Quantum Christoffel Nonlinear Magnetization

Abstract

The Christoffel symbol is an essential quantity in Einstein's general theory of relativity. We discover that an electric field can induce a nonlinear magnetization in quantum materials, described by a Christoffel symbol defined in the Hilbert space of quantum states (quantum Christoffel symbol). Quite different from the previous scenarios, this orbital magnetization does not need spin-orbit coupling and inversion symmetry breaking. Through symmetry analysis and first-principles calculations, we identify a number of point groups and 2D material candidates (e.g., BiF$_3$, ZnI$_2$, and Ru$_4$Se$_5$) that host this quantum Christoffel nonlinear magnetization. More importantly, this nonlinear magnetization allows the quantum Christoffel symbol to be probed by optical techniques such as magneto-optical Kerr spectroscopy or transport measurements such as tunneling magneto-resistance. This quantum Christoffel nonlinear magnetization gives a paradigm of how geometry dictates physics.

Figures (3)

• Figure 1: In Einstein's general relativity, the Christoffel symbol $\Gamma_{ij}^k$ describe how basis vectors $\boldsymbol{e}_i$ transform across the coordinates $\boldsymbol{x}$ of the curved spacetime. We find a nonlinear magnetization $\mathbf{M}$ in response to an electric field $\mathbf{E}$ can be described by a quantum Christoffel symbol $\boldsymbol{\Gamma}$ in momentum space ($k_x$-$k_y$), in terms of $\mathbf{M}\propto \mathbf{E} \mathbf{E} \int \boldsymbol{\Gamma}$. A heuristic picture is as follows. One role of the electric field $\mathbf{E}$ is to induce a curved quantum space, described by $\boldsymbol{\Gamma}$. The other role of $\mathbf{E}$ is to drive a "loop current" of electron orbital motion in the curved quantum space, leading to the magnetization $\mathbf{M}$ that is quadratic in $\mathbf{E}$.
• Figure 2: (a) Lattice structure of the 2D BiF$_3$ grown along the [001] crystallographic direction. (b) Band structure of the 2D BiF$_3$ along the high symmetry line with and without spin-orbit coupling (SOC). (c) First-principles calculation results of the nonlinear magnetization coefficient $\alpha_{zxx}$ as a function of the Fermi energy $\varepsilon_F$, by using Eq. (\ref{['Eq: alpha_full']}) (with and without SOC) and also with the Haldane-like perturbation $\mathcal{H}'$ Eq. (\ref{['Eq: Hp']}) ($t'$ = 2 meV), respectively. In the calculation with $\mathcal{H}'$, the hopping parameter $t'$ = 2 meV, relaxation time $\tau$ = 100 ps, and additional electric field $\mathbf{E}$ = 0.02 V/$\mu$m. (d) First-principles calculation results of $\alpha_{zxx}$ as a function of the Fermi energy $\varepsilon_F$, by using Eq. (\ref{['Eq: alpha_CS']}) (Christoffel) and Eq. (\ref{['Eq: alpha_full']}) (Total), respectively. The non-Christoffel contribution is much smaller than the contribution described by the quantum Christoffel symbol. (e) Kerr rotation angle $\theta_\text{K}$ as a function of photon energy $\hbar \omega$, at Fermi level -0.25 eV and -0.10 eV, respectively. The Kerr angle measures how much a linearly polarized light is rotated after being reflected from the sample and is a direct probe of magnetization.
• Figure 3: Nonlinear magnetization coefficient $\alpha_{zxx}$ for the effective model [Eq. \ref{['Eq: model']}]. (a) $\alpha_{zxx}$ as a function of the Fermi energy $\varepsilon_F$ without particle-hole symmetry ($\lambda = 0.5$ eV$\cdot$nm$^2$), for the contribution from the quantum Christoffel nonlinear magnetization and non-Christoffel contribution. (b) $\alpha_{zxx}$ as a function of the particle-hole symmetry breaking parameter $\lambda$ at $\varepsilon_F = 0.2$ eV and $t = 0.9$ eV$\cdot$nm$^2$. (c) $\alpha_{zxx}$ as a function of $\varepsilon_F$ for the particle-hole symmetric case ($\lambda = 0$) at several values of $t$, which breaks the $\mathcal{C}_{x,y}^2$ rotational and $\mathcal{M}_{x,y}$ mirror reflection symmetries. (d) $\alpha_{zxx}$ as a function of $t$ at $\lambda = 0$ and $\varepsilon_F = 0.2$, 0.3, and 0.4 eV. Insets in (a) and (c) compare the band structures without and with particle-hole symmetry, respectively. Other model parameters are $v = 1$ eV$\cdot$nm and $m = 0.1$ eV.