# A Lattice Non-Perturbative Hamiltonian Construction of 1+1D Anomaly-Free Chiral Fermions and Bosons - on the equivalence of the anomaly matching condit

A non-perturbative Hamiltonian construction 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 finite quantum Hamiltonian system realizing onsite symmetry $G$ defined on a 1D spatial lattice with a continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$?" Our answer is "yes." In particular, we show that the 3$_L$-5$_R$-4$_L$-0$_R$ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and charge-4, and two right-moving fermions of charge-5 and charge-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 interactions between fermions with intermediate strength. We show how to design such proper interactions by looking for interaction terms with extra symmetries. In general, we show that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on 1D lattice with onsite symmetry, by using a quantum Hamiltonian with a continuous time, if we include properly-designed interactions between matter fields. We comment on the new ingredients and the differences of ours comparing to Eichten-Preskill and Chen-Giedt-Poppitz models, and suggest modifying Chen-Giedt-Poppitz model to have successful mirror-decoupling. As an additional remark, we show a topological non-perturbative proof on the equivalence relation between 't Hooft anomaly matching conditions and the boundary fully gapping rules of U(1) symmetry.

**Comments:**39 pages, 9 figures. Sec IV and Appendix C,D,E with a proof of 't Hooft anomaly matching condition (ABJ's U(1) anomaly) = boundary fully gapping rules, Appendix B on Ginsparg-Wilson fermions as SPT edge states with a non-onsite symmetry. v3: many new updates. Special thanks to John Preskill and Erich Poppitz

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