NonPerturbative Regularization of 1+1D AnomalyFree Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules
Abstract
A nonperturbative lattice regularization of chiral fermions and bosons with anomalyfree symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask "whether there is a local shortrange quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial lattice with continuous time, such that its low energy physics produces a 1+1D anomalyfree chiral matter theory of symmetry $G$?" In particular, we show that the 3$_L$5$_R$4$_L$0$_R$ U(1) chiral fermion theory, with two leftmoving fermions of charge3 and 4, and two rightmoving fermions of charge5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multifermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)anomalyfree chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffered from NielsenNinomiya theorem's fermiondoubling, if we include properlydesigned interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to GinspargWilson fermion, EichtenPreskill, and ChenGiedtPoppitz (CGP) models, and suggest modifying CGP model to have successful mirrordecoupling. As an additional remark, we show a topological nonperturbative proof on the equivalence relation between the 't Hooft anomaly matching conditions and the boundary fully gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) of U(1) symmetry. Our proof holds universally independent from Hamiltonian or Lagrangian/path integral formulation of quantum theory.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 arXiv:
 arXiv:1307.7480
 Bibcode:
 2013arXiv1307.7480W
 Keywords:

 High Energy Physics  Lattice;
 Condensed Matter  Strongly Correlated Electrons;
 High Energy Physics  Phenomenology;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 42 pages. Sec IV and Appendix C,D,E with a proof of 't Hooft anomaly matching condition = boundary fully gapping rules, Appendix B on GinspargWilson fermions as SPT edge states with a nononsite symmetry. A variant lattice model but with the same topological nonperturbative derivation in arXiv:1807.05998. v4: Refinement. Special thanks to Erich Poppitz, John Preskill and Edward Witten