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In the $1+1\mathrm{D}$ultralocal 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. Published by the American Physical Society2025 

We study the nonlinear $$\sigma$$-model in $${(d+1)}$$-dimensional spacetime with connected target space $$K$$ and show that, at energy scales below singular field comfigurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $$d$$-representations of a discrete $$d$$-group $$\mathbb{G}^{(d)}$$ (i.e. the emergent symmetry is the dual of the invertible $$d$$-group $$\mathbb{G}^{(d)}$$ symmetry). The $$d$$-group $$\mathbb{G}^{(d)}$$ is determined such that its classifying space $$B\mathbb{G}^{(d)}$$ is given by the $$d$$-th Postnikov stage of $$K$$. In $(2+1)$D and for finite $$\mathbb{G}^{(2)}$$, this symmetry is always holo-equivalent to an invertible $${0}$$-form---ordinary---symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $$\sigma$$-model spontaneously breaks this symmetry, and when $$\mathbb{G}^{(d)}$$ is finite, it is described by the deconfined phase of $$\mathbb{G}^{(d)}$$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $$\sigma$$-model in $${(3+1)}$$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $$\mathbb{C}P^{N-1}$$ nonlinear $$\si$$-models in the large-$$N$$ limit.