Lattice fermions as spectral graphs
arXiv:2112.13501 · doi:10.1007/JHEP02(2022)104
Abstract
We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized four-dimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.
29 pages, 11 figures
References in corpus (16)
- Four-dimensional graphene and chiral fermions
- Creutz Fermions on an Orthogonal Lattice
- Broken Symmetries from Minimally Doubled Fermions
- Single flavor staggered fermions
- Index Theorem and Overlap Formalism with Naive and Minimally Doubled Fermions
- Aoki Phases in the Lattice Gross-Neveu Model with Flavored Mass terms
- Search for Fermion Actions on Hyperdiamond Lattices
- Pairs of chiral quarks on the lattice from staggered fermions
- Classification of Minimally Doubled Fermions
- Phase structure for lattice fermions with flavored chemical potential terms
- Improved lattice fermion action for heavy quarks
- QCD phase diagram with 2-flavor lattice fermion formulations
- Varieties and properties of central-branch Wilson fermions
- Strong-coupling Analysis of Parity Phase Structure in Staggered-Wilson Fermions
- On the suitability of the Brillouin action as a kernel to the overlap procedure
- Supersymmetric Gauge Theory on the Graph