Showing papers in "Journal of Physics A in 2011"

A 'missing' family of classical orthogonal polynomials

Abstract: We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = −1. We also show that these polynomials provide a nontrivial realization of the Askey–Wilson algebra for q = −1.

Private randomness expansion with untrusted devices

Abstract: Randomness is an important resource for many applications, from gambling to secure communication. However, guaranteeing that the output from a candidate random source could not have been predicted by an outside party is a challenging task, and many supposedly random sources used today provide no such guarantee. Quantum solutions to this problem exist, for example a device which internally sends a photon through a beamsplitter and observes on which side it emerges, but, presently, such solutions require the user to trust the internal workings of the device. Here, we seek to go beyond this limitation by asking whether randomness can be generated using untrusted devices—even ones created by an adversarial agent—while providing a guarantee that no outside party (including the agent) can predict it. Since this is easily seen to be impossible unless the user has an initially private random string, the task we investigate here is private randomness expansion. We introduce a protocol for private randomness expansion with untrusted devices which is designed to take as input an initially private random string and produce as output a longer private random string. We point out that private randomness expansion protocols are generally vulnerable to attacks that can render the initial string partially insecure, even though that string is used only inside a secure laboratory; our protocol is designed to remove this previously unconsidered vulnerability by privacy amplification. We also discuss extensions of our protocol designed to generate an arbitrarily long random string from a finite initially private random string. The security of these protocols against the most general attacks is left as an open question.

Diffusion with optimal resetting

Abstract: We consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r. We consider several generalisations of the model of M. R. Evans and S. N. Majumdar (2011), Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601: (i) a space dependent resetting rate r(x) ii) resetting to a random position z drawn from a resetting distribution P(z) iii) a spatial distribution for the absorbing target PT(x). As an example of (i) we show that the introduction of a non-resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target. We address the problem of optimal resetting, that is, minimising the mean time to absorption for a given target distribution. For an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows.

Frame-like geometry of double field theory

Abstract: We relate two formulations of the recently constructed double field theory to a frame-like geometrical formalism developed by Siegel. A self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar. This curvature scalar can be used to define an action, and it is shown that this action is equivalent to that of double field theory.

Nonlinear self-adjointness and conservation laws

Abstract: The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the strict self-adjointness (definition 1) and quasi-self-adjointness introduced earlier by the author. It is shown that the equations possessing nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint form. For example, the heat equation ut − Δu = 0 becomes strictly self-adjoint after multiplying by u−1. Conservation laws associated with symmetries are given in an explicit form for all nonlinearly self-adjoint partial differential equations and systems.

The effective temperature

Abstract: This review presents the effective temperature notion as defined from the deviations from the equilibrium fluctuation–dissipation theorem in out-of-equilibrium systems with slow dynamics. The thermodynamic meaning of this quantity is discussed in detail. Analytic, numeric and experimental measurements are surveyed. Open issues are mentioned.

Invisibility in $\mathcal {PT}$-symmetric complex crystals

Abstract: Bragg scattering in sinusoidal -symmetric complex crystals of finite thickness is theoretically investigated by the derivation of exact analytical expressions for reflection and transmission coefficients in terms of modified Bessel functions of first kind. The analytical results indicate that unidirectional invisibility, recently predicted for such crystals by coupled-mode theory (Z Lin et al 2011 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.106.213901), breaks down for crystals containing a large number of unit cells. In particular, for a given modulation depth in a shallow sinusoidal potential, three regimes are encountered as the crystal thickness is increased. At short lengths the crystal is reflectionless and invisible when probed from one side (unidirectional invisibility), whereas at intermediate lengths the crystal remains reflectionless but not invisible; for longer crystals both unidirectional reflectionless and invisibility properties are broken.

T-systems and Y-systems in integrable systems*

Abstract: T- and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov–Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analog of L-operators in KP hierarchy, Stokes phenomena in 1D Schrodinger problem, AdS/CFT correspondence, Toda field equations on discrete spacetime, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ansatze, quantum transfer matrix method and so forth. This review is a collection of short reviews on these topics which can be read more or less independently.

Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories

Abstract: In these lectures, I give a pedagogical presentation of some of the recent progress in supersymmetric Chern–Simons-matter theories, coming from the use of localization and matrix model techniques. The goal is to provide a simple derivation of the exact interpolating function for the free energy of ABJM theory on the three-sphere, which implies in particular the N3/2 behavior at strong coupling. I explain in detail part of the background needed to understand this derivation, like holographic renormalization, localization of path integrals and large N techniques in matrix models.

Bounds in 4D conformal field theories with global symmetry

Abstract: We explore the constraining power of OPE associativity in 4D conformal field theory with a continuous global symmetry group. We give a general analysis of crossing symmetry constraints in the 4-point function , where phi is a primary scalar operator in a given representation R. These constraints take the form of 'vectorial sum rules' for conformal blocks of operators whose representations appear in R circle times R and R circle times (R) over bar. The coefficients in these sum rules are related to the Fierz transformation matrices for the R circle times R circle times (R) over bar circle times (R) over bar invariant tensors. We show that the number of equations is always equal to the number of symmetry channels to be constrained. We also analyze in detail two cases-the fundamental of SO(N) and the fundamental of SU(N). We derive the vectorial sum rules explicitly, and use them to study the dimension of the lowest singlet scalar in the phi x phi(dagger) OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of phi and approaches 2 in the limit dim(phi) -> 1. For several small groups, we compute the behavior of the bound at dim(phi) > 1. We discuss implications of our bound for the conformal technicolor scenario of electroweak symmetry breaking.

The Darboux transformation of the derivative nonlinear Schrödinger equation

Abstract: The n-fold Darboux transformation (DT) is a 2 × 2 matrix for the Kaup–Newell (KN) system. In this paper, each element of this matrix is expressed by a ratio of the (n + 1) × (n + 1) determinant and n × n determinant of eigenfunctions. Using these formulae, the expressions of the q[n] and r[n] in the KN system are generated by the n-fold DT. Further, under the reduction condition, the rogue wave, rational traveling solution, dark soliton, bright soliton, breather solution and periodic solution of the derivative nonlinear Schrodinger equation are given explicitly by different seed solutions. In particular, the rogue wave and rational traveling solution are two kinds of new solutions. The complete classification of these solutions generated by one-fold DT is given.

On the observability and asymmetry of adiabatic state flips generated by exceptional points

Abstract: In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This ‘adiabatic flip’ effect is an outcome of the appearance of a degeneracy known as an ‘exceptional point’ inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.

Nonassociative gravity in string theory

Abstract: In an on-shell conformal field theory approach, we find indications of a three-bracket structure for target space coordinates in general closed string backgrounds. This generalizes the appearance of noncommutative gauge theories for open strings in two-form backgrounds to a putative noncommutative/nonassociative gravity theory for closed strings probing curved backgrounds with non-vanishing three-form flux. Several aspects and consequences of the three-bracket structure are discussed and a new type of generalized uncertainty principle is proposed.

Disordered topological insulators: a non-commutative geometry perspective

Abstract: The progress in the field of topological insulators is impetuous, being sustained by a suite of exciting results on three fronts: experiment, theory and numerical simulation. Very often, the theoretical characterizations of these materials involve advanced and abstract techniques from pure mathematics, leading to complex predictions which nowadays are tested by direct experimental observations. Many of these predictions have already been confirmed. What makes these materials topological is the robustness of their key properties against smooth deformations and onset of disorder. There is quite an extensive literature discussing the properties of clean topological insulators, but the literature on disordered topological insulators is limited. This review deals with strongly disordered topological insulators and covers some recent applications of a well-established analytic theory based on the methods of non-commutative geometry and developed for the integer quantum Hall effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chern number, and to discuss the physical properties protected by these invariants. Working with two explicit two-dimensional models, one for a Chern insulator and one for a quantum spin-Hall insulator, we first give an in-depth account of the key bulk properties of these topological insulators in the clean and disordered regimes. Extensive numerical simulations are employed here. A brisk but self-contained presentation of the non-commutative theory of the Chern number is given and a novel numerical technique to evaluate the non-commutative Chern number is presented. The non-commutative spin-Chern number is defined and the analytic theory, together with the explicit calculation of the topological invariants in the presence of strong disorder, is used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.

Non-geometric fluxes, asymmetric strings and nonassociative geometry

Abstract: We study closed bosonic strings propagating both in a flat background with constant H-flux and in its T-dual configurations. We define a conformal field theory capturing linear effects in the flux and compute scattering amplitudes of tachyons, where the Rogers dilogarithm plays a prominent role. For the scattering of four tachyons, a fluxed version of the Virasoro–Shapiro amplitude is derived and its pole structure is analysed. In the case of an R-flux background obtained after three T-dualities, we find indications for a nonassociative target-space structure which can be described in terms of a deformed tri-product. Remarkably, this product is compatible with crossing symmetry of conformal correlation functions. We finally argue that the R-flux background flows to an asymmetric CFT.

Basics of generalized unitarity

Abstract: We review generalized unitarity as a means for obtaining loop amplitudes from on-shell tree amplitudes. The method is generally applicable to both supersymmetric and non-supersymmetric amplitudes, including non-planar contributions. Here, we focus mainly on Yang–Mills theory, in the context of on-shell superspaces. Given the need for regularization at loop level, we also review a six-dimensional helicity-based superspace formalism and its application to dimensional and massive regularizations. An important feature of the unitarity method is that it offers a means for carrying over any identified tree-level property of on-shell amplitudes to loop level, though sometimes in a modified form. We illustrate this with examples of dual conformal symmetry and a recently discovered duality between color and kinematics. This article is an invited review for a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Scattering amplitudes in gauge theories’.

Loop amplitudes in gauge theories: modern analytic approaches

Abstract: This article reviews on-shell methods for analytic computation of loop amplitudes, emphasizing techniques based on unitarity cuts. Unitarity techniques are formulated generally but have been especially useful for calculating one-loop amplitudes in massless theories such as Yang?Mills theory, QCD and QED.

Families of quasi-rational solutions of the NLS equation and multi-rogue waves

Abstract: We construct a multi-parametric family of the solutions of the focusing nonlinear Schr?dinger equation (NLS) from the known results describing the multi-phase almost-periodic elementary solutions given in terms of Riemann theta functions. We give a new representation of their solutions in terms of Wronskians determinants of order 2N composed of elementary trigonometric functions. When we perform a special passage to the limit when all the periods tend to infinity, we obtain a family of quasi-rational solutions. This leads to efficient representations for the Peregrine breathers of orders N = 1,?2,?3 first constructed by Akhmediev and his co-workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue-wave formation in the frame of the NLS model first explained by V B Matveev in 2010. Our formulation allows us to isolate easily the second- or third-order Peregrine breathers from ?generic? solutions and also to compute the Peregrine breathers of orders 2 and 3 easily with respect to other approaches. In the cases N = 2,?3, we obtain the comfortable formulas to study the deformation of a higher Peregrine breather of order 2 to the three rogue-wave or order 3 to the six rogue-wave solutions via the variation of the free parameters of our construction.

Some mathematical properties of superoscillations

Abstract: In this paper, we give a possible mathematical setting for superoscillations. We define the set of superoscillation in terms of the uniform convergence of functions on such a set and study the problem of the approximation of a function by superoscillating functions.

Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative

Abstract: In this paper, the solution of a fractional diffusion equation with a Hilfer-generalized Riemann–Liouville time fractional derivative is obtained in terms of Mittag–Leffler-type functions and Fox's H-function. The considered equation represents a quite general extension of the classical diffusion (heat conduction) equation. The methods of separation of variables, Laplace transform, and analysis of the Sturm–Liouville problem are used to solve the fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative defined in the infinite domain can be expressed via Fox's H-function. It is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the Caputo and Riemann–Liouville sense are special cases of those diffusion equations with the Hilfer-generalized Riemann–Liouville time fractional derivative. The asymptotic behaviour of the solutions are found for large values of the spatial variable. The fractional moments of the fundamental solution of the fractional diffusion equation are obtained. The obtained results are relevant in the context of glass relaxation and aquifer problems.

Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon

Abstract: For non-Hermitian Hamiltonians with an isolated degeneracy (‘exceptional point’), a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of Bessel functions. Floquet solutions, returning exactly to their initial states (up to a constant) are found, as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous Hamiltonian. Adiabatically (i.e. for slow cycles), the multipliers of exponentially subdominant eigenstates can vary wildly, unlike those driven by Hermitian operators, which change little. These variations are explained as an example of the Stokes phenomenon of asymptotics. Improved (superadiabatic) approximations tame the variations of the multipliers but do not eliminate them.

Global weak solutions for the Novikov equation

Abstract: In the paper, we mainly study the existence and uniqueness of global weak solutions for the Novikov equation. We first recall some results and definitions on strong solutions and weak solutions for the equation. Then, we show that the equation has smooth solutions which exist globally in time, provided the initial data satisfy certain sign conditions. Finally we prove the existence and uniqueness of global weak solutions to the equation with the initial data satisfying certain sign conditions.

The smallest refrigerators can reach maximal efficiency

Abstract: We investigate whether size imposes a fundamental constraint on the efficiency of small thermal machines. We analyse in detail a model of a small self-contained refrigerator consisting of three qubits. We show analytically that this system can reach the Carnot efficiency, thus demonstrating that there exists no complementarity between size and efficiency.

Weak self-adjoint differential equations

Abstract: The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006 J. Math. Anal. Appl. 318 742?57; 2007 Arch. ALGA 4 55?60). In Ibragimov (2007 J. Math. Anal. Appl. 333 311?28), a general theorem on conservation laws was proved. In this paper, we generalize the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. We find a class of weak self-adjoint quasi-linear parabolic equations. The property of a differential equation to be weak self-adjoint is important for constructing conservation laws associated with symmetries of the differential equation.

Equilibrium statistical mechanics of bipartite spin systems

Abstract: The aim of this paper is to give an extensive treatment of bipartite mean field spin systems, pure and disordered. At first, bipartite ferromagnets are investigated, and an explicit expression for the free energy is achieved through a new minimax variational principle. Then, via the Hamilton?Jacobi technique, the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one. The same is investigated for bipartite spin-glasses. By the Borel?Cantelli lemma we obtain the control of the high temperature regime, while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation, uniquely defined by a minimax variational principle again. We also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one-party model counterparts. For the sake of completeness, we show further that at zero temperature the replica symmetric entropy becomes negative and, consequently, such a symmetry must be broken. The treatment of the fully broken replica symmetry case is deferred to a forthcoming paper. As a first step in this direction, we start deriving the linear and quadratic constraints to overlap fluctuations.

Konishi operator at intermediate coupling

Abstract: Thermodynamic Bethe Ansatz equations for two-particle states from the sector proposed by Arutyunov, Suzuki and the author are solved numerically for the Konishi operator descendent up to 't Hooft's coupling λ ≈ 2046. The data obtained is used to analyze the properties of Y-functions and address the issue of the existence of the critical values of the coupling. In addition, we find a new integral representation for the BES dressing phase which substantially reduces the computational time.

Conformal Galilei groups, Veronese curves and Newton–Hooke spacetimes

Abstract: Finite-dimensional nonrelativistic conformal Lie algebras spanned by polynomial vector fields of Galilei spacetime arise if the dynamical exponent is z = 2/N with N = 1,2,.... Their underlying group structure and matrix representation are constructed (up to a covering) by means of the Veronese map of degree N. Suitable quotients of the conformal Galilei groups provide us with Newton-Hooke nonrelativistic spacetimes with a quantized reduced negative cosmological constant � = −N.

Simulating quantum dynamics on a quantum computer

Abstract: We explicitly show how to simulate time-dependent sparse Hamiltonian evolution on a quantum computer, with complexity that is close to linear in the evolution time. The complexity also depends on the magnitude of the derivatives of the Hamiltonian. We propose a range of techniques to simulate Hamiltonians with badly behaved derivatives. These include using adaptive time steps, adapting the order of the integrators, and omitting regions about discontinuities. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian, and accounts for all computational resources. We explicitly determine the number of bits of output that this oracle needs to provide, and show how to efficiently perform the required 1-sparse unitary operations using these bits. We also account for discretization error in the time, as well as accounting for Hamiltonians that are a sum of terms that are sparse in different bases.

An action principle for Vasiliev’s four-dimensional higher spin gravity

Abstract: We provide Vasiliev's fully nonlinear equations of motion for bosonic higher spin gauge fields in four spacetime dimensions with an action principle. We first extend Vasiliev's original system with differential forms in degrees higher than 1. We then derive the resulting duality-extended equations of motion from a variational principle based on a generalized Hamiltonian sigma-model action. The generalized Hamiltonian contains two types of interaction freedoms: one, a set of functions that appears in the Q-structure of the generalized curvatures of the odd forms in the duality-extended system; and the other, a set depending on the Lagrange multipliers, encoding a generalized Poisson structure, i.e. a set of polyvector fields of rank 2 or higher in target space. We find that at least one of the two sets of interaction-freedom functions must be linear in order to ensure gauge invariance. We discuss consistent truncations to the minimal type A and B models (with only even spins), spectral flows on-shell and provide boundary conditions on fields and gauge parameters that are compatible with the variational principle and that make the duality-extended system equivalent, on-shell, to Vasiliev's original system.

Scattering amplitudes: the most perfect microscopic structures in the universe

Abstract: This review gives an overview of many of the recent developments in understanding the structure of relativistic scattering amplitudes in gauge theories ranging from QCD to super-Yang?Mills theory, as well as (super)gravity. I also provide a pedagogical introduction to some of the basic tools used to organize and illuminate the color and kinematic structure of amplitudes. This is an invited review introducing a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Scattering amplitudes in gauge theories'.