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 with continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry G?" In particular, we show that the 3L-5R-4L-0R 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 suffered 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. As an additional remark, we show a topological non-perturbative 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. Contents I. Introduction and Summary2Fermion model 6 III. From a continuum field theory to a discrete lattice model 9 A. Free kinetic part and the edge states of a Chern insulator 9 1. Kinetic part mapping and RG analysis 9 2. Numerical simulation for the free fermion theory with nontrivial Chern number 10 B. Interaction gapping terms and the strong coupling scale 11 IV. Topological Non-Perturbative Proof of Anomaly Matching Conditions = Boundary Fully Gapping Rules 13 A. Bulk-Edge Correspondence -2+1D Bulk Abelian SPT by Chern-Simons theory 13 B. Anomaly Matching Conditions and Effective Hall Conductance 15 C. Anomaly Matching Conditions and Boundary Fully Gapping Rules 16 1. Physical picture 17 2. Topological non-perturbative proof 18 3. Perturbative arguments 19 4. Preserved U(1) N/2 symmetry and a unique ground state V. General Construction of Non-Perturbative Anomaly-Free chiral matter model from SPT VI. Conclusion Acknowledgments A. C, P , T symmetry in the 1+1D fermion theory B. Ginsparg-Wilson fermions with a non-onsite U(1) symmetry as SPT edge states 1. On-site symmetry and non-onsite symmetry 2. Ginsparg-Wilson relation, Wilson fermions and non-onsite symmetry C. Proof: Boundary Fully...

In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems, since the low energy dynamics is directly encoded in the partition functions. More generally, symmetries, anomalous symmetries, non-invertible gravitational anomalies, and their combinations, can all be viewed as constraints on low energy dynamics. In this paper, we demonstrate that they all can be viewed uniformally and systematically as pure (non-invertible) gravitational anomalies.

Defining a Chiral Fermion Theory on a lattice has presented an ongoing challenge both in Condensed Matter physics and in Lattice Gauge Theory. In this paper, we demonstrate that a chiral free-fermion theory can live on an ultra-local spacetime lattice if we allow the Lagrangian to be non-hermitian. Rather than a violation of unitarity, the non-hermitian structure of our Lagrangian arises because time is discrete, and we show that our model is obeys an elementary unitarity condition: namely, that the norm of the two-point functions conserves probability. Beyond unitarity, our model displays several surprising properties: it is formulated directly in Minkowskian time; it has exactly Lorentz invariant dynamics for all frequencies and momenta (in the large volume limit); and it is free from all gauge anomalies, despite the prediction from field theory that it should suffer one. We show that our model is a discrete time description of a single chiral edge mode of several recently proposed 2 + 1d Floquet models. That the chiral edge can be treated without the rest of the 2 + 1d system, even when coupled to a gauge field, implies that the Floquet models are radically different from Integer Quantum Hall models, which also support chiral edge modes. Furthermore, the Floquet results imply that our model can be physically realized, which presents an opportunity for gauge theories to be simulated in a condensed matter or cold atom context. Our results present a solution to the 'Chiral-fermion problem:' a chiral field theory can indeed be defined on an ultra-local spacetime lattice, and we address how our model avoids several no-go arguments.

We study quantum spin systems with a proper combination of geometric frustration, spin-orbit coupling and ferromagnetism. We argue that such a system is likely to be in a chiral spin state, a fractional quantum Hall (FQH) state for bosonic spin degrees of freedom. The energy scale of the bosonic FQH state is of the same order as the spin-orbit coupling and ferromagnetism -overall much higher than the energy scale of FQH states in semiconductors.

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