Emergent curved space and gravitational lensing in quantum materials

Abstract

We show that an effective gravitational field naturally emerges in quantum materials with long-wavelength spin (or pseudospin) textures. When the itinerant electrons' spin strongly couples to the background spin texture, it effectively behaves as a spinless particle in a curved space, with the curvature arising from quantum corrections to the electron's spin orientation. The emergent curved space gives rise to the electron lensing effect, an analog of the gravitational lensing. The lensing effect can appear in systems without (emergent) magnetic fields, such as those with coplanar spin textures. Our work shows that novel ``gravitational'' phenomena generically appear in quantum systems due to nonadiabaticity, opening new research directions in quantum physics.

Figures (7)

• Figure 1: Schematic illustration of emergent curved space: a spinful electron in a slowly varying spin texture is equivalent to a spinless electron in a curved space, whose metric encodes the "stretching" of real-space distances by the spin texture.
• Figure 2: The radial spiral spin texture (Eq. \ref{['eq:spin_cone']}) and a corresponding electron geodesic (solid curve). The effectively curved space gives rise to gravitational lensing of electrons around the origin.
• Figure 3: (a) Two geodesics on a cone. (b) Deflection due to gravitational lensing when the curved metric and the curvature are localized in a gray region. The deflection angle $\varphi$ is given by the curvature $K$ enclosed by the blue and orange lines.
• Figure 4: Example geodesics of the cone geometry with $\nu = 0.3$ (corresponding to a deficit angle $\approx 0.24\pi$) from Eq. \ref{['eq:geodesics_v2']}. These satisfy $r_0 = 2, \theta_0 = \pi/2$, and varying $\phi_0$. As the geodesics approach the origin, they become the union of two rays.
• Figure 5: Geodesics on a cone and the corresponding net. The left panel shows geodesics on the cone intersecting at points A and B, with interior angles $\theta_1$ and $\theta_2$. The right panel shows the net obtained by cutting the cone along line OA, where points A$_1$ and A$_2$ both represent the same point A on the cone. The angle $\Delta\Theta$ denotes the cone’s angle deficit.