Is there a conflict between causality and diamagnetism?

TL;DR

The paper addresses whether causality, as encoded in the Kramers-Kronig relations, forbids diamagnetism. It shows that the apparent conflict arises from incomplete accounting of the full multipole content in linear response; when electric dipole, magnetic dipole, electric quadrupole, diamagnetic, and crucially electric dipole–octopole channels are included, the resulting $\mathrm{Im}[\varepsilon(\omega)\mu(\omega)]$ remains nonnegative and $\mu(\omega)\to 1$, $\varepsilon(\omega)\to 1$ as $\omega\to\infty$ due to Thomas–Reiche–Kuhn sum rules. The mechanism assigns the diamagnetic response to an instantaneous angular-momentum term $\left\langle \mathbf{L}_{\text{dia}}\right\rangle = \int d^3x\, \mathbf{x}\times[\mathbf{E}^{\parallel}\times\mathbf{B}]$, ensuring causality is preserved even when $\mathrm{Im}\chi(\omega)$ is negative in some bands. This resolves the paradox for insulators and informs macroscopic QED modeling of diamagnetic media and metamaterials, with potential extensions to conductors in weak-field regimes.

Abstract

There is a long-standing apparent conflict between the existence of diamagnetism and causality as expressed through the Kramers-Kronig relations. In essence, using causality arguments, along with a small number of seemingly well-justified assumptions, one can show that diamagnetism is impossible. However, experiments show diamagnetic responses from magnetic media. We present a resolution to this issue, which also explains the absence of observed dia-electric responses in media. In the process, we expose some of the short-comings in earlier analyses that have kept the paradox alive.

Figures (2)

• Figure 1: Energy level diagram for two example, off-resonant, transitions. In this example, transitions between $\left|{g}\right\rangle$ and $\left|{e_1}\right\rangle$ can be mediated by both the electric dipole operator, as well as the electric octopole operator. A dipole transition is, however, not possible between $\left|{g}\right\rangle$ and $\left|{e_2}\right\rangle$, which is dominated by a combination of the magnetic dipole and electric quadrupole operators. All these combinations are required for a self-consistent model of the diamagnetic response.
• Figure 2: (a) Illustrative example of the permeability $\mu(\omega)$ for a strongly diamagnetic medium, using $\omega_{eg}=\{1,2,3\}$, $\Delta^{eg}_\text{e-dip} = \{2,0,0\}$, $\Delta^{eg}_\text{m-dip} = \{0,1/16,0\}$, $\Delta^{eg}_\text{quad} = \{0,0,1/64\}$, $\Delta^{eg}_\text{dip-oct} = \{65/64,0,0\}$, $\Delta^{eg}_\text{dia} = \{0,0,9\}$ and with $\gamma_{e} = \{0.18,0.05,0.04\}$, all in units of the plasma frequency $\omega_p = \sqrt{\rho_0 q^2/m}$. Here $\rho_0$ is the emitter density of the medium, $q$ the electron charge and $m$ is the mass. The exact form of the transition strength definitions can be found in Appendix \ref{['app:rotAvg']}, but we note that the coefficients are here chosen phenomenologically. Inset displays $\varepsilon(\omega)\mu(\omega)$ for the same medium. (b) Model atom-like medium as calculated from first principles, using all allowed transitions between $0<{n_x, n_y, n_z} \leq 6$, where $L_x = 0.2/2\pi \; \omega_p^{-1} = L_z$, $L_y = 0.4/2\pi \;\omega_p^{-1}$, $m = 500\,\omega_p$, $\rho_0 = 16000 \,\omega^3_p$ and $q = 1/\sqrt{32}$. Here all transitions have $\gamma_e = 0.15 \omega_p$. Note that this medium is weakly diamagnetic, as is expected for atoms.