Lorentz violation and momentum-space geometric phases

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>C</a:mi> <a:mi>P</a:mi> <a:mi>T</a:mi> </a:mrow> </a:math> violation and one preserving <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mi>C</c:mi> <c:mi>P</c:mi> <c:mi>T</c:mi> </c:mrow> </c:math> symmetry. Some alternative topological invariants are also obtained. In certain scenarios with large Lorentz violation, the physical vacuum is associated with a topological phase.