PhD Thesis Defenses

Anomaly and Topology


Chiral anomalies and topological phases of matter form the basis of the research presented in this dissertation. The chiral anomaly is considered both in the context of magnetic Weyl semimetals and in the context of non-Hermitian Dirac actions. Topological phases of matter play a role in this work through the research on Weyl semimetals and in the formulation of local topological markers.

The simplest example of magnetic Weyl semimetals consist of two Weyl cones separated in momentum space by a magnetisation vector which acts as an axial gauge field. We describe the emergence of axial electromagnetic fields by considering a magnetic field driven domain wall in this magnetisation. The parallel axial magnetic and axial electric fields give rise to the axial anomaly, and in turn to the chiral magnetic effect; a nonequilibrium current located at the domain wall. The chiral magnetic effect is a source of electromagnetic radiation, and a measurement of this radiation would provide evidence of the existence of the axial anomaly.

Electronic manipulation of domain walls is a central objective in spintronics. We describe how the axial anomaly, in terms of external electromagnetic fields, acts as a torque on the domain wall, and allows for electric control of the equilibrium configuration of the domain wall. We show how the axial anomaly is used to flip the chirality of the domain wall by tuning the electric field. Measuring the change in domain wall chirality constitutes a signal of the axial anomaly. We also describe how the Fermi arc boundary states of the Weyl semimetal at the domain wall result in an effective hard axis anisotropy which allows for large domain wall velocities irrespective of the intrinsic anisotropy of the material.

Our interest in non-Hermitian chiral anomalies stems from the existence of topological phases of matter in non-Hermitian models. We evaluate the chiral anomaly for a non-Hermitian Dirac theory with massless fermions with complex Fermi velocities coupled to non-Hermitian axial and vector gauge fields. The anomaly is compared with the corresponding anomaly of a Hermitianised and an anti-Hermitianised action derived from the non-Hermitian action. We find that the non-Hermitian anomaly does not correspond to the combined anomalous terms derived from the Hermitianised and anti-Hermitianised theory, as would be expected classically, resulting in new anomalous terms in the conservation laws for the chiral current.

Local topological markers are real space expressions of topological invariants evaluated by local expectation values and are important for characterising topology in noncrystalline structures. We derive analytic expressions for local topological markers for strong topological phases of matter in odd dimensions, by generalising the formulation of the even dimensional local Chern marker. This is not a straightforward task since the topological invariants in odd dimensions are basis dependent. Our solution is to express the invariants in terms of a family of parameter dependent projectors interpolating between a trivial state and the topological state of interest. The odd dimensional invariant is therefore expressed as a Chern character integrated over the combined space of the odd dimensional Brillouin zone and the one dimensional parameter space. As a result, we provide an easy-to-use chiral marker for symmetry classes with a chiral constraint, and a Chern-Simons marker for symmetry classes with either time reversal symmetry (in three dimensions) or particle hole symmetry (in one dimension). These markers are readily extended to interacting systems by considering the topological equivalence between a gapped one-particle density matrix of the interacting state and a projector corresponding to a free fermion state.