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 both be 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 were an internal 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 that 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 gaplessness local in the parameter space.
We critically examine the issue raised by Nagaosa and Oshikawa [J. Phys. Soc. Jpn. 65 (1996) 2241] on how Fujikawa's Jacobian arising from chiral anomaly contributes a topological term to the derivative expansion of the 1d Hubbard model at half filling, thus governing the behaviour of the spin gap. We find that when the nonabelian nature of the system is properly accounted for, the scheme requires modification. The modified result is shown to agree with nonabelian bosonization. Extensions to spin-Peierls systems are outlined.
The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a
D-dimensional system under modular transformation, or modular anomaly, signals the presence of a
(D+1)-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a
(2+1)-D and a
(3+1)-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in
(2+1)-D and
(3+1)-D are modular invariant. However, we show in
(3+1)-D that when the edge theory is coupled to a background
U(1) gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a
(4+1)-D bulk. Using the modular anomaly, we find that the edge theory of a
(4+1)-D insulator with space-time inversion symmetry
(PT) and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.