Zero modes and geometric phase for 2D Weyl fermions on Lifshitz backgrounds

TL;DR

The paper analyzes massless Weyl fermions on a 2+1D Lifshitz background with metric anisotropy, identifying curvature-induced geometric phases via the spin connection and a SUSY-based framework for zero modes. Using the Dirac phase method, the authors derive a holonomy operator $U(\mathcal{C})$ and a Wilson loop $W(\mathcal{C})$ that depend on the dynamical exponent $z$, scale $\ell$, and path period $T$, revealing curvature-driven valley mixing through a non-Abelian phase. The Weyl equations are recast into radial SUSY-like equations with central potentials $V_{\pm}(r)$ and a superpotential $W(r)$, yielding an exact, normalizable zero mode $\tilde{\psi}_0(r)$ for $\alpha<1$ where $\alpha=1-1/z$ and localization is controlled by $\tilde{\kappa}$. These results bridge holographic Lifshitz gravity with planar condensed-matter phenomena, suggesting curvature-induced transport effects and edge-state-like behavior in 2D Weyl systems, with potential connections to Hall physics and Aharonov–Bohm-type phenomena in curved geometries.

Abstract

Here we investigate analytical properties of Weyl fermions in (2+1)-dimensional Lifshitz spacetimes. In particular, we are interested in obtaining geometric phases and verifying the existence of well-behaved fermionic zero modes. Using the Dirac phase method, we show how geometric phases naturally arise from the coupling between the fermionic fields and the Lifshitz geometry. We also present exact solutions of the zero modes by rewriting the Weyl equation as a system of supersymmetric equations.