Non-Perturbative Regularization of 1+1D Anomaly-Free Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules

Abstract

A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask "whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial lattice, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$?" In particular, we propose that the 3${}_L$-5${}_R$-4${}_L$-0${}_R$ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 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 multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free 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 suffering from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed 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 Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. We show a topological non-perturbative proof of the equivalence between $G$-symmetric 't Hooft anomaly cancellation conditions and $G$-symmetric gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multi-U(1) symmetry. We expect our result holds universally regardless of spatial Hamiltonian or Lagrangian/spacetime path integral formulation. Numerical tests are demanding tasks but highly desirable for future work.

Comments: 41 pages. Sec IV and Appendix C,D,E with a proof of 't Hooft anomaly matching condition = boundary fully gapping rules, Appendix B on Ginsparg-Wilson fermions as SPT edge states with a non-onsite symmetry. A variant model in arXiv:1807.05998. Numerical simulation later verified in: arXiv:2202.12355. Special thanks to Erich Poppitz, John Preskill and Edward Witten. Accepted by Physical Review B