Attendees of the meeting are listed below. For attendees giving a talk, click on their talk title to see the abstract.
|Remy Adderton||The N-1 coupled Temperley-Lieb algebra structure of the N-state super-integrable chiral Potts chain
Abstract: The N-state super-integrable chiral Potts chain is a remarkable generalisation of the more widely studied quantum Ising chain in a transverse field. In this talk I will show that this N-state model has an algebraic structure given by N-1 coupled Temperley-Lieb algebras. This is analogous to the well known relation between the quantum Potts chain and the Temperley-Lieb algebra. Both periodic and open boundary conditions are considered. I will also discuss some representations of the coupled Temperley-Lieb algebra in the one- and two-boundary cases.
|Murray Batchelor||Some correlations in the quantum Ising chain
Abstract: The quantum Ising chain in a transverse field is arguably one of the most well known exactly solved models. Among the several reasons for this is because the model is solved in terms of free fermions via the Jordan-Wigner transformation. I will describe some calculations for spin correlations in this model with open boundary conditions, and discuss how these results compare with those obtained previously for the more usual periodic boundary conditions. It will become evident that these calculations form the starting point for something more general.
|Vladimir Bazhanov||Towards Canonical Quantization of Non-Linear Sigma-Models
Abstract: In this talk we revisit the problem of canonical quantization of two-dimensional non-linear sigma models (NLSM) in two dimensions. We unravel the integrable structure of the O(3) NLSM and its one-parameter deformation – the sausage model, by resolving the long-standing “non-ultralocality” problem. Our consideration is based on the continuous version of the the Quantum Inverse Scattering Method enhanced by a powerful ODE/IQFT correspondence, which connects stationary states of Integrable QFT models with special solutions of classical integrable equations. This approach leads to new efficient methods for computation of vacuum eigenvalues of the continuous analogs of quantum transfer-matrices for the O(3)/sausage NLSM.
|Nicholas Beaton||Partition function zeros of adsorbing walk models
Abstract: The zeros of the size-n partition functions for a statistical mechanical model can be used to help understand the critical behaviour of the model as n→∞. I will discuss recent work on the partition function zeros of adsorbing walk models, including Monte Carlo results of Janse van Rensburg for self-avoiding walks in two and three dimensions, and exact results by the two of us for adsorbing Dyck paths.
|Chris Bradly||Critical exponents of the adsorption transition of lattice polymers in two and three dimensions
Abstract: Adsorption of polymers to a surface is a well-studied problem with applications to adhesion, surface coating, chromatography and wetting. For polymers grafted to an attractive surface there is a phase transition between a high-termperature desorbed phase, where the polymer is extended in the d-dimensional bulk, and a low-temperature adsorbed phase, where the polymer is restricted to the (d-1)-dimensional surface. The critical behaviour of the transition between these phases is governed by several exponents including the crossover exponent φ and shift exponent 1/δ. In particular, it is believed that φ may be a superuniversal exponent, having the exact value 1/2 in all dimensions. However, recent numerical evidence for the three-dimensional case challenges this notion, suggesting that φ ≠ 1/2 and also that φ ≠ 1/δ, in contrast to the two-dimensional case. Unfortunately, the many results of numerical studies appear to be inconsistent with each other and thus the value of these exponents is still an open question. We present new numerical results for simulations of lattice polymers in the presence of an impermeable adsorbing surface. We use the flatPERM algorithm – an athermal, flat histogram, chain-growth method – to simulate polymers modelled as self-avoiding walks and trails on a bounded lattice. To avoid bias from different lattice models we look at self-avoiding walks on the hex, square and cubic lattices and self-avoiding trails on the square and cubic lattices, in all cases up to length 1024. For each of these lattice models we use finite-size scaling analysis to explore several different ways of estimating the critical temperature and hence the critical exponents. The resulting spread in values for φ and 1/δ around the expected value of 1/2 indicates that there is a significant systematic error in finite-size simulations of lattice polymers at the adsorption transition. This far exceeds the statistical error usually reported. We conclude that in three dimensions the numerical evidence does not preclude the possibility that φ=1/δ=1/2.
|Joshua Capel||Equitable presentation for the Racah algebra R(4) and orthogonal polynomials for su(1,1)
Abstract: This talk will briefly discuss how the Racah polynomials naturally arise as the interbasis expansion coefficients between different positive discrete representations of the Racah algebra R(3), and how this generalises to the Racah algebra R(4). This gives rise to the bivariate Racah polynomials discover by Tratnik  which are diagonalised by a three-term and nine-term operator, but it also gives rise to some newer bivariate Racah polynomials diagonalised by a 9-term and 6-term operator (Joint work with Sarah Post).
|Wee Chaimanowong||Seiberg-Witten Theory and Frobenius Manifolds
Abstract: In 2004, Dubrovin proposed a connection between Seiberg-Witten theory and Frobenius manifolds. More precisely, a certain Frobenius manifold was constructed such that its odd period coordinates are related to A and B-periods on a Seiberg-Witten curve. However, it is not well-understood why such a connection exists. In this talk, I will explore how one might arrive at a Frobenius manifold naturally from Seiberg-Witten theory and how we might generalize this result.
|Zeying Chen||Exact solution to 2 species TASEP
Abstract: The exact solution to the single species totally asymmetric simple exclusion process (TASEP) has been discovered in 1997, known as the Schütz’s formula. In this talk, we would like to give the solution to the two species TASEP, by using the Bethe ansatz. We will derive the Green’s function to the two species TASEP, and some asymptotic observable obtained by using this Green’s function.
|Maria Eugenia Gabach Clement||Geometrical inequalities for black holes and ordinary objects.
Abstract: In this talk we review some of the relations that General Relativity sets among physical parameters of black holes and ordinary objects. These geometrical inequalities show, for instance, that a black hole of given mass can not rotate too fast, or that an ordinary object of given size can not have too much electric charge. We present the well known results in this area and the basic questions that are still unanswered.
|Nathan Clisby||Monte Carlo studies of dense polymers
Abstract: I will describe how to efficiently implement connectivity changing moves which allow for rapid Monte Carlo sampling of dense polymer configurations, such as Hamiltonian paths. I will then give an overview of some important features of dense polymers, such as the occurrence of long-range correlations within the chains and the effect of entanglement on dynamics, and explain how efficient Monte Carlo algorithms will help us to understand them.
|Jan de Gier|
|Holger Dullin||Toric, Semi-toric, and beyond
Toric integrable systems with their momentum maps whose images are rational polytopes are well understood. More recently San Vu Ngoc and Alvaro Pelayo classified semi-toric system in two degrees of freedom with additional singularities of focus-focus type. I will briefly review both toric and semi-toric systems. Then I will show that semi-toric systems can always be deformed such that the global circle action remains intact, but the focus-focus point is replaced by an elliptic-elliptic point and possibly additional singularities, some of which are hyperbolic (joint work with Alvaro Pelayo ). This deformation is inspired by the Hamiltonian Hopf bifurcation, well known in dynamical systems. This mechanism can be used to deform toric systems into semi-toric systems, and also to deform semi-toric systems into hyperbolic semi-toric systems. I will discuss some examples of such deformations (joint work with Joachim Worthington ) and the behaviour of the joint spectrum of the corresponding integrable quantum systems.
 Holger R. Dullin and Alvaro Pelayo. Generating hyperbolic singularities in semitoric systems via Hopf bifurcations. Journal of Nonlinear Science, 26(3), pp 787-811, 2016.
 Holger R. Dullin and Joachim Worthington. The polygonal invariant of a deformed spin-oscillator with hyperbolic singularities. (in preparation)
|Angela Foerster||Multi-well quantum tunneling models
In this work we present a general construction of integrable models for boson tunneling in multi-well systems , with a particular emphasis in the three and four well cases. Algebraic aspects of this construction are discussed. As an application we show how to engineer an atomtronic switching device by breaking the integrability of the triple well system . This is done by acting with an external field in one of the wells. By increasing its value allows for tuning the system from the switched-on configuration through to switched-off, with a precise level of control.
 Quantum integrable multi-well tunneling models, L. Ymai, A. Tonel, A. Foerster, J .Links, J. Phys. A50 (2017) 264001
 Control of tunneling in a triple-well system: an atomtronic switching device K. Wilsmann, L. Ymai, A. Tonel, J. Links and A. Foerster, arXiv:1710.05831
|Alexander Garbali||Lattice integrable stochastic processes
Abstract: Using a certain twisting procedure XXZ-type integrable Hamiltonians can be turned into Markov matrices which define integrable stochastic processes. Integrability provides numerous tools to study non-equilibrium characteristics of these processes giving new insights about their universality class (typically Kardar-Parisi-Zhang). Famous examples of these processes include Asymmetric Simple Exclusion Process (ASEP) and Zero Range Process (ZRP). I will give an algebraic description of ASEP, ZRP and their multi-species generalizations. In the most general setting of the multi-species ZRP process I will discuss the steady state vector and its relation to the Macdonald theory of symmetric functions. This information is important for computing current and density profiles as well as observable quantities.
|Andrus Giraldo||Shilnikov-like Chaos in Two Nonlinear Coupled Photonic Nanocavities
Recent experiments have shown that two coupled photonic crystal (PhC) nanocavities exhibit spontaneous symmetry breaking and bistable behaviour. In particular, bistability has been extensively studied in the last decades due to its usefulness for optical memories and logical switching. Theoretically, bistability and symmetry breaking have been observed in the Bose–Hubbard model, which describes the dynamics of the two coupled PhC nanocavities.
We consider an extension of the Bose–Hubbard model for the slowly varying amplitudes A and B of the electric fields in each nanocavity. It is given as a set of two complex ordinary differential equations determined by a photon lifetime τ, linear couplings between the cavities κ and γ, a detuning from the cavity resonance δ, and a coherent driving term f. Previous work on this model has centered in delimiting regions in the (κ,f)-parameter plane where symmetric and asymmetric continuous-wave solutions exist.
Our work focusses on the overall dynamics of this extended Bose–Hubbard and, in particular, the existence and disappearance of self-pulsations. As more energy is pumped into the system, represented by an increase of the coherent driving term f, self-pulsations arise from Hopf bifurcations, which then disappear in sequences of homoclinic bifurcations. In particular, we find chaotic Shilnikov bifurcations and the appearance of chaotic attractors. They imply complicated dynamics taking place in the individual cavities, or both of them simultaneously. We present the overal changes of this system, from simple to chaotic dynamics, as a function of the coherent driving term and the detunning from the cavity resonance. More specifically, we present the bifurcation diagram in the (f,δ)-plane as the other parameter values are fixed at experimental values. Our global bifurcation analysis of the two coupled photonic crystal nanocavities predicts types of (chaotic) dynamics well within the range of future experiments
|Jens Grimm||Universal finite-size scaling above the upper critical dimension
Abstract: Finite-size scaling (FSS) is a fundamental physical theory within statistical mechanics, describing the asymptotic approach to the thermodynamic limit of finite systems in the neighbourhood of a critical phase transition. It is well-known that models of critical phenomena typically possess an upper critical dimension, dc, such that in dimensions d ≥ dc, their thermodynamic behaviour is governed by critical exponents taking simple mean-field values. In contrast to the simplicity of the thermodynamic behaviour, the theory of FSS in dimensions above dc is surprisingly subtle, and remains the subject of ongoing debate. In this talk, we will clarify recent claims regarding the correct FSS behaviour on hypercubic lattices with free and periodic boundaries, and provide a consistent scaling picture for both boundary conditions. Our findings are supported by Markov-chain Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices.
|Anthony Guttmann||Eulerian orientations
Abstract: Eulerian orientations are a class of maps with structure. 4-valent Eulerian orientations are a particular case of the 6-vertex model when all vertices are equally weighted. We discuss the solution of this problem, as well as the more general case when orientations are counted by edges.
|Robert Alexander Henry|
|Mark Holmes||Limiting behaviour of discrete-measure-valued processes
Abstract: We’ll give a survey of ongoing and previous work of various authors describing the limiting behaviour of the voter model, critical lattice trees, and other statistical mechanics models in high dimensions.
|Phillip Isaac||Number parity effects in a bosonic tunneling model
Abstract: We present a bosonic tunneling model and analyse it along a critical line in parameter space. In this special case, the energy spectrum is computed analytically, and several key features are discussed which on the particle number being odd or even.
|Iwan Jensen||Partition function zeros of the Ising model
Abstract: We present selected results from our recent studies of the partition function zeros of the Ising model on several lattices, with various boundary conditions, with and without a magnetic field and bond/site disorder. Apart from lots of pretty pictures we will also present some precise quantitative results.
|Kazuya Kawasetsu||The characters of relaxed highest-weight modules over affine Kac-Moody algebras
Abstract: The relaxed highest-weight modules over affine Kac-Moody algebras play an important role in the Creutzig-Ridout Verlinde formula for admissible affine vertex algebras. In this talk, we compute the characters of the irreducible relaxed highest-weight modules over the affine Kac-Moody algebra sl2 induced from the dense irreducible modules over sl2, using Mathieu’s coherent families. We show that the characters are “coherent”, that is, they are the product of a q-series and a formal delta function in z. If time allows, we will also consider the characters of relaxed highest-weight modules over osp(1|2) and sl3.
|Andrew Keane||State-dependent delays in the El Niño Southern Oscillation system
Abstract: Delay differential equations (DDEs) have been used successfully in the past to model climate systems at a conceptual level. An important aspect of these models is the existence of feedback loops that feature a delay time, usually associated with the time required to transport energy through the atmosphere and/or oceans across the globe. So far, the values of the delays are generally assumed to be constant. Recent studies have demonstrated that even simple DDEs with nonconstant delay times, which change depending on the state of the system, can produce surprisingly rich dynamical behaviour. Here, we identify physical arguments for the existence of such state-dependent delays in a DDE model for the El Niño Southern Oscillation climate system. We then conduct a bifurcation analysis by means of continuation software to investigate the effects of state-dependent delays on the dynamics of the system.
|Zimin Li||A tale of twin peaks: spectral statistics of the quantum Rabi model
Abstract: The quantum Rabi model describes the interaction between light and matter, more specifically, between a two-level atom (qubit) and a single bosonic mode of a quantised light field. A key parameter in the model is the strength of the coupling between the qubit and the light field. The quantum Rabi model is of newfound interest as a number of experiments and quantum simulations have begun to probe the physics of the previously elusive ultrastrong and deep strong coupling regimes where the much simpler Jaynes-Cummings model no longer applies. Recent studies have revealed that the energy spectrum of this model is made up of three distinct regimes: (i) the perturbative ultrastrong coupling regime which comprises the Jaynes-Cummings model, (ii) a region where non-perturbative ultrastrong and non-perturbative deep strong coupling regimes coexist, and (iii) the perturbative deep strong coupling regime. In this talk I will discuss progress on calculating the spectral statistics of the energy level spacings in this model and how the spectral statistics relates to this classification of three distinct regimes. I will also discuss results for the probability distribution of an analogous mathematical model. Our results give a more or less complete understanding of the spectral properties of the quantum Rabi model.
|Woei Chet Lim||Explicit G1 spike solutions in General Relativity
Abstract: We present a family of new explicit solutions of Einstein’s field equations with stiff fluid. They admit one spacelike Killing vector field, and are generated by applying Stephani’s transformation (i.e. stiff fluid version of Geroch’s transformation) on Bianchi type V spatially homogeneous solutions. The most interesting phenomenon exhibited is the intersecting spikes near the initial singularity.
|Jon Links||Quantum phase diagram for a model of an ultracold gas of atoms and molecules.
Abstract: I will give a description of the quantum phase diagram for a model describing the interconversion of bosonic atoms and homonuclear diatomic molecules. The analysis will include semiclassical approximations, results from direct numerical diagonalisation, and exact formulae obtained from Bethe Ansatz techniques.
|Tianshu Liu||Affine osp(1|2) and its coset construction
Abstract: Conformal field theory is an essential tool of modern mathematical physics with applications to string theory and to the critical behaviour of statistical lattice models. The symmetries of a conformal field theory include all angle-preserving transformations. In two dimensions, these transformations generate the Virasoro algebra, a powerful symmetry that allows one to calculate observable quantities analytically. The construction of one family of conformal field theories from the affine Kac-Moody algebra sl(2) were proposed by Kent in 1986 as a means of generalising the coset construction to non-unitary Virasoro minimal models, these are known as the Wess-Zumino-Witten models at admissible levels. This talk aims to illustrate, with the example of the affine Kac-Moody superalgebra osp(1|2) at admissible levels, how the representation theory of a vertex operator superalgebra can be studied through a coset construction. The method allows us to determine key aspects of the theory, including its module characters, modular transformations and fusion rules.
|Matt Majic||Spheroidal harmonics for point source interaction with a sphere
We propose a powerful approach to solve Laplace’s equation for point sources near a spherical object. The central new idea is to use prolate spheroidal solid harmonics, which are separable solutions of Laplace’s equation in spheroidal coordinates, instead of the more natural spherical solid harmonics. Using electrostatics as an example, we motivate this choice and show that the resulting series expansions converge much faster. This improvement is discussed in terms of the singularity of the solution and its analytic continuation. The benefits of this approach are further illustrated for a specific example: the calculation of modified decay rates of light emitters close to nanostructures in the quasistatic approximation. We expect the general approach to be applicable with similar benefits to the solution of Laplace’s equation for other geometries and to other equations of mathematical physics.
We will also discuss the extension to the internal potential involving radially inverted offset prolate spheroidal harmonics.
PHYSICAL REVIEW E 95, 033307 (2017), arXiv:1711.09551 (internal solution)
|Vladimir Mangazeev|| Stochastic zero range processes related to Uq(A_n(1))
Abstract: We show that the quantum R matrix for symmetric tensor representation of the Uq(A_n(1)) algebra satisfies the sum rule required for its stochastic interpretation. It gives rise to an integrable zero range process of n classes of particles in one dimension. For n=2 we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of q-bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.
|Ian Marquette||Algebraic derivation of spectrum of superintegrable models
Abstract: Quadratic and more generally finitely generated polynomial algebras appear to be naturally related with integrals of quantum superintegrable models. These models have many properties that make them interesting from mathematics and physics perspective. In particular they are connected with various special special function and orthogonal polynomials. In recent years, it has been shown that two dimensional models are related to the full Askey scheme of orthogonal polynomials, exceptional orthogonal polynomials, Painleve transcendents. I will review some recent results on higher dimensional superintegrable systems and their symmetry algebras that take the form of higher rank polynomial algebras with an embedded structure. I will also discuss how the spectrum can be derived.
|Makoto Narita||On Global properties of Gowdy spacetimes in the Brans-Dicke theory
Abstract: Recent results show that standard singularity theorem that holds when an energy condition is applied in general relativity also holds when that energy condition is applied to the Bakry-Emery-Ricci tensor which naturally arises in the scalar-tensor theory of gravity. The theory is one of the generalized theories of gravitation and is a low energy effective superstring theory. Thus it is important to investigate global behavior of solutions to the gravitational field equations in the theory. We study the global properties of the Gowdy spacetime generated by Cauchy data on T3 in the Brans-Dicke theory which is one of the scalar-tensor theory of gravity. We show that the past boundaries of the maximal Cauchy developments of Gowdy initial data sets are asymptotically velocity-terms dominated singularities. The Kretschmann scalar blows up on the boundary. Thus the maximal Cauchy development cannot extend beyond the boundary and our result shows that the validity of the strong cosmic censorship conjecture.
|Paul Norbury||A new cohomology class on the moduli space of stable curves
Abstract: We define a collection of cohomology classes on the moduli space of curves. We prove that a generating function for the intersection numbers involving these new cohomology classes is a tau function of the KdV hierarchy. This is analogous to the theorem conjectured by Witten and proven by Kontsevich that a generating function for intersection numbers on the moduli space of curves is a tau function of the KdV hierarchy.
|Jeremy Nugent||Superintegrable systems in 4d
Abstract: In this talk we investigate pp wave metrics and their superintegrability properties.
|Susumu Osawa||Mass Renormalization in the Nelson Model
Abstract: The asymptotic behavior of the effective mass meff(Λ) of the so-called Nelson model in quantum field theory is considered, where Λ is an ultraviolet cutoff parameter of the model. Let m be the bare mass of the model. It is shown that for sufficiently small coupling constant |α| of the model, meff(Λ)/m can be expanded as meff(Λ)/m= 1+∑n=1∞ an(Λ) α2n. A physical folklore is that an(Λ)=O( [logΛ](n-1)) as Λ→∞. It is rigorously shown that 0<limΛ→∞a1(Λ)
|Luca Parisi||Evolutionary Algorithms for Margin Maximisation of Support Vector Machine
Abstract: The capability of support vector machines to classify highly non-linear, multi-dimensional data heavily relies on the choice of kernel cost parameters. Appropriate kernel function parameters influence the determination of the decision boundary. Support vector machine uses hyperplanes as a decision boundary for data classification. Particularly, the selection of an optimal hyperplane determines the accuracy, sensitivity and specificity of such learning-based classification algorithms. Given that evolutionary algorithms can help minimise the cost function, we have formulated a mathematical approach that uses evolutionary algorithms to optimise kernel function parameter selection of support vector machines. This algorithmic implementation can lead to a considerable improvement in maximising the margin of the optimal hyperplane, accounting for higher true positive and true negative rates, thus maximising the classification accuracy.
|Natasa Pavlovic||From quantum many particle systems to nonlinear dispersive PDE, and back
Abstract: Analysis of large systems of interacting particles is a key for predicting and understanding various phenomena arising in different contexts, from physics (in understanding e.g. boson stars) to social studies (when modeling social networks). Since the number of particles is usually very large one would like to understand qualitative and quantitative properties of such systems of particles through some macroscopic, averaged characteristics. In order to identify macroscopic behavior of multi-particle systems, it is helpful to study the asymptotic behavior when the number of particles approaches infinity, with the hope that the limit will approximate properties observed in the systems with a large finite number of particles. An example of an important phenomenon that describes such macroscopic behavior of a large system of particles is the Bose-Einstein condensation (BEC), which is a state of the matter of a dilute Bose gas at very low temperatures when the gas moves as a single particle. Although the BEC was predicted in early days of quantum mechanics by Bose and Einstein, the first experimental realization came in 1995. Mathematical models have been developed to understand such phenomena. Those models connect large quantum systems of interacting particles and nonlinear PDE that are derived from such systems in the limit of the number of particles going to infinity. In this talk we will focus on developments that connect a quantum many particle system of bosons and the nonlinear Schrodinger equation.
|Ellen Powell||Gaussian multiplicative chaos measures
Abstract: I will talk about recent progress in the construction of general Gaussian multiplicative chaos measures. In particular, I will discuss new universality results in the critical regime, and also some special constructions when the underlying Gaussian field is a multiple of the Gaussian free field.
|Thomas Quella||Symmetry protected topological phases and selection rules for topological phase transitions
Abstract: We show that phase transitions between different symmetry protected topological phases are constrained to obey certain selection rules. This is explained for Haldane phases of SU(N) spin chains which lead to selection rules for renormalization group flows between SU(N) WZW models.
|Narrendar RaviChandran||Bifurcation Analysis of a FitzHugh-Nagumo-based Tetrahedral Neuronal Network Modelled under External Electrical Stimulation
Abstract: Recent advances in minimally invasive brain stimulation have allowed to treat nervous system disorders more effectively. In this study, we developed a mathematical model of a FitzHugh-Nagumo-based simplistic representation of a cortical neuronal network, modelled as a tetrahedron. Cortical fast spiking neurons are bi-stable resonators, undergoing Andronov-Hopf bifurcation. The effects of an external electrical stimulation were computationally modelled on such a neuronal network and studied via a bifurcation analysis. The activation dynamics of a single axon was represented by ordinary differential equations, as a codimension-1 bifurcation case. The external stimulation perturbed the neuronal network, inducing individual neurons to transit from their resting state to an excited one, causing action potential. Although bifurcation analysis has been conducted on single and two-neuron models extensively, the proposed mathematical model provides further understanding on the dynamic behaviour of multi-neuronal networks. Furthermore, such tetrahedral neuronal models can have direct scalability for improving physiological accuracy in representing cortical activity. Ultimately, this study shows potential of computational models for testing the effectiveness of deep brain stimulation.
|Christopher Raymond||Generalised Galilean W-algebras
Abstract: We consider two dimensional conformal field theories from the perspective of their symmetry algebras. The Galilean contraction is a technique for constructively generating new algebraic structures from known symmetry algebras. We outline a generalisation of the contraction method which leads to new infinite families of CFTs, and discuss some properties of the resulting theories.
|David Ridout||sl(3) weight modules and higher-rank logarithmic CFT
Abstract: Rational conformal field theories (CFTs), such as the Wess-Zumino-Witten (WZW) models, are generally constructed from irreducible highest-weight modules. On the other hand, logarithmic CFTs are constructed from modules that are not all irreducible but also need not be highest-weight. Non-highest-weight modules appear, in particular, for the logarithmic analogues of the WZW models, sometimes called the fractional level WZW models. Recently, we have obtained a fairly complete picture of the modules arising for fractional level sl(2) models. In this talk, I will try to outline this picture and discuss current work that aims to extend this to the sl(3) case.
|Tanja Schilling||On the "generalized Generalized Langevin Equation" and its application to the crystallization process
Abstract: As a researcher in statistical physics, one may often be interested in reducing the complexity of a many-particle system to the study of a set of relevant observables (for instance, the system could be a undercooled melt and the aim could be to develop a model for the transition to the crystalline phase). If the system is in equilibrium, a systematic way to derive an equation of motion for the “relevant” observables from the microscopic dynamics has been known for some time as the “Mori-Zwanzig” formalism, which leads to the Langevin equation. In contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable will have. We adapt Mori-Zwanzig formalism to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. We also derive a fluctuation-dissipation-like relation which relates the memory kernel and the autocorrelation function of the fluctuating force. In addition, we show how to relate the Taylor expansion of the memory kernel to experimental data, thus allowing to construct the equation of motion from direct measurements. We finally apply the method to the crystallization process.
|Sebastian Schuster||Blueprint formulae for electromagnetic analogue space-times
Abstract: Research on astrophysical and other general relativistic space-times is often stymied by the smallness of the most interesting effects. However, borrowing wave equations from other areas of physics, as long as it is possible to describe them in geometric terms, it is also possible to formulate analogue spacetimes – analogues of these astrophysical scenarios which may be easier to analyze. Laboratory experiments are now starting to get first results from experiments based on this analogue spacetime approach. Here we present an algorithm for generating such analogue spacetimes in the context of electromagnetic media. This allows us to demonstrate how (using on-demand bespoke permittivity, permeability and magneto-electric effects) to mimic any given space-time.
|Yibing Shen||Ground-state energy for the p + ip pairing Hamiltonian interacting with its environment
Abstract: We first study the p + ip Hamiltonian isolated from its environment and consider the case of large particle number. A continuum limit approximation is applied to compute the ground-state energy. We discuss the evolution of the solution curve, and the limitations of this approach. We then consider an alternative approach which generalises to accommodate interaction with the environment.
|Qian-Qian Shi||Dualities and Ground State Phase Diagrams for Quantum Spin-s XYZ Models
Abstract: Dualities and ground state phase diagrams are discussed for quantum XYZ chains with arbitrary spin s. It is shown that dualities reproduce the ground state phase diagrams for quantum half-integer spin XYZ chains, such as spin s=1/2, 3/2, but not for quantum integer spin XYZ chains, such as spin s=1, 2, in which the so-called symmetry protected topological phase–Haldane phase occurs. Dualities discussed for the one-dimensional quantum XYZ chains with aritrary spin s may be extended to quantum XYZ models with arbitrary spin s on a bipartite lattice in any spatial dimensions, such as square and honeycomb lattices.
|Jessica Silva||Thermodynamics and Gravity
Abstract: Tolman’s relation, for the temperature of a self-gravitating fluid, is broadly accepted within the general relativity community. However that’s still an intriguing concept since it contradicts naive versions of the zeroth law of thermodynamics. Based on that, we will discuss how thermodynamics emphasises the great distinction between gravity and other forces.
|Steve Siu||Quantum Hamiltonian Reduction for relaxed affine sl2 modules
Abstract: The Virasoro algebra and its representation theory is central to the study of Conformal Field Theory. It is known that the highest-weight modules of the Virasoro algebra can be obtained from the affine sl2 highest-weight modules through the Quantum Hamiltonian Reduction. In this talk we will discuss a possible generalisation of this result by considering the Quantum Hamiltonian Reduction of the relaxed highest-weight modules of affine sl2.
|William Stewart||On the twisted sector of Wess-Zumino-Witten models
Abstract: In this talk I will introduce the Wess-Zumino-Witten (WZW) models of two dimensional conformal field theory. I will focus on the integrable representations of affine Lie algebras that appear in the spectrum of a WZW model at positive integer level. I will then construct the twisted sector of the WZW models using outer automorphisms of finite dimensional simple Lie algebras. The resulting symmetry algebra is a twisted affine Lie algebra. I will analyse the restrictions imposed by singular vectors on the allowed modules in the twisted sector. For the affine sl(3) WZW model at positive integer levels, I will demonstrate that the allowed modules are restricted to the integrable highest weight A(2,2)-modules. If time permits, I will extend these arguments to a general WZW model at positive integer level and propose that the allowed modules in the twisted sector are likewise restricted to the integrable highest weight modules of the corresponding twisted affine Lie algebra.
|Aiden Suter||W-algebra Representation Theory
Abstract: 2-dimensional conformal quantum field theories (CFTs) have an infinite-dimensional symmetry algebra known as the Virasoro algebra. Affine W-algebras were first studied as generalisations of the Virasoro algebra and were created by including additional fields which were compatible with the Virasoro field. The W3-algebra is an example of such an algebra and is generated by the Virasoro field and an additional “W” field. Key physical properties of CFTs are determined by correlation functions which are constructed from representations of the symmetry algebra for the theory. In my research, we are interested in systematically constructing non-simple indecomposable representations of the W3-algebra which are of interest in the study of logarithmic CFT. We aim to achieve this by two methods. The first is to generalise similar constructions for the Virasoro algebra which has been more extensively studied. The other is to use the vertex operator algebra (VOA) formalism of W-algebras. In particular we employ the use of the Zhu algebra – an associative algebra constructed from a VOA which shares the same representations as the VOA.
|Arthur Suvorov||Isospectral neutron stars in general relativity
Abstract: In 1966, Kac posed the mathematical problem of ‘can one hear the shape of a drum?’. The problem effectively asks whether or not knowledge of the spectrum of the Laplace operator (‘hearing’ the quasi-normal frequencies) allows one to uniquely identify the geometry of the space (‘drum’). The answer turns out to be no, meaning that one cannot always uniquely determine the geometry by listening to the quasi-normal modes; several drums may sound the same. In general relativity, axial perturbations of a neutron star are modelled by solving the Laplace-Beltrami (wave) equation over the background Tolman-Oppenheimer-Volkoff spacetime. As for the drum problem, we show that two static neutron stars, with distinct fluid profiles, can admit the same quasi-normal mode spectrum. We discuss possible implications of this with regards to identifying neutron star properties from gravitational wave and quasi-normal mode observations.
|Luc Vinet||Fractional Revival in Spin Networks and Graphs
Abstract: Fractional revival occurs when a wave packet is reproduced periodically at various locations under its time evolution. Engineering this phenemenon can be used to generate entanglement. For instance, if the dynamics of a spin chain can be arranged so that a state initially at one end is revived in a balanced way at both ends, a maximally entangled state is obtained. The situation where there is only one revival site corresponds to state transfer, another important task. We shall discuss how spin chains can be designed so that fractional revival occurs. A model with next-to-nearest interactions will be presented in this connection. Coherent transport and fractional revival on graphs other than paths will also be discussed. In particular, the relations between quantum walks on graphs of the ordinary and generalized Hamming schemes and certain analytic spin chains will be stressed. The important role played by orthogonal polynomials in the analyses will also be underscored.
|Alessandra Vittorini Orgeas|
|Andrew Waldron||Contact Geometry and Quantum Mechanics
Abstract: We present a generally covariant approach to quantum mechanics. Generalized positions, momenta and time variables are treated as coordinates on a fundamental “phase-spacetime” manifold. Dynamics are encoded by giving phase-spacetime a contact structure. BRST quantization then yields a physical Hilbert space whose elements satisfy a parallel transport equation on a certain vector bundle over phase-spacetime. The inner product of solutions both reproduces and generalizes the Wigner functions of standard quantum mechanics.
|Jingbo Wang||Quantum Walk and Quantum Computation
Abstract: Quantum walk has shown much potential as a general framework for developing novel quantum computation algorithms. The efficiency of these algorithms depends on interference between the multiple paths that are simultaneously traversed by a quantum walker. It also hinges on local interactions and intrinsic quantum correlations such as entanglement, if multiple quantum walkers are involved. As such, quantum walk has become a subject of intense theoretical and experimental studies. An increasingly pressing challenge is to demonstrate quantum supremacy of quantum-walk-based algorithms over classical computation; this requires an efficient decomposition of the quantum walk operators in order to implement the prescribed algorithm on a quantum computer. In this talk, I will discuss the design principles of efficient quantum circuit for quantum walks on a wide range of undirected and directed Graphs, aiming to provide some intuition on how such decomposition is derived.
|Mark Wilson||Analysing signed networks, Ising spin glass models, and fullerene graphs using the frustration index
Abstract: Structural balance in networks with positive and negative edges has become a focus in graph theory and network science. In signed networks, the frustration index determines distance of a network from a state of structural balance making it a key measure for analysing positive and negative interactions under different contexts. Originally suggested decades ago, the complexity involved in computing the frustration index has restricted its usage. We develop new discrete optimisation models to compute this measure in decent-sized instances. Our extended optimisation models include several speed-up techniques involving prioritised branching and valid inequalities that improve the branch and bound algorithm. The speed-up techniques make our models capable of processing graphs with thousands of nodes and edges in seconds on inexpensive hardware. We discuss some applications of the proposed models in sociology, physics, chemistry, finance, international relations, and biology. The frustration of such signed networks has different meanings and interpretations depending on the context. Our datasets include several social networks and biological networks that are modelled as signed graphs. We also discuss numerical results on the frustration index in networks of formal alliances and antagonisms between countries as well as financial portfolio networks. Molecular graphs of carbon and Ising spin glass models are among other applications of this NP-hard graph optimisation problem. The findings unify the applications of a graph-theoretical measure in understanding complex structures in several fields of research. Aref S, Mason AJ, Wilson MC. An exact method for computing the frustration index in signed networks using binary programming. Preprint arXiv:1611.09030 (2016). Aref S, Mason AJ, Wilson MC. Computing the line index of balance using integer programming optimisation. Preprint arXiv:1710.09876 (2017).
|Sungha Yun||Free Surface Waves under a Pressure Distribution - Critical Surface Tension Case
Abstract: We consider forced surface waves under a pressure distribution on an incompressible, inviscid fluid in a two dimensional channel on a horizontal rigid flat bottom and nonzero surface tension on the free surface. It has been known that a nondimensional wave speed, called Foude number, is near 1 and a nondimensional surface tension, called Bond number, is near 1/3, the KdV theory fails and a time dependent fifth order KdV equation, called Kawahara equation, can be derived to model the wave motion on the free surface. In this paper a time dependent Kawahara equation with a forcing is studied both theoretically and numerically. Existence theorem and new types of numerical results are presented.
|Xin Zhang||Retrieve the Bethe states of quantum integrable models solved via the off-diagonal Bethe Ansatz
Abstract: Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin-1/2 chain as examples, we show that for a given inhomogeneous T-Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.
|Huan-Qiang Zhou||Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer's Principle
Abstract: Fidelity mechanics, a scheme to investigate critical phenomena in quantum many-body physics, is formulated as an analogue of the four laws in thermodynamics and black hole mechanics, thus unveiling a formal connection between critical points and black holes. Rich physics is unveiled even for prototypical models in quantum many-body systems, such as one-dimensional quantum XY model,quantum transverse Ising chain in a longitudinal field and quantum XYZ chain.
|Zongzheng Zhou||Unified correlation function behaviour on high-dimensional tori
Abstract: It was generally believed that above upper critical dimensions, the thermodynamic behaviour of the correlation functions of Ising model and self-avoiding walk (SAW) behave the same as the Green’s function of the simple random walk (SRW). However, this connection naturally broke on the torus. For Ising and SAW on the high-dimensional torus, asymptotically the correlation function is a power-law for the short distance and reaches a plateau for the large distance. However, Green’s function for SRW on the torus is infinity since it is recurrent on any dimensions. In this project, we work on repairing this broken connection by considering a random-length random walk.
|Paul Zinn-Justin||Schubert puzzles and quantum integrability
Abstract: About 10 years ago, I suggested the idea that Schubert calculus, a branch of enumerative geometry that deals with configurations of vector subspaces of a vector space, could be interpreted as a “scattering process” in a quantum integrable model. These ideas beautifully came to fruition in recent collobaration with A. Knutson. I shall describe how a broad class of problems generalizing Schubert’s original setting are solved by “puzzles”, which are nothing but partition functions in disguise of some integrable models, based on the series of algebras A2, D4, E6, E8.