Generalized Statistics on Lattices

The statistics of particles and extended excitations, such as loops and membranes, are fundamental to modern condensed matter physics, high-energy physics, and quantum information science, yet a comprehensive lattice-level framework for computing them remains elusive. In this work, we develop a universal microscopic method to determine the generalized statistics of Abelian excitations on lattices of arbitrary dimension, and demonstrate it by deriving the statistics of particles, loops, and membranes in up to three spatial dimensions. Our approach constructs a sequence of local unitary operators whose many-body Berry phase encodes the desired statistical invariant. The required sequence is generated automatically from the Smith normal form of locality constraints and therefore needs no extra physical input. We prove that the resulting invariants are quantized, provide an algorithm that computes them efficiently, and show how they unify familiar braiding and fusion data of particles while also uncovering new self- and mutual-statistics of loop and membrane excitations. We further demonstrate that each statistical invariant corresponds to an 't Hooft anomaly of a generalized symmetry; we show that a non-trivial invariant both (i) obstructs gauging that symmetry and (ii) forbids any short-range-entangled (symmetry-preserving) ground state. This establishes a precise connection between microscopic lattice anomalies and many-body dynamics, providing a generalization of the Lieb-Schultz-Mattis theorem that constrains a wide class of quantum lattice systems.