Abstract
The non-regularizability of free fermion field theories, which is the root of various quantum anomalies, plays a central role in particle physics and modern condensed matter physics. In this paper, we generalize the Nielsen-Ninomiya theorem to all minimal nodal free fermion field theories protected by the time reversal, charge conservation, and charge conjugation symmetries. We prove that these massless field theories cannot be regularized on a lattice.
Highlights
The non-regularizability of massless free fermion field theories is the origin of various quantum anomalies
A famous example is the Nielsen-Ninomiya[1] theorem, namely, Weyl nodes with net chirality cannot be realized by any chargeconserved lattice model in three dimensions
Weyl nodes with net chirality can appear on the boundary of a 4D charge-conservation-protected topological insulator (The free fermion topological classification of this 4D topological insulator is Z.)
Summary
The non-regularizability of massless free fermion field theories is the origin of various quantum anomalies. Under charge conservation and time-reversal (T 2 = −1) symmetries, Dirac cones with net vorticity cannot be realized by any lattice model They can appear on the boundary of a 3D topological insulator. The low energy field theory describing the boundary of on-site symmetry protected topological states (SPTs) cannot be regularized on a lattice. In other words, they can not be realized as finite-range tight-binding models where the symmetry acts on the degrees of freedom on each lattice site independently. The purpose of this paper is to prove the following folklore, namely: Any symmetry-protected minimal nodal free-fermion field theory cannot be regularized on a lattice. Because of the length of this proof, it is left to Appendix B and Appendix C
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