Browsing by Author "Murugan, Jeffrey"

Now showing 1 - 15 of 15
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    A study of circuit Complexity for Coherent States
    (2022) Tladi, Mpho; Haque, Shajid; Murugan, Jeffrey; Weltman, Amanda
    Computational complexity is a popular quantity in quantum information theory. It has made huge strides in recent years in the study of black hole dynamics. A brief definition of complexity is the measure of how difficult it is to implement a task. For a quantum system, complexity evaluates the difficulty of preparing a quantum state from a given reference state by unitary transformations. However, in the dual gravity theory complexity has a geometric meaning. In some black hole context, Leonard Susskind and collaborators proposed two holographic conjectures. The Complexity=Volume (CV) states that complexity of the boundary field theory is dual to the volume of a co dimension one maximal surface that extends to the boundary of the Ads space. Complexity=Action (CA) posits that complexity of the boundary is the same as the action evaluated as an action on patch in the bulk defined as the Wheeler De Witt patch. In recent years, these two conjectures have initiated an extensive study of complexity. This thesis is also motivated by these conjectures and will investigate complexity in the field theory side of the story. Specifically, we will explore the complexity for coherent states. We will start with a review of different methods of computing complexity. Finally, we then investigate the complexity for coherent states by using the methods of circuit complexity and operator complexity
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    Achieving baseline states in sparsely connected spiking-neural networks: stochastic and dynamic approaches in mathematical neuroscience
    (2015) Antrobus, Alexander Dennis; Murugan, Jeffrey; Ellis, George F R
    Networks of simple spiking neurons provide abstract models for studying the dynamics of biological neural tissue. At the expense of cellular-level complexity, they are a frame-work in which we can gain a clearer understanding of network-level dynamics. Substantial insight can be gained analytically, using methods from stochastic calculus and dynamical systems theory. This can be complemented by data generated from computational simulations of these models, most of which benefit easily from parallelisation. One cubic millimetre of mammalian cortical tissue can contain between fifty and one-hundred thousand neurons and display considerable homogeneity. Mammalian cortical tissue (or grey matter") also displays several distinct firing patterns which are widely and regularly observed in several species. One such state is the "input-free" state of low-rate, stochastic firing. A key objective over the past two decades of modelling spiking-neuron networks has been to replicate this background activity state using "biologically plausible" parameters. Several models have produced dynamically and statistically reasonable activity (to varying degrees) but almost all of these have relied on some driving component in the network, such as endogenous cells (i.e. cells which spontaneously fire) or wide-spread, randomised external input (put down to background noise from other brain regions). Perhaps it would be preferable to have a model where the system itself is capable of maintaining such a background state? This a functionally important question as it may help us understand how neural activity is generated internally and how memory works. There has also been some contention as to whether driven" models produce statistically realistic results. Recent numerical results show that there are connectivity regimes in which Self-Sustained, Asynchronous, Irregular (SSAI) firing activity can be achieved. In this thesis, I discuss the history and analysis of the key spiking-network models proposed in the progression toward addressing this problem. I also discuss the underlying constructions and mathematical theory from measure theory and the theory of Markov processes which are used in the analysis of these models. I then present a small adjustment to a well known model and provide some original work in analysing the resultant dynamics. I compare this analysis to data generated by simulations. I also discuss how this analysis can be improved and what the broader future is for this line of research.
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    Artificial Neural Networks as a Probe of Many-Body Localization in Novel Topologies
    (2022) Beetar, Cameron; Murugan, Jeffrey; Rosa, Dario; Weltman, Amanda
    We attempt to show that artificial neural networks may be used as a tool for universal probing of many-body localization in quantum graphs. We produce an artificial neural network, training it on the entanglement spectra of the nearest-neighbour Heisenberg spin1/2 chain in the presence of extremal (definitely ergodic/localizing) disorder values and show that this artificial neural network successfully qualitatively classifies the entanglement spectra at both extremal and intermediate disorder values as being in either the ergodic regime or in the many-body-localizing regime, based on known results. To this network, we then present the entanglement spectra of systems having different topological structures for classification. The entanglement spectra of next-to-nearest-neighbour (J1 − J2, and, in particular, Majumdar-Ghosh) models, star models, and bicycle wheel models - without any further training of the artificial neural network - are classified. We find that the results of these classifications - in particular how the mobility edge is affected - are in agreement with heuristic expectations. This we use as a proof of concept that neural networks and, more generally, machine learning algorithms, endow physicists with powerful tools for the study of many-body localization and potentially other many-body physics problems.
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    Chaos and Scrambling in Quantum Small Worlds
    (2020) Hartmann, Jean-Gabriel Keiser; Murugan, Jeffrey; Shock, Jonathan
    In this thesis, we introduce a novel class of many-body quantum system, which we term ‘quantum small worlds'. These are strongly-interacting systems that interpolate between completely ordered (nearest-neighbour, next-to-nearest-neighbour etc.) and completely random interactions. They are systems of quantum spin particles in which the network topology is given by the Watts-Strogatz model of network theory. As such, they furnish a novel laboratory for studying quantum systems transitioning between integrable and non-integrable behaviour. Our motivation is to understand how the dynamics of the system are affected by this transition, particularly with regards to the ability of the system to scramble (quantum) information, and potential emergence of chaotic behaviour. Our work begins with a review of the relevant literature regarding algebraic graph theory and quantum chaos. Next, we introduce the model by starting from a well understood integrable system, a spin- 1 2 Heisenberg, or Ising, chain. We then inject a small number of long-range interactions and study its ability to scramble quantum information using two primary devices: the out-of-time-order correlator (OTOC) and the spectral form factor (SFF). We find that the system shows increasingly rapid scrambling as its interactions become progressively more random, with no evidence of quantum chaos as diagnosed by either of these devices.
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    Entanglement entropy, the Ryu-Takayanagi prescription, and conformal maps
    (2017) Grant-Stuart, Alastair; Murugan, Jeffrey; Shock, Jonathan
    We define and explore the concepts underpinning the Ryu-Takayanagi prescription for entanglement entropy in a holographic theory. We begin by constructing entanglement entropy in finite-dimensional quantum systems, and defining the boundary at infinity of a bulk spacetime. This is sufficient for a naïve application of the Ryu-Takayanagi prescription to some simple examples; nonetheless, we review the general theory of minimal submanifolds in Riemannian ambient manifolds in order to better characterise the objects involved in the prescription. Finally, we explore the symmetries of the boundary theory to which the prescription applies, and thereby extend the aforementioned examples. Throughout, emphasis is placed on making explicit the mathematical structures that are taken for granted in the research literature.
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    Études on fuzzy geometry and cosmology
    (2007) Murugan, Jeffrey; Ellis, George F R
    We investigate various aspects of noncommutative geometry and fuzzy field theory and their relations to string theory. In particular, we study the BPS and non-BPS solutions of the CJPN nonlinear sigma model on the noncommutative plane in some detail and show among other things that a class of its solitonic excitations may be built from bound states of noncommutative scalar solitons. We then go on to construct a fuzzy extension of the semilocal SU(N)a x U(l)L Yang-Mills-Riggs model. We find that not only does this noncommutative model support a large class of BPS vortex solutions but, unlike in the commutative model, these are exact solutions of the BPS equations. We also study the large coupling limit of the semilocal model and demonstrate conclusively the metamorphosis of the semilocal vortex to an appropriate degree instanton of the fuzzy CJPN model. In the second part of this work, we study the perpendicular intersection of Dl- and D7-branes in type liB string theory and the fuzzy 6-sphere that resolves the singularity of the intersection. We demonstrate the equivalence of the D7 and dual D-string descriptions by computing the energy, charge and radial profiles of the solution in each description. We conclude the thesis with a foray into cosmology by constructing a realisation of a recently proposed singularity-free inflating universe. We discuss the basic characteristics of this model and show that none are at odds with current observations.
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    Geometrical and nonperturbative aspects of low dimensional field theories
    (2000) Murugan, Jeffrey; Barashenkov, Igor
    We present a collection of results on solitons in low-dimensional classical field theory. We begin by reviewing the geometrical setting of he nonlinear ơ-model and demonstrate the integrability of the theory in two-dimensions on a symmetric target manifold. After reviewing the construction of soliton solutions in the 0(3) ơ-model we consider a class of gauged nonlinear ơ-models on two-dimensional axially-symmetric target spaces. We show that, for a certain choice of self-interaction, these models are all self-dual and analyze the resulting Bogomol'nyi equations in the BPS limit using techniques from dynamical systems theory. Our analysis is then extended to topologically massive gauge fields. We conclude with a deviation into exploring links between four-dimensional self-dual Yang-Mills equations and various lower-dimensional field theories. In particular, we show that at the level of equations of motion, the Euclidean Yang-Mills equations in light-cone coordinates reduce to the two-dimensional nonlinear ơ-model.
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    The KLT relations in unimodular gravity
    (2016) Burger, Daniel; Weltman, Amanda; Murugan, Jeffrey; Ellis, George F R
    Here we initiate a systematic study of some of the symmetry properties of unimodular gravity, building on much of the known structure of general relativity, and utilizing the powerful technology developed in that context, such as the spinor helicity formal-ism. In particular, we show, up to five-points and tree-level, that the KLT relations of perturbative gravity hold for trace free or unimodular gravity. This work is in conjunction with a paper written with A. Welman, J. Murugan and G.F.R. Ellis (ARXIV: 1511.08517)
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    On the local and global properties of information manifolds
    (2015) Clingman, Tslil; Murugan, Jeffrey
    In the first part of the work, we show a general relation between the spatially disjoint product of probability density functions and the sum of their Fisher information metric tensors. We then utilise this result to give a method for constructing the probability density functions for an arbitrary Riemannian Fisher information metric tensor given its associated Nash embedding. We note further that this construction is extremely unconstrained, depending only on certain continuity properties of the probability density functions and a select symmetry of their domains. In the second part of the work, with the aim of understanding the global, algebrao-topological nature of information manifolds, we exhibit some of the necessary category theory required to effect a discussion of homological algebra in a general setting.
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    Quantum states on spheres in the presence of magnetic fields
    (2019) Slayen, Ruach Pillay; Murugan, Jeffrey; Shock, Jonathan
    The study of quantum states on the surface of various two-dimensional geometries in the presence of strong magnetic fields has proven vital to the theoretical understanding of the quantum Hall effect. In particular, Haldane’s seminal study of quantum states on the surface of a compact geometry, the sphere, in the presence of a monopole magnetic field, was key to developing an early understanding of the fractional quantum Hall effect. Most of the numerous studies undertaken of similar systems since then have been limited to cases in which the magnetic fields are everywhere constant and perpendicular to the surface on which the charged particles are confined. In this thesis, we study two novel variations of Haldane’s spherical monopole system: the 'squashed sphere’ in the presence of a monopole-like magnetic field, and the sphere in the presence of a dipole magnetic field. In both cases the magnetic field is neither perpendicular nor constant with respect to the surface on which the charged particles are confined. Furthermore, the spherical dipole system has vanishing net magnetic flux. For the 'squashed sphere’ system we find the lowest Landau level single-particle Hilbert space, and it is shown that the effect of the squashing is to localise the particles around the equator. For the spherical dipole system we find the entire single-particle Hilbert space and energy spectrum. We show that in the strong-field limit the spectrum exhibits a Landau level structure, as in the spherical monopole case. Unlike in the spherical monopole case, each Landau level is shown to be infinitely degenerate. The emergence of this Landau level structure is explained by the tendency of a strong dipole field to localise particles at the poles of the sphere.
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    Searching for self-duality in non-maximally supersymmetric backgrounds
    (2017) Tarrant, Justine Alecia; Murugan, Jeffrey
    Fermionic T-duality is the generalisation to superspace of bosonic T-duality (i.e. to include fermionic degrees of freedom). Originally, T-duality described the equivalence relation between two physical theories, each living on a different background. However, this thesis is concerned with fermionic T-duality and its role in self-duality. The goal is to determine whether AdS backgrounds with less than maximal supersymmetry are self-dual. A background is said to be self-dual if, after a specific sequence of bosonic and fermionic T-duality transformations, the original background is recovered. Self-dual backgrounds are of great interest due to their link to integrability. Fermionic T-duality has played a pivotal role in proving that the maximally supersymmetric background AdS₅ × S⁵ is self-dual. This background is also known to be integrable, therefore, when it was shown to be self-dual, the hypothesis that self-duality implied integrability, and vice-versa, was born. We investigate how far this hypothesis may be stretched for a number of AdS backgrounds, for which integrability has already been determined. The following backgrounds were considered: AdS₂ × S² × T⁶ and AdSd × Sᵈ XT(¹⁰⁻³ᵈ) (d = 2; 3). This question of self-duality was approached in two ways. In the first approach we show that these less supersymmetric backgrounds are self-dual by working with the supergravity fields and using the fermionic Buscher procedure derived by Berkovits and Maldacena. In the second approach, we verify the self-duality of Green-Schwarz supercoset σ-models on AdSd × Sᵈ (d = 2; 3) backgrounds. Furthermore, we prove the self-duality of AdS₅ × S⁵ without gauge fixing K-symmetry. We show that self-duality is a property which holds for the exceptional backgrounds, where the need to T-dualise along one of the spheres arises, again. Nature is not supersymmetric, therefore learning how to do physics in AdS₅ × S⁵ is not enough. In order to understand theories like Quantum Chromodynamics, we need to systematically break the supersymmetry present in our toy models. In this regard, it is easy to appreciate the significance of studying backgrounds with less than maximal supersymmetry.
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    Some aspects of the mass deformed ABJM theory
    (2014) Mohammed, Asadig; Murugan, Jeffrey
    In this thesis, we discuss some aspects of the Aharony, Bergman, Jafferis & Maldacena (ABJM) theory. In particular, encouraged by the recent construction of fuzzy sphere solutions in the ABJM theory, we re-analyze the latter from the perspective of a Matrix-like model. In particular, we argue that a vortex solution exhibits properties of a supergraviton, while a kink represents a 2-brane. Other solutions are also consistent with the Matrix-type interpretation. We study vortex scattering and compare with graviton scattering in the massive ABJM background, however our results are inconclusive. We speculate on how to extend our results to construct a Matrix theory of ABJM. We also present an embedding of the 3-dimensional relativistic Landau-Ginzburg model for condensed matter systems in an N = 6, U(N) × U(N) Chern-Simons-matter theory (the ABJM model) by consistently truncating the latter to an abelian effective field theory encoding the collective dynamics of O(N) of the O(N²) modes. In fact, depending on the VEV on one of the ABJM scalars, a mass deformation parameter μ and the Chern-Simons level number k, our abelianization prescription allows us to interpolate between the abelian Higgs model with its usual multi-vortex solutions and a φ⁴ theory. We sketch a simple condensed matter model that reproduces all the salient features of the abelianization. In this context, the abelianization can be interpreted as giving a dimensional reduction from four dimensions. Finally we present ansätze that reduce the mass-deformed ABJM model to gauged Abelian scalar theories, using the fuzzy sphere matrices Gα. One such reduction gives a Toda system, for which we find a new type of nonabelian vortex. Another gives the standard Abelian-Higgs model, thereby allowing us to embed all the usual (multi-)vortex solutions of the latter into the ABJM model. By turning off the mass deformation at the level of the reduced model, we can also continuously deform to the massive φ⁴ theory in the massless ABJM case. In this way we can embed the Landau-Ginzburg model into the AdS/CFT correspondence as a consistent truncation of ABJM. In this context, the mass deformation parameter μ and a field VEV <φ> act as g and gc respectively, leading to a well-motivated AdS/CMT construction from string theory. To further this particular point, we propose a simple model for the condensed matter field theory that leads to an approximate description for the ABJM abelianization. Finally, we also find some BPS solutions to the mass-deformed ABJM model with a spacetime interpretation as an M2-brane ending on a spherical M5-brane.
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    A study of holographic superconductors
    (2009) Umeh, Obinna; Murugan, Jeffrey
    The proposal that the physics of quantum critical phase transition in strongly coupled condensed matter systems can be described by a gravitational theory within the frame work of gauge/gravity correspondence is investigated more extensively for s-wave superconductors. We consider a gravitational theory with a black hole solution in anti de Sitter spacetime, coupled to an Abelian-Higgs system in (d + 1)-dimensions. A wide range of negative mass squared for the scalar field that satisfied the Brietenlolmer-Freedman stability bound and the unitarity bound are considered in the probe limit. The dependence of the some of the physical quantities on the scaling dimensions of the dual condensates were thoroughly investigated. We observe that the holographic superconductors can be consistently classified into two, based on the scaling dimensions and the charge of the dual condensates. Holographic superconductors of dimension λ- exhibit features of type II superconductors while those of dimension λ+ show features of type 1. The validity of this classification was confirmed by solving the bulk equations of motion perturbatively near the quantum critical point in order to calculate the superconducting characteristic lengths at a fixed charge q. The results show that there is a critical scaling dimension beyond which a holographic superconductor behave as type I and below this value it is a type II. The properties of holographic superconductors presented in this report are in qualitative agreement with the Ginzburg-Landau theory.
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    Topics in Gauge/Gravity Duality
    (2014) Rughoonauth, Nitin; Murugan, Jeffrey
    The gauge theory/gravity correspondence encompasses a variety of di_erent specific dualities. We investigate various topics in the context of Super–Yang- Mills/type IIB string theory and superconformal Chern-Simons-matter/type IIA string theory dualities. We carry out a rather extensive study of the type IIA AdS3_S3_S3_S1 Green- Schwarz superstring, up to quadratic order in fermions. We discuss issues related to fixing its _-symmetry and show the one-loop finiteness of two-point functions of bosonic fields. We then perform a Hamiltonian analysis and compare SU(2) string states with predictions from the conjectured Bethe equations. Furthermore, we show that, at least at tree-level, the two-body S-matrix is reflectionless. We then concern ourselves with extending Mikhailov’s construction of giant gravitons from holomorphic functions to include meromorphic functions, which lead to giants with non-trivial topologies in AdS5_S5. We explore what topological configurations giants, whose dynamics preserve a certain amount of supersymmetry, assume. We are particularly interested in solutions created by a localised modification of a set of intersecting spherical giant gravitons, as this seems the most tractable limit. We finally explore some aspects of holographic particle-vortex duality, in particular its realisation in the ABJM model and a possible relation to Maxwell duality in AdS4. We formulate a symmetric version of the transformation that acts as a self-duality, show how to embed it as an abelian duality in the (2+1)-dimensional, N = 6 super–Chern-Simons-matter theory that is the ABJM model, and speculate on a possible non-abelian extension.
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    Towards a holographic description of pulsar glitch mechanism
    (2015) Misra, Anuj; Murugan, Jeffrey
    This work aims to review the progress in understanding the underlining physics of pulsar glitches: beginning from the pedagogical development of the subject to eventually motivating the use of AdS/CFT techniques in studying a certain class of condensed matter systems. The foundation of this work is built upon the Gross Pitaevskii (GP) model of super-fluidity applied to the interior matter of neutron stars, where the condensate wave function acts as the order parameter of the macroscopic coherence theory. The excitation modes of the field equations are found to be solitonic vortices, which then go on to present a theoretical basis to the plausible theories of pulsar glitches involving vortex dynamics. The second major thrust of this thesis is in reviewing the application of AdS/CFT in study of strongly-coupled condensed matter systems, with special attention to the models of holographic superfluidity that admit vortex-like solutions. The basic identification of the characteristic free energy configuration of global vortices in the AdS/CFT prescription enables to motivate its use in studying the pulsar glitch mechanism. The last part of this work traces the conclusions of this review and attempts to present the current state-of-progress of the field with its extensive domain of purview and open lines of inquiry.