The six-vertex model of statistical mechanics is a prototypical model for the area with lots of different applications. Initially motivated by modelling the formation of ice it has for instance be used in proving the Alternating Sign Matrix Conjecture in enumerative combinatorics.

Lattice models in statistical mechanics

One of the most celebrated models in statistical mechanics is the (two-dimensional square-lattice) Ising model which describes ferromagnetism, and which was solved analytically by Lars Onsager in 1944. His solution was based on an infinite-dimensional algebra, now called the Onsager algebra, which today is understood to be related to the loop algebra of sl(2), a particular example of a Kac-Moody algebra although the wok of Kac and Moody came much later.

Baxter’s work in the 1980s introduced the method of commuting transfer matrices based on solutions of the quantum Yang-Baxter equation and his subsequent solution of the eight-vertex model revolutionised much of the area showing that exact solutions are possible in a much larger class of statistical lattice models. Similar to the case of the Ising model infinite-dimensional algebras play a pivotal role: solutions to the quantum Yang-Baxter equation can be classified in terms of so-called “quantum groups” or q-deformed enveloping algebras of Kac-Moody algebras introduced by Drinfel’d and Jimbo in the mid 1980’s. This links this area of statistical lattice models firmly to the area of representation theory in mathematics and there is a mutual cross-fertilisation of techniques covering besides quantum groups a wealth of different types of other algebras, such as Hecke or Temperley-Lieb algebras.

Modern mathematical aspects of these lattice models pursued here in Glasgow are a description of the underlying combinatorics of lattice models, many of them have partition functions which are special either because of their algebraic or geometric applications, for example in describing quantum cohomology and Gromov-Witten invariants, or their links with enumerative combinatorics.

Physically these lattice models have received much attention because many of them exhibit phase transitions near so-called critical points where the parameters are tuned to certain special values. At the critical point the system can be described effectively in terms of two-dimensional conformal field theories. The long-term goal is to compute exact (analytic) expressions for correlation functions. For example, in the Ising model the probability to find two spins aligned is important to understand the correlations in the system and one celebrated result is that the corresponding correlation function is a solution of the Painleve VI equation.

Selected Publications

Korff, C. (2021) Cylindric Hecke characters and Gromov-Witten invariants via the asymmetric six-vertex model. Communications in Mathematical Physics, 381(2), pp. 591-640. (doi: 10.1007/s00220-020-03906-x)
Korff, C. (2013) Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra. Communications in Mathematical Physics, 318(1), pp. 173-246. (doi: 10.1007/s00220-012-1630-9)

Korff, C. (2005) A Q-operator identity for the correlation functions of the infinite XXZ spin-chain. Journal of Physics A: Mathematical and General, 38(30), pp. 6641-6657. (doi: 10.1088/0305-4470/38/30/002)
Korff, C. (2003) Auxiliary matrices for the six-vertex model at qN = 1 and a geometric interpretation of its symmetries. Journal of Physics A: Mathematical and General, 36(19), pp. 5229-5266. (doi: 10.1088/0305-4470/36/19/305)

Grants & Workshops

Integrable systems and symmetric functions, LMS workshop, March 2023, Glasgow
Integrable models and deformations of vertex algebras via symmetric functions; EPSRC standard research grant (May 2022- April 2026)

Quantum integrable systems and infinite dimensional algebras; Royal Society University Research Fellowship
Discrete integrable systems and the crystal limit of quantum integrable models; British Council Cooperation Research Award with the U of Tokyo joint with the ISMP group (Glasgow 2008-11)
New developments in correlation functions of integrable lattice models; EPSRC Mathematical Sciences Small Grant (City U, 2005)