Any rigorous theorem on quantum many-body systems is useful.

An important theorem for quantum spin systems was proven by Hastings. It is a generalization of the Lieb–Schultz–Mattis theorem to dimensions larger than one. It concerns spin-1/2 systems with

**one spin per unit cell**on a two-dimensional lattice with periodic boundary conditions (on a torus). The theorem states that if there is**no symmetry breaking**then the ground state is separated from the first excited state by an**energy gap**that vanishes in the thermodynamic limit. Hence, under these specific conditions one cannot have a singlet ground state with a non-zero energy gap.There is a nice discussion of the theorem in some lecture notes by Matthew Fisher, which contains the Figure below.

## No comments:

## Post a Comment