Speaker: Hal Tasaki (Gakushuin University)

Date & Time: 2:00 - 4:00 pm, July 11, 2019.

Room: Seminar Room A&D (Engineering Building #6)

Title: Quantum Spin Chains and von Neumann Algebra: A Lieb-Schultz-Mattis type theorem without continuous symmetry

Abstract: By a Lieb-Schultz-Mattis type theorem, we mean a no-go theorem that rules out the existence of a unique gapped ground state in a certain class of quantum many-body systems. The original Lieb-Schultz-Mattis theorem, which was proved in 1961 for the antiferromagnetic Heisenberg chain, and its various extensions rely essentially on the U(1) symmetry of the models. Recently it was realized that Lieb-Schultz-Mattis type statements can be shown for one-dimensional quantum many-body systems that have discrete on-site symmetry whose action forms a nontrivial projective representation of the symmetry group. We here present a fully rigorous version of such a Lieb-Schultz-Mattis type theorem for one-dimensional quantum spin systems. Rather surprisingly (at least to the present speaker), the operator algebraic formulation of spin chains seems to be mandatory for the proof. In particular the notion of the von Neumann algebra (especially the Cuntz algebra) plays an essential role in the proof.

I start by reviewing the original Lieb-Schultz-Mattis theorem and its proof, and then discuss the statement and the (basic strategy of the) proof of the new theorem. The new theorem can be stated as a no-go theorem for the existence of a translation invariant area-law states (which is not necessarily a ground state).

The talk is based on a joint work with Yoshiko Ogata in arXiv:1808.08740 (Comm. Math. Phys. 2019).