_1^2$ vs. $B$ field, shown for (a) low $T$ and (b) intermediate $T$. In the small $B$ regime and for both temperatures, $\Omega$ approaches the linear extrapolation determined by $M$ (Eq. \ref{['eq:M_main_result']}) evaluated at $B = 0$. At larger magnetic fields, the inclusion of the quadratic correction associated with $\chi$ (Eq. (\ref{['Eqn:Chiformula']})--(\ref{['Eqn:ChiInt']})) at $B = 0$ further improves the agreement with $\Omega$, providing a non-trivial test of Eqs. (\ref{['eq:M_main_result']})--(\ref{['Eqn:ChiInt']}). The parameters used are identical to those in Fig. (\ref{['fig:intLLspectrum']}). The quantum oscillations visible at low $T$, manifested as the oscillatory behavior in $\Omega$, cannot be captured by the effective description based on zero-field $M$ and $\chi$ and are therefore beyond the scope of our present theory.