The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy
states from a lattice model cannot be a trivial symmetric insulator if the
filling per unit cell is not integral and if the lattice translation symmetry
and particle number conservation are strictly imposed. In this paper, we
compare the one-dimensional gapless states enforced by the LSM theorem and the
boundaries of one-higher dimensional strong symmetry-protected topological
(SPT) phases from the perspective of quantum anomalies. We first note that,
they can be both described by the same low-energy effective field theory with
the same effective symmetry realizations on low-energy modes, wherein
non-on-site lattice translation symmetry is encoded as if it is a local
symmetry. In spite of the identical form of the low-energy effective field
theories, we show that the quantum anomalies of the theories play different
roles in the two systems. In particular, We find that the chiral anomaly is
equivalent to the LSM theorem, whereas there is another anomaly, which is not
related to the LSM theorem but is intrinsic to the SPT states. As an
application, we extend the conventional LSM theorem to multiple-charge
multiple-species problems and construct several exotic symmetric insulators. We
also find that the (3+1)d chiral anomaly provides only the perturbative
stability of the gapless-ness local in the parameter space.Comment: 14 + 3 pages, 1 figure. (The first two authors contributed equally to
the work.