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Orbital Magnetization of Interacting Electrons

Xi Chen, Zhi-Da Song

Abstract

We derive an exact expression for the orbital magnetization of electrons with short-range interactions (such as density-density interactions) in terms of exact zero-frequency response functions of the zero-field system. The result applies to weakly and strongly correlated electrons at zero and finite temperature, provided that the local grand potential density only depends on local thermodynamic parameters. We benchmark the formula for non-interacting and weakly-coupled electrons. To zeroth and first orders in the interaction strength, it agrees with the modern theory of orbital magnetization and its recent generalization to self-consistent Hartree-Fock bands. Our work provides an exact framework of interacting orbital magnetization beyond mean-field treatments, and paves the way for quantitative studies of strongly correlated electrons in external magnetic fields.

Orbital Magnetization of Interacting Electrons

Abstract

We derive an exact expression for the orbital magnetization of electrons with short-range interactions (such as density-density interactions) in terms of exact zero-frequency response functions of the zero-field system. The result applies to weakly and strongly correlated electrons at zero and finite temperature, provided that the local grand potential density only depends on local thermodynamic parameters. We benchmark the formula for non-interacting and weakly-coupled electrons. To zeroth and first orders in the interaction strength, it agrees with the modern theory of orbital magnetization and its recent generalization to self-consistent Hartree-Fock bands. Our work provides an exact framework of interacting orbital magnetization beyond mean-field treatments, and paves the way for quantitative studies of strongly correlated electrons in external magnetic fields.
Paper Structure (2 sections, 40 equations, 2 figures)

This paper contains 2 sections, 40 equations, 2 figures.

Table of Contents

  1. Calculation of $\tilde{M}_z$ for non-interacting electrons
  2. Diagrammatic calculation of $k_U =\partial \tilde{M}_z/{\partial U}|_{U=0, \mu}$

Figures (2)

  • Figure 1: Comparison between the first order derivative of auxiliary OM $\tilde{M}_z$ to interaction strength $U$ calculated from Hartree-Fock bands ($k_{HF}$) and the weak-coupling expansion ($k_U$), as well as the result from permuting $x\leftrightarrow y$ in \ref{['eq:OM']} plus an additional minus sign ($k_U^{\rm yx}$). Panels (a) and (b) show the agreement between $k_U$ and $k_{HF}$ at system size $L=40$ for a Chern insulator ($\mu = 0$, $C=-1$) and a metal ($\mu = 1.5$), respectively. We deliberately choose a relatively small $L$ to make the two lines distinguishable to the eye. The small difference is due to finite-size effect. Panels (c) and (d) display the finite-size scaling of relative differences at low temperature ($k_BT=0.1$) corresponding to (a) and (b) respectively, plotting the relative difference between $k_{HF}$ and $k_U$, as well as between $k_{HF}$ and $k_{U}^{\rm yx}$ against $L$. The data confirm that both differences vanish in the thermodynamic limit.
  • Figure S1: (a) A standard bare interaction vertex. (b-c) Hartree and Fock contributions to first order self-energy correction of the propagator. (d) The zeroth-order diagram for $\mathcal{I}$. First order self-energy correction to $\mathcal{I}$ should be dressing (b-c) to the two fermion lines in this diagram. (e-f) Bare vertex contribution to the first order result of $\mathcal{I}$. (g) Four different connected contractions for the zeroth order result of $\mathcal{J}$.