Heidelberg University

More physics in less dimensions: topological phase transitions, critical phenomena far from thermal equilibrium, and Goldstone-modes in 2D colloidal monolayers

Peter Keim, University of Konstanz

Abstract:

Dimensionality strongly affects physical phenomena, and often the behavior in lower dimension is richer in variety - this ranges from Quantum-Hall-Effect, recurrence probabilities of random walks, the lack of symmetry breaking in D < 3, over the glass transitions where theories become exact in infinite dimension to melting by topological defects in D=2 (Nobel Price 2016).

I will discuss how dimensionality affect phase transitions? Whereas nucleation of super cooled fluids in 3D is a first order transition, the melting process in 2D system has been a matter of debate for long. Meanwhile the so called Kosterlitz-Thouless-Halperin-Nelson-Young-theory (KTHNY-theory), predicting two continuous transitions is well accepted. It describes the transitions to the high symmetry phases by the dissociation of two kinds of topological defects, thus the translational and orientational order is destroyed at different temperatures. The intermediate hexatic phase is a fluid but has six-fold orientational order.

Another topic is about cooling a system through a second order phase transition at a nonzero rate. This offers the possibility to investigate spontaneous symmetry breaking out of equilibrium. Since continuous transitions are accompanied by critical slowing down of order parameter fluctuations, the system has to fall out of equilibrium in the vicinity of the transition. The freeze out time depends on the cooling rate at sets the length scale of symmetry broken domains. The scenario is described by the Kibble-Zurek mechanism which was originally used to describe symmetry breaking of the Higgs field shortly after Big-Bang or vortex formation in quantum fluids.

As colloidal systems are excellent model systems to observe those self-organization processes on single particle resolution, I will explain the above phenome with some experiments, including the colloidal monolayer we use. The observalbes are the trajectories of a large ensemble of particles which serves half of the full set of phase-space on all relevant time scales. Thus the experiments are very close to computer simulations and results can easily be compared to theory. The tools are correlation functions and structure factors and the whole set of statistical mechanics.

Finally I will discuss if and how crystals and global magnetism can exist in low dimensions at all. This is given in the framework of Mermin-Wagner-Hohenberg-theorem which is a rigid proof that symmetry breaking in 1D and 2D can not exist. How does this compare to 2D melting theories? This is discussed from an experimental point of view, including the dualism between (lack of) broken symmetries and random walks in 2D.