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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. 

ABSTRACT Bounded count data are commonly encountered in environmental studies. This paper examines two environmental applications illustrating their relevance. The first investigates the effect of winter malnutrition on mule deer (Odocoileus hemionus) fawn mortality. The second application analyzes public perceptions of environmental issues using data from the Eurobarometer 95.1 survey (March–April 2021), which includes a question rating the perceived severity of climate change on a scale from 1 to 10. Together, these studies demonstrate the need for flexible bounded count models in environmental research. In this context, the binomial and beta‐binomial (BB) models are widely used for bounded count data, with the BB model offering the advantage of accounting for overdispersion. However, atypical observations in real‐world applications may hinder the performance of the BB model and lead to biased or misleading inferences. To address this limitation, we propose the contaminated beta‐binomial (cBB) distribution (cBB‐D), which introduces an additional BB component to accommodate atypical observations while preserving the mean and variance structure of the BB model. The cBB‐D thus captures both overdispersion and contamination effects in bounded count data. To incorporate explanatory variables, we further develop the contaminated BB regression model (cBB‐RM), in which none, some, or all cBB parameters may depend on covariates. The proposed models are applied to two environmental datasets, complemented by a sensitivity analysis on simulated data to assess the influence of atypical observations on parameter estimation. The methodology is implemented in the open‐sourcecBBpackage forR, available athttps://github.com/arnootto/cBB. 

This paper presents a search for massive, charged, long-lived particles with the ATLAS detector at the Large Hadron Collider using an integrated luminosity of $$140~fb^{−1}$$ of proton-proton collisions at $$\sqrt{s}=13$$~TeV. These particles are expected to move significantly slower than the speed of light. In this paper, two signal regions provide complementary sensitivity. In one region, events are selected with at least one charged-particle track with high transverse momentum, large specific ionisation measured in the pixel detector, and time of flight to the hadronic calorimeter inconsistent with the speed of light. In the other region, events are selected with at least two tracks of opposite charge which both have a high transverse momentum and an anomalously large specific ionisation. The search is sensitive to particles with lifetimes greater than about 3 ns with masses ranging from 200 GeV to 3 TeV. The results are interpreted to set constraints on the supersymmetric pair production of long-lived R-hadrons, charginos and staus, with mass limits extending beyond those from previous searches in broad ranges of lifetime