Interacting bosons in one dimension and their relation to the KPZ universality class.

The Lieb-Liniger model is one of the simplest examples of an exactly soluble many body quantum system. It consists of bosons in one dimension interacting via a delta function potential. It is of great interest in its own right, but in the last few years it has attracted attention because a number of problems in the Kadar–Parisi–Zhang (KPZ) universality class can be mapped on to it and this has lead to new insights into the KPZ.

The talk will start with a brief sketch of how the KPZ is related to the Lieb-Liniger model. The connection is made by mapping to the directed polymer in a random medium and then using replicas. However, most of the talk will be a step-by-step analysis of the Lieb-Liniger model. It provides a very simple example of the use of the Bethe ansatz and gives some insight into quantum integrability.

A good reference on the Lieb-Liniger is:

• Sutherland, Bill. Beautiful models: 70 years of exactly solved quantum many-body problems. World Scientific Publishing Co Inc, 2004.

Two recent reviews on its use for solving stochastic and disordered problems are.

• Quastel, Jeremy, and Herbert Spohn. "The one-dimensional KPZ equation and its universality class." Journal of Statistical Physics 160.4 (2015): 965-984.
• Dotsenko, Viktor S. "Universal randomness." Physics-Uspekhi54.3 (2011): 259.