announcement page by the host, Keiji Saito
Oct. 1, 2019 / Last Modified: Oct. 10, 2019
I gave a series of lectures on SPT phases in quantum spin chains at Keio University. Hand written notes and the list of references are available.The notes and the reference list have been updated after the lectures.
Hal Tasaki
Oct. 7, 13:00-14:30, 14:45-16:15, 16:30-18:00
Oct. 8, 13:00-14:30, 14:45-16:15, 16:30-18:00
Room 211, Building 12, Yagami Campus, Keio Univeristy
The discovery of Haldane phenomena in antiferromagnetic quantum spin chains, which brought Haldane the 2016 Nobel prize in physics, further led to the proposal by Gu and Wen of the notion of symmetry protected topological (SPT) phase. The theory of SPT phases in spin chains was further refined by Pollmann, Turner, Berg, and Oshikawa, who defined topological indices within the formulation of matrix product states (MPS). Very recently, in 2018 and 2019, fully rigorous index theorems that characterize SPT phases in quantum spin chains were developed by Ogata. The new index theorems apply to any spin chains with a unique gapped ground state, and free from MPS ansatz or any unproved assumptions. Ogata's theory makes an essential (and elegant) use of sophisticated notions from operator algebraic formulation of quantum spin systems on infinite lattices.
The purpose of these lectures is to explain the essence of Ogata's index theorems along with necessary background. I do not assume background in operator algebraic formulation or in modern topological condensed matter physics. You don't need to know what MPS or SPT are. Basic knowledge about quantum spin system will be sufficient.
Tentative plan of the lectures is as follows. On the first day, I will start by discussing the AKLT model, a prototypical model that exhibits Haldane phenomena, and then discuss a topological phase transition that takes place when we modify the model. We then introduce the formulation of MPS and discuss the index theorem by Pollmann, Turner, Berg, and Oshikawa. (I think I won't have time to discuss "physics" of Haldane phenomena in detail.) The second day should start from a careful introduction (to the beginners) of the operator algebraic formulation of quantum spin systems. I will then try to describe Ogata's index theorem in a manner which is both rigorous and intuitively clear. I wish to convince the audience of the strength and beauty of her approach.