(Fairly) Current Research Interests:

3D Plaquette Ising Model, Fractons etc

The 3D Plaquette Ising Model is a recurring theme. It first crossed the horizon in 1997 when Mariano Baig, Dominic Espriu, R.K.P.C. Malmini (a former PhD student) and myself discovered that it had a first order phase transition. This was bad news at the time, since we were looking for continuum limits which require a continuous phase transition. This was motivated by the genesis of the model in the work of Savvidy and Wegner on Gonihedric string and surface models.

Nonetheless, the dynamical properties of the model, which appeared to be glassy even in the absence of any quenched disorder, were interesting and were pursued with Adam Lipowski when he visited Heriot-Watt for a year in 1998-9.

More recent (>2014) work with Marco Mueller and Wolfhard Janke of Leipzig University found non-standard scaling behaviour at the first order transition since the the low temperature phase is highly degenerate as the result of a subsystem symmetry in the model, intermediate between a global and a local (gauge) symmetry.

Such subsystem symmetries have recently (>2016) been implicated in
fractonic models, in which quasiparticle excitations display restricted (or even zero) mobility. Indeed, gauging the subsystem symmetry in the quantum 3D Plaquette Ising Model gives rise to a canonical fractonic model, the X-cube model, in much the same way as gauging the global symmetry of the standard quantum 2D Ising model gives rise to the toric code.


(Fairly) Recent Papers


Past Research Interests

Broadly, my interests lie in statistical mechanics, particularly lattice models of spin systems, though I started my own random walk in theoretical particle physics (see the PhD section). Encountering Clive Baillie at Caltech, where we were both postdocs, led to a diversion into discretized models of random surfaces and spin models on dynamical lattices that continued after Clive moved to U.C.Boulder and I (eventually) ended up at Heriot-Watt.

Discretized Models of Random Surfaces

Discretized random surface models can be thought of either as models of real membranes, e.g. lipid bilayers or cell membranes, if they are embedded in three dimensions or, more esoterically, of (euclidean) string worldsheets. For the latter, the idea is to find a suitable lattice action and identify a continuum limit to define a string theory.



Spin Models on Dynamical Random Lattices

On the other hand, spin models models living on suitable dynamical (connectivity) lattices give a toy model for gravity + matter systems that incorporates the back reaction between matter and geometry. In the 2D case some can be solved exactly via matrix model methods and critical exponents extracted for comparison with the numerical studies.



Spin Models on Quenched Random Lattices

It is also possible to think about fixing, or freezing, the random lattices on which the spins live. There one is considering a "quenched" ensemble, familiar from disordered systems and spin glasses. Instead of the more commonly considered case of quenched bond disorder, one has quenched connectivity disorder in this case. A question that is interesting to address is how such quenched connectivity disorder affects phase transitions.



Spin Models on "Thin" Graphs

Spin model configurations on 2D random lattices (graphs) can be generated by the Feynman diagrams of various matrix models. The planar limit of such an NxN matrix model is given by taking N -> infinity. The opposite limit of N->1 ("thin" random graphs) is also of interest and is related to spin models on Bethe lattices and the mean field approach. A second stay at Orsay in 1993-4 as a Marie Curie fellow offered the opportunity to work on this with a PhD student there, Jean-Philippe Kownacki, which continued on returning to Heriot-Watt with Petr Plechac and others.




Balls in Boxes (ZRP)

Some forays have also been made into solvable nonequilibrium models such as the ASEP (Asymmmetric Exclusion Process) and ZRP (Zero Range Process) with Richard Blythe at Edinburgh University and Piotr Bialas and Zdzislaw Burda in Krakow. This even spilled over into "econophysics" with the Polish collaborators.



ASEP


Other long-standing interests have included partition function zeroes in statistical mechanics, scaling relations for higher order transitions, logarithmic exponents, "invisible states" and applications of information geometric ideas as orginally pioneered by Ruppeiner to the study of phase transitions in various systems, including families of black holes. Partners in crime in these enterprises have included Ralph Kenna , Brian Dolan and Wolfhard Janke, with schemes often being hatched at Wolfhard's annual CompPhys meetings in Leipzig.


Partition Function Zeroes

Lee and Yang were the first to realize that extending the parition function of a statistical mechanical model into the complex plane gave a useful way of understanding how the non-analyticities that characterize phase transitions could emerge from lattice models in the thermodynamic limit. Their work considered field driven transitions and was extended by Fisher to thermal transitions. The scaling of the density of the partition function zeroes that arises in such an approach gives a useful alternative method for extracting the critical properties of phase transitions and deciding their nature.


Logarithmic Exponents

Logarithmic corrections to scaling often appear at upper critical dimensions. Consideration of the scaling functions for partition function zeroes derives scaling relations between such terms, analogous to the more standard power law scaling relations familiar from other critical exponents.



Invisible States

Usually states in a spin model contribute to both energy and entropy, but it is possible (and even physically reasonable) to have "invisible" states which only contribute entropically.



Information Geometry

The notion of Information Geometry, the study of the geometrical structure of families of probability distributions, is a long-standing one in statistics. Ruppeiner and Weinhold were the first to suggest that these ideas could also be applied in thermodynamics and statistical mechanics. The geometry of the parameter space of a statistical mechanical model in this approach gives an alternative way of characterizing phase transitions.



Pain

On the admin side I survived coordinating an EC HCM (Human Capital and Mobility) network grant: "Analytical and Numerical Investigations of Random Geometries" (1993-1996) with only marginal hair loss and was foolish enough to take on coordinating another EC IHP (Improving Human Potential - they like their TLAs) network grant "Discrete Random Geometries: From Solid State Physics to Quantum Gravity" and an associated ESF (European Science Foundation) network "Geometry and Disorder" for the period 2000-2003/4 but I cunningly passed the role of network co-ordinator onto Renate Loll for the Framework 6 network ENRAGE (2005-2009).