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., ${\mathrm{BiF}}_3$, ${\mathrm{ZnI}}_2$, and ${\mathrm{Ru}}_4{\mathrm{Se}}_5$) 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 magnetoresistance. This quantum Christoffel nonlinear magnetization gives a paradigm of how geometry dictates physics.

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• Geometric & topological phases
• Quantum transport
• Two-dimensional electron system
• Methods in transport
• Wannier function methods