Universal Curvature Force on Dislocations from a Cartan Geometric Defect Action

TL;DR

This work proposes a unified Cartan-geometric framework in which lattice defects are described by a Riemann–Cartan geometry: dislocations correspond to torsion $T^a$ and disclinations to curvature $R^a{}_b$, while phase defects are captured by a U(1) gauge sector. A defect action $S[e,\omega] = \int [ \alpha T^a \wedge *T_a + \beta R^a{}_b \wedge *R^b{}_a + \gamma e^a \wedge R_{ab} \wedge e^b ]$ with a crucial mixed term induces dynamic coupling between torsion and curvature, yielding two Euler–Lagrange equations that govern defect dynamics. Key results include canonical field configurations for screw/edge dislocations and wedge disclinations, a universal curvature-induced Magnus-like transverse force on moving dislocations, and curvature-mediated reconnection/annihilation of dislocation lines, all without ad hoc inputs; coupling to U(1) phase defects enriches the framework to describe vortex–defect interactions. The approach offers a principled, gauge-theoretic route to modeling defect networks with potential experimental tests in colloidal crystals, mechanical metamaterials, and strained 2D materials, enabling first-principles predictions of defect dynamics and interactions.

Abstract

We develop a unified Cartan geometric framework where dislocations and disclinations correspond to torsion and curvature of the material coframe connection, respectively, and phase defects emerge as U(1) vortices. This single action principle produces coupled equations of motion and conservation laws governing these defects. Our theory predicts a universal Magnus-like force exerted by curvature on moving dislocations, as well as disclination-driven reconnection events. These phenomena offer experimentally testable signatures in colloidal crystals and mechanical metamaterials.

Figures (5)

• Figure 1: Schematic geometry of coframe $e^a$ and spin connection $\omega^{ab}$.
• Figure 2: (a) Torsion density $T^3(x,y)$ for a screw dislocation, approximated by a narrow Gaussian to visualize the core; the analytic result is $T^3=b\,\delta^{(2)}(r)\,dx\wedge dy$. (b) Helical displacement illustrating $e^3=dz+\tfrac{b}{2\pi}d\theta$ and $z(\theta)=\tfrac{b}{2\pi}\theta$ (two full turns shown). The central axis denotes the defect line.
• Figure 3: Geometric displacement field of edge dislocation. Quiver plot shows the in-plane displacement field $\mathbf{u} = (u_x, u_y)$ from the coframe Eq. \ref{['E32']}. The red dot marks the dislocation core at the origin. The field exhibits characteristic $1/r$ decay with circulation corresponding to Burgers vector $\mathbf{b} = b\hat{x}$.
• Figure 4: Connection $\omega^1_2 = \Theta d\theta$ for wedge disclination. The circulatory field around the core (red dot) produces curvature $R^1_2 = 2\pi \Theta \delta^{(2)}(r)$ upon differentiation.
• Figure 5: Magnus-like configurational force on a moving dislocation. The color map shows the magnitude of the scalar product $|\mathbf{F}_\perp \cdot \mathbf{v}|$ of the Magnus-type force and velocity. The thick blue contour marks the locus where the scalar product is exactly zero, demonstrating that the configurational force is strictly transverse to the dislocation velocity for all directions in the slip plane. Black arrows indicate sample velocity directions $\mathbf{v}$ for reference.