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.)

[1] Two soluble models of an antiferromagnetic chain, Elliott Lieb, Theodore Schultz, Daniel Mattis, Annals of Physics Volume 16, Issue 3, December 1961, Pages 407-466

[2] Lieb-Schultz-Mattis in higher dimensions, M. B. Hastings, Phys. Rev. B 69, 104431 – Published 29 March 2004

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    $\begingroup$ I would say the original proof in [1] is pretty rigorous. $\endgroup$
    – lcv
    Oct 21, 2018 at 1:39

1 Answer 1


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.

  • $\begingroup$ thanks very much +1, I did not know this Ref $\endgroup$
    – wonderich
    Oct 21, 2018 at 14:17
  • $\begingroup$ My paper [2] was already a rigorous proof of the higher dimensional LSM theorem. I do not know how it is described now, but multiple times at conferences 10-15 years ago I heard Nachtergaele refer to his paper with Sims as "a guide" (his words) to my paper. It is true that I was not as well-versed in the mathematical style of writing then, and it was valuable of NS to give the proof in form suitable for CMP, but I would hope that the community would see past that to recognize that all steps in later proofs were already present in [2]. $\endgroup$ Oct 7, 2020 at 15:15

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