Integrability in Statistical Mechanics and Condensed Matter

Universality is the property that some macroscopic quantities of a system are independent of most of its microscopic details, and depend only on very few global aspects. As such, the search for universality is a universal trend in theoretical physics, even in all of science. Knowledge of which quantities are universal and on which parameters they depend turns out to be an extremely powerful tool in theoretical investigations. Once a quantity is known to be universal property of a physical system, a model of it is no longer required to be very accurate, it is only required to capture the universality class.

This freedom can be used to select a model such that no further approximations are necessary to calculate the macroscopic properties of the model from its defining parameters. In practice this approach has been proven extremely successful in a wide range of phenomena in statistical physics. The approach has been developed in the theory of phase transitions and for quantum chains. It has strong connections with and has even been instrumental in modern developments in mathematics like knot theory, quantum groups and Rogers-Ramanujan identities. While the ITFA fosters this development in the person of Warnaar, the contribution of the other players, Schoutens and Nienhuis, is to extend its use to quite different low-dimensional phenomena such as polymers, transport in random media, quasi-crystals, conduction in high magnetic field and models for traffic flow.

Generalized statistics and combinatorics

One of the intrinsic quantum numbers of elementary particles is their spin. According to the parity of its spin (integer or half-integer) a particle is either termed a boson or fermion. Since fermions obey Fermi's exclusion principle, the statistical properties of fermions and bosons are fundamentally different. In recent years it has become clear that many of the quasi-particles essential in the description of condensed matter and statistical systems obey statistical rules that are incompatible with those of either bosons or fermions. This has led Haldane to the introduction of fractional statistics, a topic of much current interest.

In a number of recent research projects, Schoutens has elaborated on Haldane's work and has managed to enlarge its scope. He studied realizations of fractional statistics in so-called conformal field theories. This led to various generalizations of Haldane's ideas, including the conceptually new phenomenon of `non-abelian exclusion statistics'. In addition, applications to edge theories for various quantum Hall systems were developed. Further work in this direction is in progress.

Interestingly, fractional statistics is not only of importance in the quasi-particle description physical systems, but has many deep connections with problems in combinatorics and number theory. The famous Rogers-Ramanujan identities, for example, can be shown to describe a one-dimensional system of quasi-particles obeying fractional statistics. The present research of Warnaar, carried out in collaboration with Schilling (and the number theorist G. E. Andrews), concerns the application of the principle of generalized statistics and fermionic counting, to problems in combinatorics. Some intriguing questions that seem tractable using the quasi-particle picture are that of finding higher-rank as well a fractional level Bailey lemma's, proving positivity problems related to ribbon tableaux and partition with ``fractional'' hook-differences (the Borwein conjectures) and the proof of various lattice-path results related to plane partitions. Warnaar and collaborators intend to address these quantum enumeration problems using insights gained by the study of quasi-particle models based on generalized statistics.

Quasi-crystals and packings

The modeling of quasi-crystals follows typically one of two lines of thought. (i) The first is the principle that the quasi-crystal is related to an ideal quasiperiodic structure in the same way as a crystal relates to a lattice. (ii) The second is that the non-crystallographic symmetry as well as possible quasi-periodicity is the result of thermal averaging. This second approach gives rise to statistical models for quasi-crystals. In the group of Nienhuis several of such models have been constructed such that they are integrable, i.e. that the thermal averaging can be performed explicitly. These new results have attracted a great deal of attention. Based on ideas of universality presented above, this may lead to further insight in the properties of generic quasi-crystals. One key ingredient, however, is missing: the precise knowledge which of the properties of these models are actually universal. This is one of the main questions of further research in this area. We intend to calculate a number of physical quantities, which, even though the models are solvable, are still a challenge. In particular we intend to calculate experimentally accessible properties such as the diffraction pattern and investigate its universality content. For the same reason we intend to elaborate on boundary effects which are experimentally important when the samples are relatively small, and which may contain several universal keys. Experimental realizability of these models directly (rather than via the argument of universality) by means of trapped colloidal suspensions are also being investigated. We hope to gain a better understanding of the partially revealed connections of the quasi-crystal models to field theories with SU(n) symmetry in which the n is connected to the non-crystallographic symmetry of the system.

Low dimensional dynamical systems

The mathematics of classical equilibrium statistical models, of quantum models and of classical dynamical models is remarkably similar. The translation involves often no more than a simple transformation of space(-time). As a result, techniques developed for quantum - and for statistical lattice models can be used to solve one-dimensional dynamical flow problems. An interesting application of these models is traffic flow. The understanding of traffic flow has gained enormously in the last decade from simulations of relatively simple models. In a collaboration between Amsterdam (Nienhuis) and Utrecht (Ernst, De Gier) we have started an investigation into versions of those traffic models which can actually be solved analytically. An analytic solution makes the application of these models in traffic forecast and effective re-routing enormously more efficient when compared to the current simulational state of the art.