  Anomalies are a powerful way to gain insight into possible lattice regularizations of a quantum field theory. In this work, we argue that the continuum anomaly for a given symmetry can be matched by a manifestly-symmetric, local, lattice regularization in the same spacetime dimensionality only if (i) the symmetry action is offsite, or (ii) if the continuum anomaly is reproduced exactly on the lattice. We consider lattice regularizations of a class of prototype models of QCD: the (1+1)-dimensional asymptotically-free Grassmannian nonlinear sigma models (NLSMs) with a $\theta$ term. Using the Grassmannian NLSMs as a case study, we provide examples of lattice regularizations in which both possibilities are realized. For possibility (i), we argue that Grassmannian NLSMs can be obtained from $\mathrm{SU}(N)$ antiferromagnets with a well-defined continuum limit, reproducing both the infrared physics of $\theta$ vacua and the ultraviolet physics of asymptotic freedom. These results enable the application of new classical algorithms to lattice Monte Carlo studies of these quantum field theories, and provide a viable realization suited for their quantum simulation. On the other hand, we show that, perhaps surprisingly, the conventional lattice regularization of $\theta$ vacua due to Berg and L\"uscher reproduces the anomaly exactly on the lattice, providing a realization of the second possibility. 
Lattice regularizations of $\theta$ vacua: Anomalies and qubit models