Applied Mathematics Research Centre

The AMRC hosts a series of seminars on subjects in statistical physics and fluid dynamics.

**Thermodynamics of a confined supercooled liquid***Wednesday May 8, 2013, 15:00 h, DH Seminar room*

We present a computer study on the thermodynamics of a supercooled liquid confined to a spherical cavity by pinning the particles outside this cavity. This method allows us to explore point-to-set correlations and length scales in the glassy liquid. We find that for small cavities and low temperatures the mobile particles are effectively frozen by the pinned particles. However, upon raising the temperature, the system undergoes a "melting" transition where the particles lose memory of their initial positions. This "melting" transition is sharper, and takes place at lower temperatures for larger cavities. We discuss these results in the context of the random first order transition theory, and the existence of a thermodynamic length scale in this glassy system.

**Magnetization Plateau and Entanglement in ****Antiferromagetic Metal-containing Polymers**Friday May 3

TBA

**The Nature of the Ordered Phase of Spin Glasses***Wednesday May 1, 2013, 15:00 h, DH Seminar room*

For nearly 30 years an argument has raged as to the nature of the low-temperature phase of spin glasses. There are two main theories. The first is the theory of Parisi which is based upon the idea that the ordered phase is like that found in mean-field theory: it has a large number of pure states arranged in an ultrametric topology and has broken replica symmetry, (RSB). The other theory is that of the droplet model which is based on simple scaling and renormalization group ideas and has but two pure states, (just as in a ferromagnet). The low-temperature phase has replica symmetry.

According to RSB theory there is a phase transition line, the de Almeida-Thouless line, when a field is applied to the spin glass. At temperature above the line there is the replica symmetric paramagnetic phase. Below it there is a phase with RSB. However, according to the droplet picture there is no phase transition at all in the presence of a field, just like for a ferromagnet whose phase transition is removed by a field. Thus a huge effort has been made by experimentalists and simulators to find if there is such a phase transition line. Some of this work will be reviewed but unfortunately the conclusions to be drawn from these studies remains controversial.

In my talk I shall present arguments that the de Almeida-Thouless line exists in more than six dimensions: that is, RSB applies only when the dimensionality d > 6. These arguments involve both RG ideas and results derived from a 1/m expansion of an m-component spin glass model.

**Thin Graphs and (17) invisible states***Friday April 26, 2013, 16:00 h, DH Seminar room*

The order of a phase transition is usually determined by the nature of the symmetry breaking at the phase transition point and the dimension of the model under consideration. For instance, q-state Potts models in two dimensions display a second order, continuous transition for q = 2,3,4 and first order for higher q.

Tamura et al recently introduced Potts models with "invisible" states which contribute to the entropy but not the internal energy and noted that adding such invisible states could transmute continuous transitions into first order transitions. This was observed both in a Bragg-Williams type mean-field calculation and 2D Monte-Carlo simulations. It was suggested that the invisible state mechanism for transmuting the order of a transition might play a role where transition orders inconsistent with the usual scheme had been observed.

In this talk we discuss an alternative mean-field approach employing 3-regular random ("thin") graphs also displays this change in the order of the transition as the number of invisible states is varied, although the number of states required to effect the transmutation, 17 invisible states when there are 2 visible states, is much higher than in the Bragg-Williams case. The calculation proceeds by using the equivalence of the Potts model with 2 visible and r invisible states to the Blume-Emery-Griffiths (BEG) model, so a by-product is the solution of the BEG model on thin random graphs.

**Exploring Critical Loop Gases with Worms***Thursday April 25, 2013, 15:30 h, DH Seminar room*

TBA