# Further Developments of Lieb-Schultz-Mattis theorem in Mathematics

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.)

• I would say the original proof in [1] is pretty rigorous. – lcv Oct 21 '18 at 1:39

## 1 Answer

A "rigorous proof" of [2] in the OP has been published by Nachtergaele and Sims, Commun. Math. Phys. 276 (2007) 437-472.

For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size $$L$$, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form $$(C\log L)/L$$. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings.

• thanks very much +1, I did not know this Ref – wonderich Oct 21 '18 at 14:17