Quantized axial charge of staggered fermions and the chiral anomaly

Abstract

In the 1+1D ultra-local lattice Hamiltonian for staggered fermions with a finite-dimensional Hilbert space, there are two conserved, integer-valued charges that flow in the continuum limit to the vector and axial charges of a massless Dirac fermion with a perturbative anomaly. Each of the two lattice charges generates an ordinary U(1) global symmetry that acts locally on operators and can be gauged individually. Interestingly, they do not commute on a finite lattice and generate the Onsager algebra, but their commutator goes to zero in the continuum limit. The chiral anomaly is matched by this non-abelian algebra, which is consistent with the Nielsen-Ninomiya theorem. We further prove that the presence of these two conserved lattice charges forces the low-energy phase to be gapless, reminiscent of the consequence from perturbative anomalies of continuous global symmetries in continuum field theory. Upon bosonization, these two charges lead to two exact U(1) symmetries in the XX model that flow to the momentum and winding symmetries in the free boson conformal field theory.

8 pages plus appendices; v2: expanded discussions on the non-abelian algebra and added references; v3: published version

References in corpus (13)

• Non-Invertible Duality Defects in 3+1 Dimensions
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• Composite fermion duality for half-filled multicomponent Landau Levels
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities
• 't Hooft anomalies for staggered fermions
• Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models
• Chiral basis for qubits and spin-helix decay

Cited by in corpus (13)

• Lattice T-duality from non-invertible symmetries in quantum spin chains
• LSM and CPT
• Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions
• ${\mathbb{Z}}_2$ gauging and self-dualities of the $XX$ model and its cousins
• Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Minimal-doubling and single-Weyl Hamiltonians
• Symmetry-Enforced Fermi Surfaces
• Anomaly of conserved and nonconserved axial charges in Hamiltonian lattice gauge theory
• Anomalies of global symmetries on the lattice
• Parity anomaly from LSM: exact valley symmetries on the lattice
• Lattice fermion formulation via Physics-Informed Neural Networks: Ginsparg-Wilson relation and Overlap fermions
• Equivalence class of Emergent Single Weyl fermion lattice models in 3 dimensions: gapless superconductors and superfluids versus chiral fermions