(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
D. A. Johnston, M. Mueller and W. Janke, Plaquette Ising models, degeneracy and scaling, Eur. Phys. J. Special Topics, 226, Issue 4, 749–764 (2017), [arXiv:1612.00060]
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.
C. Baillie, D. Johnston and R. Williams, Crumpling in Dynamically Triangulated Random Surfaces with Extrinsic Curvature: Nucl. Phys. B335, 469-501, (1990)
C. Baillie, D. Johnston and R. Williams, Non-Universality in Dynamically Triangulated Random Surfaces with Extrinsic Curvature: Mod. Phys. Lett. A5, 1671-8, (1990)
C. Baillie, D. Johnston and R. Williams, Crumpling Dynamically Triangulated Random Surfaces in Higher Dimensions: Phys. Lett. B243, 358-364, (1990)
C. Baillie, D. Johnston and R. Williams, Computing Aspects of Simulating Dynamically Triangulated Random Surfaces: Comp. Phys. Comm. 58-65, 105, (1990)
C. Baillie G. Kilcup and D. Johnston, Computational Status and Prospects of Lattice Calculations in High Energy Physics: Jour. of Supercomputing, 277-300, (1990)
C. Baillie, S. Catterall, D. Johnston and R. Williams, Further Investigations of the Crumpling Transition in Dynamically Triangulated Random Surfaces: Nucl. Phys. B348, 543, (1991)
C. Baillie, D. Johnston, Crumpling Dynamically Triangulated Random Surfaces in Two Dimensions: Phys. Lett. B258, 346-52, (1991)
C. Baillie and D. Johnston, Weak Self Avoidance and Crumpling in Random Surfaces with Extrinsic Curvature: Phys. Lett. B273, 380-8, (1991)
C. Baillie and D. Johnston, Strong Self-Avoidance and Crumpling in Random Surfaces with Extrinsic Curvature : Phys. Lett. B283, 55-62, (1992)
C. Baillie and D. Johnston, A Modified Steiner Fuctional String Action: Phys. Rev. D45, 3326-30, (1992)
C. Baillie and D. Johnston, An Effective Model for Crumpling in Two-Dimensions: Phys. Rev. D46, 4761-4, (1992)
C. Baillie and D. Johnston, Crossover Between Weakly and Strongly Self-Avoiding Random Surfaces: Phys. Lett. B295, 249-55, (1992)
C. Baillie, D. Espriu and D. Johnston, Steiner Variations on Random Surfaces: Phys. Lett. B305, 109-114, (1993)
C. Baillie and D. Johnston, Freezing Fluid Random Surfaces: Phys. Rev. D48, 5025-8, (1993)
C. Baillie and D. Johnston, Smooth Random Surfaces from Tight Immersions?: Phys. Rev. D49, 4139-43, (1994)
C. Baillie, A. Irback, W. Janke and D. Johnston, Scaling in Steiner Random Surfaces : Phys. Lett. B325, 45-50, (1994)
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.
C. Baillie and D. Johnston, A Numerical Test of KPZ Scaling: Potts Models Coupled to 2D Quantum Gravity: Mod. Phys. Lett. A7, 1519-31, (1992)
C. Baillie and D. Johnston, Multiple Potts Models Coupled to Two-Dimensional Quantum Gravity: Phys. Lett. B286, 44-52, (1992)
C. Baillie and D. Johnston, The XY Model Coupled to Two-Dimensional Quantum Gravity: Phys. Lett. B291, 233-40, (1992)
D. Johnston, Ising (anti-)ferromagnet on dynamical triangulations and quadrangulations: Phys. Lett. B314, 69-73, (1993)
C. Baillie and D. Johnston, 2D O(3) model coupled to 2D Quantum Gravity: Phys. Rev. D49, 603-6, (1994)
C. Baillie and D. Johnston, Damaging 2D Quantum Gravity: Phys. Lett. B326, 51-6, (1994)
D. Johnston, Frustrating and Diluting Dynamical Lattice Ising Spins: Phys. Lett. B336, 229-36, (1994)
C. Baillie and D. Johnston, Square Gravity: Phys. Lett. B357, 287-294, (1995)
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.
D. Johnston, Zero Potts Models Coupled to Two-Dimensional Gravity: Phys. Lett. B277, 405-10, (1992)
C. Baillie, K. Hawick and D. Johnston, Quenching 2D Quantum Gravity: Phys. Lett. B328, 284-90, (1994)
C. Baillie, W. Janke and D. Johnston, Softening of Phase Transitions on Quenched Random Gravity Graphs: Phys. Lett. B388, 14-20, (1996)
W. Janke and D. Johnston, The Wrong Kind of Gravity, Phys. Lett. B460 271-275, (1999)
W. Janke and D. Johnston, Ising and Potts Models on Quenched Random Gravity Graphs, Nucl. Phys. B578 681-698, (2000)
W. Janke and D. Johnston, Non-self-averaging in autocorrelations: Ising and Potts Models on Quenched Random Gravity Graphs, J. Phys. A33 2653-2662, (2000)
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.
C. Baillie, D. Johnston and J-P. Kownacki, Ising Spins on Thin Graphs: Nucl. Phys. B432, 551-570, (1994)
C. Baillie, W. Janke, D. Johnston and P. Plechac, Spin Glasses on Thin Graphs: Nucl. Phys. B450 [FS], 730-752, (1995)
C. Baillie, N. Dorey, W. Janke and D. Johnston, The Villain Model on Thin Graphs: Phys. Lett. B369, 123-9, (1996)
C. Baillie, D. Johnston, E. Marinari and C. Naitza, Dynamic Behavior of Spin Glasses on Quenched phi3 graphs: J. Phys. A29, 6683-6691, (1996)
D. Johnston and P. Plechac, Potts Models on Feynman Diagrams: J. Phys. A30, 7349-7363, (1997)
D. Johnston and P. Plechac, Equivalence of Ferromagnetic Spin Models on Trees and Random Graphs: J. Phys. A31 475-482, (1998)
D. Johnston, The Yang Lee Edge Singularity on Feynman Diagrams: J. Phys. A31 5461-5469, (1998)
D. Johnston, Thin Animals, J. Phys. A31 9405-9417, (1998)
D. Johnston and P. Plechac, Vertex Models on Feynman Diagrams, Phys. Lett. A248 37-45, (1998)
D. Johnston, A Potts/Ising Correspondence on thin graphs, J. Phys. A32 5029-5036, (1999)
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.
P. Bialas, Z. Burda and D. Johnston, Condensation in the Backgammon Model: Nucl. Phys. B493, 505-516, (1997)
P. Bialas, Z. Burda and D. Johnston, Balls in Boxes and Quantum Gravity: Nucl. Phys. B (Proc. Suppl.) 63A-C 763-765, (1998)
P. Bialas, Z. Burda and D. Johnston, Phase Diagram of the Mean Field Model of Simplicial Quantum Gravity, Nucl. Phys. B542 413-424, (1999)
P. Bialas, L. Bogacz, Z. Burda and D. Johnston, Finite Size Scaling in the Balls in Boxes Model, Nucl. Phys. B575 599-612, (2000)
Z. Burda, D. Johnston, J. Jurkiewicz, M. Kaminski, M.A. Nowak, G. Papp, I. Zahed, Wealth Condensation in Pareto Macro-Economies, Phys. Rev. E65, 026102, (2002)
Z. Burda, D. Johnston, J. Jurkiewicz, M. Kaminski, M.A. Nowak, G. Papp, I. Zahed, Wealth Condensation and “Corruption” in a Toy Model, Acta Phys. Pol. 36 2442, (2005)
R. Blythe W. Janke, D. Johnston and R. Kenna, The Grand Canonical Asymmetric Exclusion Process and the One-Transit Walk, J. Stat. Mech. P06001, (2004)
R. Blythe W. Janke, D. Johnston and R. Kenna, Dyck Paths, Motzkin Paths and Traffic Jams, J. Stat.Mech. P10007, (2004)
R. Blythe W. Janke, D. Johnston and R. Kenna, Continued Fractions and the Partially Asymmetric Exclusion Process, J. Phys. A42 325002, (2009)
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.
B. Dolan, W. Janke, D. Johnston and M. Stathakopoulos, Thin Fisher Zeroes, J. Phys. A34 6211-6223, (2001)
W. Janke, D. Johnston and M. Stathakopoulos, Fat Fisher Zeroes, Nucl. Phys. B614 494-512, (2001)
B. Dolan and D. Johnston, 1D Potts, Lee-Yang Edges and Chaos, Phys. Rev. E65, 057103, (2002)
W. Janke, D. Johnston and M. Stathakopoulos, A Kertesz Line on Planar Random Graphs?, J. Phys. A35, 7575, (2002)
W. Janke, D. Johnston and R. Kenna, New Methods to Measure Phase Transition Strength, Nucl. Phys. B (Proc. Suppl.), (2002)
W. Janke, D. Johnston and R. Kenna, Phase Transition Strength through Density of General Distributions of Zeros, Nucl. Phys. B [FS] 682/3, 618-634, (2004)
W. Janke, D. Johnston and R. Kenna, Properties of Higher Order Phase Transitions, Nucl. Phys. B 736, 319, (2005)
W. Janke, D. Johnston and R. Kenna, Properties of phase transitions of higher order, Proceedings of Science (Lattice 2005), 244, (2005)
W. Janke, D. Johnston and R. Kenna, Phase Transition Strength from General Distributions of Zeroes, Computer Physics Communications 169, proceedings of CCP-2004, 457-461, (2005)
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.
W. Janke, D. Johnston and R. Kenna, Scaling Relations for Logarithmic Corrections, Phys. Rev. Lett. 96 115701, (2006)
W. Janke, D. Johnston and R. Kenna, Self-consistent Scaling Theory for Logarithmic Correction Exponents, Phys. Rev. Lett. 97, 155702, (2006)
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.
D. A. Johnston and R. P. K. C. M. Ranasinghe, Potts Models with (17) Invisible States on Thin Graphs, J. Phys. A46 225001, (2013)
N. Ananikian, N. Sh. Izmailyan, D. A. Johnston, R. Kenna and R. P. K. C. M. Ranasinghe, Potts models with invisible states on general Bethe lattices, J. Phys. A46 385002, (2013)
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.
B. Dolan, D. Johnston and R. Kenna, The Information Geometry of the 1D Potts Model, J. Phys. A 35, 9025-9035, (2002)
W. Janke, D. Johnston and R.P.K.C. Malmini, The Information Geometry of the Ising Model on Planar Random Graphs, Phys. Rev. E66, 056119, (2002)
W. Janke, D. Johnston and R. Kenna, The Information Geometry of the Spherical Model, Phys. Rev. E67, 046106, (2003)
W. Janke, D. Johnston and R. Kenna, Information Geometry, One, Two, Three and Four, Acta Phys. Pol. 34 4923-4937, (2003)
W. Janke, D. Johnston and R. Kenna, Information Geometry and Phase Transitions, Physica A, 336 181-186, (2004)
W. Janke, D. Johnston and R. Kenna, Geometrothermodynamics of the Kehagias-Sfetsos Black Hole, J. Phys. A43 425206, (2010)
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).