  We analyze lattice Hamiltonian systems whose global symmetries have 't Hooft
anomalies. As is common in the study of anomalies, they are probed by coupling
the system to classical background gauge fields. For flat fields (vanishing
field strength), the nonzero spatial components of the gauge fields can be
thought of as twisted boundary conditions, or equivalently, as topological
defects. The symmetries of the twisted Hilbert space and their representations
capture the anomalies. We demonstrate this approach with a number of examples.
In some of them, the anomalous symmetries are internal symmetries of the
lattice system, but they do not act on-site. (We clarify the notion of "on-site
action.") In other cases, the anomalous symmetries involve lattice
translations. Using this approach we frame many known and new results in a
unified fashion. In this work, we limit ourselves to 1+1d systems with a
spatial lattice. In particular, we present a lattice system that flows to the
$c=1$ compact boson system with any radius (no BKT transition) with the full
internal symmetry of the continuum theory, with its anomalies and its
T-duality. As another application, we analyze various spin chain models and
phrase their Lieb-Shultz-Mattis theorem as an 't Hooft anomaly matching
condition. We also show in what sense filling constraints like Luttinger
theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our
understanding allows us to use information from the continuum theory to derive
some exact results in lattice model of interest, such as the lattice momenta of
the low-energy states.
