The LiebSchultzMattis Theorem: A Topological Point of View
Abstract
We review the LiebSchultzMattis theorem and its variants, which are nogo theorems that state that a quantum manybody system with certain conditions cannot have a locallyunique gapped ground state. We restrict ourselves to onedimensional quantum spin systems and discuss both the generalized LiebSchultzMattis theorem for models with U(1) symmetry and the extended LiebSchultzMattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests we take a modern topological point of view and prove the generalized LiebSchultzMattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locallyunique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetryprotected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locallyunique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists' point of view. We have tried to make the present article readable and almost selfcontained. We only assume basic knowledge about quantum spin systems.
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 arXiv:
 arXiv:2202.06243
 Bibcode:
 2022arXiv220206243T
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 28 pages, 8 figures, this is the final version