The Lieb-Schultz-Mattis theorem [1] and its higher-dimensional generalizations [2] says that a translation-invariant lattice model of spin-1/2's cannot allow a non-degenerate ground state preserving both spin rotational and translation symmetries.

Another way to state Lieb-Schultz-Mattis theorem is that an insulator with half-odd-integer spin per unit cell cannot have a trivial gapped ground state: In 1+1 spacetime dimension, the ground state must either break the translational symmetry (say along the $X$-direction as the lattice translational symmetry group of integer $\mathbb{Z}$) or be gapless (many low energy states in the large/infinite size volume limit of the system), while in higher dimensions the system may also spontaneously break the SO(3) spin rotational symmetry or support Topological quantum field theory (TQFT) at low energy.

There are many later developments in physics.

Question: I wonder whether there are also some developments in mathematics for rigorous proofs or other extensions of Lieb-Schultz-Mattis theorem [1]? (In particular, since Elliott H. Lieb is a mathematical physicist and professor of mathematics.)