Topological effective field theories for Dirac fermions from index theorem

Abstract

Dirac fermions have a central role in high energy physics but it is well known that they emerge also as quasiparticles in several condensed matter systems supporting topological order. We present a general method for deriving the topological effective actions of (3+1) massless Dirac fermions living on general backgrounds and coupled with vector and axial-vector gauge fields. The first step of our strategy is standard (in the Hermitian case) and consists in connecting the determinants of Dirac operators with the corresponding analytical indices through the zeta-function regularization. Then, we introduce a suitable splitting of the heat kernel that naturally selects the purely topological part of the determinant (i.e. the topological effective action). This topological effective action is expressed in terms of gauge fields using the Atiyah-Singer index theorem which computes the analytical index in topological terms. The main new result of this paper is to provide a consistent extension of this method to the non Hermitian case where a well-defined determinant does not exist. Quantum systems supporting relativistic fermions can thus be topologically classified on the basis of their response to the presence of (external or emergent) gauge fields through the corresponding topological effective field theories.

7 pages, published version

References in corpus (10)

• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Topological Field Theory of Time-Reversal Invariant Insulators
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• Electron fractionalization in two-dimensional graphenelike structures
• Protected boundary states in gapless topological phases
• Topological field theory and thermal responses of interacting topological superconductors
• 3D Dynamics of 4D Topological BF Theory With Boundary
• Abelian Chern-Simons-Maxwell theory from a tight binding model of spinless fermions
• An Index Theorem for Graphene

Cited by in corpus (9)

• Probing non-Hermitian phase transitions in curved space via quench dynamics
• (3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions
• Geometric Model of Topological Insulators from the Maxwell Algebra
• Topological Electromagnetic Effects and Higher Second Chern Numbers in Four-Dimensional Gapped Phases
• Fractonic self-duality and covariant magnetic fractons
• On the Emergent Dynamics of Fermions in Curved Spacetime
• Effective Lagrangian for Non-Abelian Two-Dimensional Topological Field Theory
• Quantum geometric tensors from sub-bundle geometry
• On geometric delusions of hexagonal structures