# Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice

### Submission summary

 As Contributors: Carlo Beenakker · Michał Pacholski Arxiv Link: https://arxiv.org/abs/2103.15615v3 (pdf) Code repository: https://doi.org/10.5281/zenodo.5556988 Date accepted: 2021-11-01 Date submitted: 2021-10-15 10:51 Submitted by: Pacholski, Michał Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approaches: Theoretical, Computational

### Abstract

The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special "staggered" grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form ${\mathcal H}\psi=E {\mathcal P}\psi$, with Hermitian tight-binding operators ${\mathcal H}$, ${\mathcal P}$, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Dear Editors, we thank the two referees for their favorable recommendation regarding the suitability of this work for SciPost Physics, and for their helpful suggestions for improvement of the manuscript.

### List of changes

In response to Referee 1 we have removed the confusing footnote 19.

The points raised by Referee 2 have been addressed as follows:

1. The systematic shift in Fig. 2 was due to an inaccurate normalization of the numerical data, which we have now corrected. We thank the referee for spotting the shift. The numerical level spacing distribution is quite close to the RMT prediction, a remaining discrepancy (below 1%) is likely due to localization effects in the disordered quantum dot.

2. A reference to Kieburg et al. has been inserted (Ref. 32). The parity dependence mentioned by the referee is for a discretization with fermion doubling, so there is no direct connection with the single-cone discretization studied here.

3. References to Verbaarschot et al. (Ref. 37), Gockeler et al. (Ref. 30), and Farchioni et al. (Ref. 31) have been inserted.

4. In the final paragraph of the concluding section we mention three topics for future research in the context of the Stacey discretization. One of these could be the discretization of both space and time, to study dynamical properties of Dirac fermions.

We have also inserted a link to a repository that contains the computer code pertaining to this research.