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Colloquium: February 11, 2015

Bernard Nienhuis*
Has E8 Been Observed in the Laboratory?
Day Time Place
February 11, 2015 Wednesday 15:40 EE01

Host: Balazs Hetenyi

ABSTRACT —  The Ising model in two dimensions is one of the simplest and most well-known many-particle models with a thermodynamic phase transition. It is a solvable model, meaning that the derivation of the collective behavior from the microscopic variables can be done exactly. For the Ising model this was done for the rst time in 1944.

The Ising model has a critical point at some temperature, in zero eld. At this critical point the scaling limit is described by a solvable conformal eld theory (CFT). The scaling behavior at near-critical temperatures is described by solvable eld theory with massive particles.

In 1989 Zamolodchikov[1] proposed a specic eld theory as the scaling limit of the critical Ising model perturbed by a magnetic eld, while keeping the temperature at its critical value. It is based on the assumption that this eld theory is solvable. The resulting eld theory, has excitations (particles) of which the masses are related to the root system of the Exceptional Lie Algebra E8. It is surprising to note that a model as simple as the Ising model can have as its scaling limit a structure as complicated as E8, an algebra with 248 dimensions.[2] This was soon followed by the discovery of a solvable lattice model of which this eld theory is the scaling limit.[3]

In 2010 experiments were done on a physical system which can be described by an Ising model in a weak magnetic eld. The measurements conrm the mass ratios that are prediced by the connection to the E8 Lie algebra.[4] In this talk I will assume no prior knowledge about Field Theory, Scaling Limit, Lie Algebra or Solvability. I will try to explain the necessary elements of these concepts.

[1] A.B. Zamolodchikov, Integrals of motion and S-matrix of the (scaled) T=Tc Ising model with magnetic eld, Int. J. Modern Physics 4 (1989), no. 16, p4235
[2] D. Borthwick and S. Garibaldi, Did a 1-dimensional magnet detect a 248-dimensional Lie algebra? Not.Amer.Math.Soc.58: p1055, (2011) (arxiv:1012.5407)
[3] S.O. Warnaar, B. Nienhuis and K.A. Seaton, New construction of solvable lattice models including an Ising model in a eld, Phys. Rev. Lett. 69, p710 (1992)
[4] R. Coldea, e.a., Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry, Science 327, p177 (2010).

*University of Amsterdam, Amsterdam, The Netherlands

The Physics Colloquia are designed to address a non-specialist, broad audience and introduce topics of contemporary research through lectures by leading experts. We warmly invite all members of the student body, including undergraduates enrolled in any programme.

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