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*Abstracts*

**OSP 2017 titles and abstracts**

**Costanza Benassi **(Warwick)

*Decay of correlations in 2d quantum systems*

The decay of correlation functions provides a great deal of information about the phase diagram of lattice systems. McBryan and Spencer proved that the two-point function decays as a power-law in the 2d classical XY model. Koma and Tasaki proved it for the quantum Hubbard model. In this talk I will discuss a recent generalisation of this result to a wide class of 2d quantum lattice systems with U(1) symmetry.

**Roberto Bondesan** (Oxford)

*Gaussian free fields at the integer quantum Hall plateau transition*

The transition between Hall conductance plateaus of the integer quantum Hall effect stands out as a possible paradigm for quantum phase transitions of Anderson-localization type.

Yet, in spite of numerous efforts and a renewed interest coming from the expanding field of topological phases, understanding this transition has so far defied an analytical solution by the methods of conformal field theory and/or the theory of integrable systems.

In this talk I will review our approach based on a supersymmetric vertex model and the construction of conformal primary fields on the lattice from highest weight vectors of the vertex model in the presence of point contacts.

The outcome of the analysis will be an effective free field description of critical wave intensities leading to a parabolic multifractality spectrum.

**James Burridge** (Portsmouth)

*What determines the geographical pattern of human dialects?*

The spatial variation in human dialects is the result of many significant historical events, but is driven on a small scale by vast numbers of individual interactions between millions of people. The study of the resulting spatial patterns of language use has become a quantitative discipline, making use of modern computational and statistical techniques. Borrowing ideas from Statistical Mechanics, here we formulate a theory which connects the language interactions of individuals to the macroscopic (geographical scale) structure of dialects. We find that transition regions between dialects experience a two-dimensional form of the surface tension effect seen in bubbles: they tend to get shorter, but this effect is mediated by variations in population density which induce curvature. Our theory predicts that the shape of coastline (or other borders), and the distribution of population act as an ``invisible hand’’ driving many complex spatial patterns of dialect use toward a small number of stable configurations. We apply our theory to Great Britain, producing results which closely resemble observations and predictions of dialectologists.

**Lax Chan** (Open)

*Substitution-based structures with absolutely continuous spectrum*

Generalising Rudin's construction, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. Several examples are discussed in detail. This includes substitutions of constant length (for any length) based on Fourier matrices.

**Claudia Clark **(UCL)

*Applications of the Stochastic Liouville-von Neumann Equation to Quantum Technology*

As nanoscale manipulations become increasingly feasible, a better understanding of quantum thermodynamics is required to improve our understanding of microscopic structures and subsequently increase the efficiency of operations on this scale. Various techniques currently exist to model the thermodynamic properties of quantum systems, but these generally rely on approximations. An exact method is the stochastic Liouville-von Neumann equation [1], based on unravelling Feynman-Vernon influence functionals [2]. We here extend its use from the one heat bath case [3] to consider the thermodynamic behaviour, including heat flow, of a system in a non-equilibrium stationary state brought about by coupling to more than one heat bath.

We develop this yet further to begin to consider using the SLN to model quantum systems, which could be used to experimentally realise quantum technologies such as heat engines and/or refrigerators. One scheme potentially amenable to modelling with the SLN equation is the recently proposed Otto refrigerator which used coupling of superconducting qubit to heat baths. Quantum thermodynamics experiments (such as electronic refrigeration) have been performed, so there is scope to ultimately compare SLN models with experimental data.

[1] Stockburger, J. T. (2004). Simulating spin-boson dynamics with stochastic Liouville–von Neumann equations. Chemical physics, 296(2), 159-169.

[2] Feynman, R. P., & Vernon, F. L. (1963). The theory of a general quantum system interacting with a linear dissipative system. Annals of physics, 24, 118-173.

[3] Schmidt, R., Carusela, M. F., Pekola, J. P., Suomela, S., & Ankerhold, J. (2015). Work and heat for two-level systems in dissipative environments: Strong driving and non-Markovian dynamics. Physical Review B, 91(22), 224303.

[4] Karimi, B., & Pekola, J. P. (2016). Otto refrigerator based on a superconducting qubit: Classical and quantum performance. Physical Review B, 94(18), 184503.

[5] Timofeev, A. V., Helle, M., Meschke, M., Möttönen, M., & Pekola, J. P. (2009). Electronic refrigeration at the quantum limit. Physical review letters, 102(20), 200801.

**Pablo de Castro** (Kings)

*Phase-separation dynamics of polydisperse colloids: a mean-field lattice-gas theory*

Phase separation in colloidal suspensions is investigated via a new dynamical theory based on the Polydisperse Lattice-Gas model. The model gives a simplified description of polydisperse colloids, incorporating a hard-core repulsion combined with polydispersity in the strength of the attraction between neighbouring particles. Our mean-field equations describe the local concentration evolution for each of an arbitrary number of species, and for an arbitrary overall composition of the system. We focus on the predictions for the dynamics of colloidal gas-liquid phase separation after a quench into the coexistence region. The critical points and the relevant spinodal curves are determined analytically, with the latter depending only on three moments of the overall composition. The results for the early-time spinodal dynamics show qualitative changes as one crosses a "quenched" spinodal that excludes fractionation, allowing only density fluctuations at fixed composition. This effect occurs for dense systems, in agreement with a conjecture by Warren that at high density fractionation is generically slow because it requires inter-diffusion of particles. We verify this conclusion by showing that the observed qualitative changes disappear when direct particle-particle swaps are allowed in the dynamics. Finally, the rich behaviour beyond the spinodal regime is examined, where the evaporation of gas bubbles with strongly fractionated interfaces causes long-lived composition heterogeneities in the liquid phase; we introduce a composition histogram method that allows such effects to be easily visualized for an arbitrary number of particle species. (Joint work with Peter Sollich)

**Carl Dettman **(Bristol)

*Connectivity of networks in complex geometries*

Wireless mesh networks consist of devices (“nodes”) that communicate directly with each other rather than via a central router or base station. This has benefits in reducing energy consumption, interference and single point of failure. Such networks are used in for example, sensor networks and the Internet of Things. Typically such networks are modelled using uniformly distributed random node locations, with the probability of a link decreasing with the mutual distance of the nodes. Popular models are the random geometric graph, in which links are made up to a specified distance threshold r0, and Rayleigh fading, in which the link probability is exp[-(r/r0)2]. There have been many results on the connectivity and other topological properties of such networks. In particular, it is known that in the limit of many nodes, connectivity is controlled by isolated nodes, which are in turn Poisson distributed.

In many scenarios, either the boundaries of the domain or the density of nodes is complex, far from uniform and with structure on many length scales. For example, the most popular and realistic human mobility models have these features. Remarkably, many of the features of uniform networks continue to hold in both smooth and fractal nonuniform node distributions, including the importance and distribution of isolated nodes. The connectivity transition itself takes place over a much broader range of node densities. These theoretical and numerical results will be discussed, as well as preliminary investigations into the boundary between this and other possible behavior. Much is yet to be understood.

**Vitaly Fain** (Bristol)

*Energy growth and scattering maps for a billiard inside a time-dependent symmetric domain close to an ellipse*

An ellipse is conjectured to be the only strictly convex curve associated to an integrable billiard. We study billiard dynamics inside an ellipse with a time-dependent periodic perturbation of the axes and and $O(\delta)$ quartic polynomial deformation of the boundary. In this situation the energy of the billiard is no longer conserved. We provide a proof of a type of Fermi acceleration: given a sufficiently large initial energy, there exists a billiard trajectory that reaches an arbitrarily larger energy value in some finite time. The proof depends on reduction of the billiard map to two Hamiltonian flows defined on the normally hyperbolic invariant manifold $\Lambda$ parametrised by energy and time in the phase space of the billiard. The two flows approximate inner and scattering maps, which are common tools that arise in the studies of Arnol'd diffusion. Melnikov type calculations imply that the scattering map is only defined on a subset of $\Lambda$ that increases with $\delta$ and becomes empty for $\delta=0$. This implies a degenerate splitting of invariant manifolds of $\Lambda$ in the first order perturbative parameters.

**Etienne Fodor** (Cambridge)

*Self-propelled particles as an active matter system*

Self-propelled particles are able to extract energy from their environment to perform a directed motion. Such a dynamics lead to a rich phenomenology that cannot be accounted for by equilibrium physics arguments. A striking example is the possibility for repulsive particles to undergo a phase separation, as reported in both experimental and numerical realizations. On a specific model of self-propulsion, we explore how far from equilibrium the dynamics operate. We quantify the breakdown of the time reversal symmetry, and we delineate a bona fide effective equilibrium regime. Our insight into this regime is based on the analysis of fluctuations and response of the particles [1]. Finally, we discuss how the nonequilibrium properties of the dynamics can also be captured at a coarse-grained level, thus allowing a detailed examination of the spatial structure that underlies departures from equilibrium [2].

[1] Phys. Rev. Lett. 117, 038103 (2016)

[2] arXiv:1610.06112, accepted for publication in Phys. Rev. X

**Ian Ford** (UCL)

*Computing nucleation barriers by simulation of molecular cluster mitosis*

The nucleation of droplets from a supersaturated vapour may be described as the clustering together of molecules and the surmounting of a thermodynamic barrier by statistical fluctuation. The free energy of formation of the cluster from monomers is one component of the barrier, and we compute this using nonequilibrium molecular dynamics simulation.

We separate a cluster into two daughter fragments and employ the Jarzynski equation to convert the required mechanical work of 'mitosis'

into a free energy. The barrier also depends on the condition of the vapour, but we find that we can proceed without the need to compute a saturated vapour density. The approach will be illustrated for clusters of water, and of caesium hydroxide.

**Nikolaos Fytas** (Coventry)

*Phase transitions in disordered systems: The example of the random-field Ising model*

In Statistical Physics, more often than not, the behavior of a strongly disordered system cannot be inferred from its clean, homogeneous counterpart. In fact, disordered systems are prototypical examples of complex entities in many aspects, mainly in the rough free-energy landscape profile. In the current talk, I will present some new results relevant to critical phenomena and universality aspects of disordered systems using as a platform the random-field Ising model at several spatial dimensionalities, below the upper critical one.

**Michal Gnacik **(Portsmouth)

*Infrequent social interaction can accelerate the spread of a persuasive idea*

We discuss the spread of a persuasive idea (the "up" spin state) through a population of continuous time random walkers in one dimension. The walkers’ movement is modelled via alpha stable Levy motions; therefore, apart from the standard Brownian motion we consider processes with jumps. Walkers adjust their opinions (the spin "up" or "down") in the gatherings, which occur according to a Poisson process, within certain interaction range. When the jump distribution is sufficiently heavy tailed, in order to increase the speed of propagation of the persuasive idea, we let the frequency of gatherings be smaller. (joint work with J. Burridge)

**John Hannay **(Bristol)

*Reconnection rate of moving vortex lines in a random wavefield*

Reconnection of moving lines in 3D is of interest in various contexts (e.g. magnetic field tubes, or superfluid vortex lines). Here the rate at which this happens is calculated exactly for perhaps the simplest model circumstance; zero lines ('vortices') of a random complex wavefield (arXiv 1702.04260).

**Rainer Klages** (Queen Mary)

*A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics*

Consider equations of motion that generate dispersion of an ensemble of particles in the long time limit. An interesting problem is to predict the diffusive properties of such a dynamical system starting from first principles. Motivated by conflicting numerical results on diffusion in polygonal billiards, we introduce an interval exchange transformation lifted onto the whole real line that mimicks deterministic diffusion in these billiards. By definition our simple map model is not chaotic, in the sense of exhibiting a vanishing Lyapunov exponent. We show analytically that it nevertheless displays a whole range of normal and anomalous diffusion under variation of a single control parameter [1].

[1] L. Salari et al., Chaos 25, 073113 (2015)

**Alexander Mozeika** (Kings)

*Spin systems on hypercubic Bethe lattices: A Bethe-Peierls approach*

We study spin systems on Bethe lattices constructed from d- dimensional hypercubes. Although these lattices are not tree-like, and therefore closer to real cubic lattices than Bethe lattices or regular random graphs, one can still use the Bethe-Peierls method to derive exact equations for the magnetization and other thermodynamic quantities. We compute phase diagrams for ferromagnetic Ising models on hypercubic Bethe lattices with dimension d = 2, 3, and 4. Our results are in good agreement with the results of the same models on d-dimensional cubic lattices, for low and high temperatures, and offer an improvement over the conventional Bethe lattice with connectivity k = 2d.

**Zohar Ringel **(Oxford)

*Classical topological paramagnets*

Topological phases of matter are one of the hallmarks of quantum condensed matter physics. One of their striking features is a bulk-boundary correspondence wherein the topological nature of the bulk manifests itself on boundaries via exotic massless phases. In classical wave phenomena analogous effects may arise; however, these cannot be viewed as equilibrium phases of matter. In this talk I'll give a set of simple rules under which robust equilibrium classical topological phenomena can exist. In particular I'll describe several simple and analytically tractable classical lattice models of Ising spins in two and three dimensions which, at suitable parameter ranges, are paramagnetic in the bulk but nonetheless exhibit some unusual long-range or critical order on their boundaries. The possibility of experimentally realizing such models would be shortly discussed.

**David Saad** (Aston)

*Optimal deployment of resources for maximizing impact in spreading processes*

The modern world can be best described as interlinked networks, of individuals, computing devices and social networks; where information and opinions propagate through their edges in a probabilistic or deterministic manner via interactions between individual constituents. These interactions can take the form of political discussions between friends, gossiping about movies, or the transmission of computer viruses. Winners are those who maximise the impact of scarce resource such as political activists or advertisements, or by applying resource to the most influential available nodes at the right time. We developed an analytical framework, motivated by and based on statistical physics tools, for impact maximisation in probabilistic information propagation on networks; to better understand the optimisation process macroscopically, its limitations and potential, and devise computationally efficient methods to maximise impact (an objective function) in specific instances.

The research questions we have addressed relate to the manner in which one could maximise the impact of information propagation by providing inputs at the right time to the most effective nodes in the particular network examined, where the impact is observed at some later time. It is based on a statistical physics inspired analysis, Dynamical Message Passing that calculates the probability of propagation to a node at a given time, combined with a variational optimisation process. The work has successfully addressed the following questions: 1) Given a graph and a propagation/infection process, which nodes are best to infect to maximise the spreading? given a limited budget, how many nodes one can infect within a given time? how long will it take one to infect all nodes? 2) Maximising the impact on a subset of particular nodes at given times, by accessing a limited number of given nodes. 3) Identify the most appropriate vaccination targets in order to isolate a spreading disease through containment and islanding of an epidemic. Numerical studies on benchmark problems show the efficacy of the method both for information propagation in time and vaccination.

**Elsen Tjhung **(Cambridge)

*Active Chiral Fluids*

Active fluids are a class of non-equilibrium systems where energy is injected into the system continuously by the constituent particles themselves. Here we study by computer simulation the dynamics of an unstructured droplet of chiral active fluid in three dimensions. This is motivated by swimming motility in some microorganism such as Toxoplasma and Plasmodium. It has been observed that these organisms swim in a helical trajectory and that the swimming motility is driven mostly by actomyosin contraction. However, the actin filaments are right-handed helices whereas the trajectory of motion is left-handed helix. Thus the microscopic chirality seems to be reversed at the macroscopic level of motion. Here, we propose a generic mechanism for such chirality reversals.

**Philipp Verpoort** (Cambridge)

*Colour-dependent interactions in the three-coloring model*

The three-coloring model was first introduced by Baxter in 1970 as the calculation of the "number of ways ... of coloring the bonds of a hexagonal lattice ... with three colors so that no adjacent bonds are colored alike". The model has since been studied in several contexts, from frustrated magnetism to superconducting devices and glassiness. In presence of interactions, when the model is no longer exactly soluble, it has already been observed that the phase diagram is highly non-trivial. In our work, we discuss the generic case of ‘color-dependent’ nearest-neighbor interactions between the vertex chiralities. We use several mappings to other soluble models, such as loop models, and Ising models, as well as numerical techniques to unveil the full phase diagram of the model. We show that it features an extremely rich phase diagram with different ordered phases which are separated by lines, sheets and even three-dimensional regions of critical points (in parameter space). We uncover different critical regimes merging into one another: c = 1/2 free fermions combining into c = 1 free bosons; c = 1 free bosons combining into c = 2 critical loop models; as well as three separate c = 1/2 critical lines merging at a seemingly supersymmetric c = 3/2 critical point.

When the three coupling constants are tuned to equal one another, transfer-matrix calculations highlight a puzzling regime where the central charge varies 'continuously' from 3/2 to 2.

**Nick Watkins** (LSE)

*Long range dependence, fractional renewal models, and Bayesian inference*

[Joint work with Christian Franzke (Hamburg), Bobby Gramacy (Virginia Tech) and Tim Graves (Arup) ]

Since the 1960s, long range dependence (LRD) as embodied by the fractional Gaussian noise and ARFIMA models, has been a well-studied mechanism for the origin of 1/f noise and the Hurst effect. This talk will draw on the authors' recent papers [Graves et al, Physica A, 2017; Watkins, arXiv, 2016; Franzke et al, Scientific Reports, 2015; Graves et al, NPG, 2015; Graves et al, arXiv 2014, to be submitted to Entropy] to discuss two new avenues of research.

The first concerns breakpoints. These have long been known to be a source of the Hurst effect, but recent research by one of us has shown that Mandelbrot had proposed a model with power law intervals between the breaks as early as 1963, and that by 1965-67 he was showing how this was an alternative non-ergodic model for 1/f noise, with consequences for model choice and time series interpretation that are increasingly becoming topical in physics and eslewhere.

The second concerns Bayesian inference when an LRD model is plausible I will discuss our recent work on a novel systematic Bayesian approach for joint inference of the memory and tail parameters in an ARFIMA model with heavy-tailed innovations.

**Martin Weigel** (Coventry)

*Fragmentation of fractal random structures*

Breakup phenomena are ubiquitous in nature and technology. They span a vast range of time and length scales, including polymer degradation as well as collision induced fragmentation of asteroids. In geology, fragmentation results in the distribution of grain sizes observed in soils; fluids break up into droplets and fluid structures such as eddies undergo fragmentation. On the subatomic scale, excited atomic nuclei break up into fragments. Practical applications, such as mineral processing, ask for optimizations according to technological requirements and efficiency considerations. More generally, a wide range of structures from transport systems to social connections are described by complex networks, whose degree of resilience against fragmentation is a recent subject of intense scrutiny.

In this talk I will give an introduction to fragmentation phenomena and show how they can be described in mean-field theory using a rate equation. Going beyond mean-field theory, I will analyze the fragmentation behavior of random clusters on the lattice. Using a combination of analytical and numerical techniques allows for a complete understanding of the critical properties of this system. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes, where the fragment size distribution encodes characteristic fingerprints of the fragmented objects.