Lorentz violation and momentum-space geometric phases

TL;DR

The paper investigates how momentum-space geometric phases, captured by Berry curvature and the first Chern number, respond to Lorentz-violating backgrounds in the SME. By analyzing two representative cases, CPT-odd $b_0$ and CPT-even $d_{0}$, it shows that large Lorentz violation can separate Weyl cones or tilt/distort them without destroying the topological charges at Weyl nodes; in some spacelike scenarios the physical vacuum can carry a quantized topological phase. The authors also derive alternative invariants from the fermion propagator to corroborate the presence and protection of Weyl-node charges. These results establish momentum-space geometric phases as gauge-invariant observables for Lorentz violation and outline pathways for future extensions to other SME operators and experimental tests.

Abstract

Geometric phases can manifest when a relativistic quantum particle moves cyclically along a loop in parameter space. The phase can be affected by the presence of a background field and can be accompanied by nontrivial topological features. The appearance of adiabatic geometric phases in momentum space is demonstrated for a Lorentz-violating Weyl fermion, where the role of the background is played by the coefficients for Lorentz violation. As explicit examples, the Berry curvature and the first Chern number are derived for two cases with large Lorentz violation, one incorporating CPT violation and one preserving CPT symmetry. Some alternative topological invariants are also obtained. In certain scenarios with large Lorentz violation, the physical vacuum is associated with a topological phase.

Figures (10)

• Figure 1: Energy eigenvalues for the timelike case \ref{['fig:dispersions-timelike-b']} before and \ref{['fig:dispersions-timelike-b-reinterpreted']} after reinterpretation.
• Figure 2: \ref{['fig:vector-potential-dirac-string']} Berry connection vector field \ref{['eq:berry-connection-u-2']}, and \ref{['fig:dirac-string-contour-plot']} contour plot of its norm.
• Figure 3: \ref{['fig:magnetic-monopole-arrows']} Vector field from a source of Berry curvature in Eq. \ref{['eq:berry-curvature-isotropic-b']}, and \ref{['fig:magnetic-monopole-contour-plot']} contour plot of its norm.
• Figure 4: 2-sphere $S_{0}^2$ around magnetic monopole with unit normal $\mathcal{n}$ and upper hemisphere $H_+$ bounded by the circle $S_{0}^1$.
• Figure 5: Energy eigenvalues for the spacelike case \ref{['fig:dispersions-purely-spacelike']} before and \ref{['fig:dispersions-purely-spacelike-reinterpreted']} after reinterpretation.