Integrable Hamiltonian Research Articles

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

Read-Green resonances in a topological superconductor coupled to a bath

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchange, which can be observed through resonances in low-energy and -momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.

A new expression of Ns versus Ef to an accurate control charge model for AlGaAs/GaAs

Semi-conductor components become the privileged support of information and communication, particularly appreciation to the development of the internet. Today, MOS transistors on silicon dominate largely the semi-conductors market, however the diminution of transistors grid length is not enough to enhance the performances and respect Moore law. Particularly, for broadband telecommunications systems, where faster components are required. For this reason, alternative structures proposed like hetero structures IV-IV or III-V [1] have been.The most effective components in this area (High Electron Mobility Transistor: HEMT) on IIIV substrate. This work investigates an approach for contributing to the development of a numerical model based on physical and numerical modelling of the potential at heterostructure in AlGaAs/GaAs interface. We have developed calculation using projective methods allowed the Hamiltonian integration using Green functions in Schrodinger equation, for a rigorous resolution “self coherent” with Poisson equation. A simple analytical approach for charge-control in quantum well region of an AlGaAs/GaAs HEMT structure was presented. A charge-control equation, accounting for a variable average distance of the 2-DEG from the interface was introduced. Our approach which have aim to obtain ns-Vg characteristics is mainly based on: A new linear expression of Fermi-level variation with two-dimensional electron gas density in high electron mobility and also is mainly based on the notion of effective doping and a new expression of AEc

Hamiltonian time integrators for Vlasov-Maxwell equations

Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.

Semisimple Lie algebras and Hamiltonian theory of finite-dimensional Lax equations with spectral parameter on a Riemann surface

The Hamiltonian property and integrability of the Lax equations with spectral parameter on a Riemann surface are considered. The operators of Lax pairs are meromorphic functions of special form on a Riemann surface of arbitrary positive genus with values in an arbitrary semisimple Lie algebra. The study combines the theory of Lax equations with spectral parameter on a Riemann surface, as proposed by I.M. Krichever in 2001, with a “group-theoretic approach.”

Averaging and spectral properties for the 2D advection–diffusion equation in the semi-classical limit for vanishing diffusivity

We consider the two-dimensional advection–diffusion equation (ADE) on a bounded domain subject to Dirichlet or von Neumann boundary conditions involving a Liouville integrable Hamiltonian. Transformation to action–angle coordinates permits averaging in time and angle, resulting in an equation that allows for separation of variables. The Fourier transform in the angle coordinate transforms the equation into an effective diffusive equation and a countable family of non-self-adjoint Schrödinger equations. For the corresponding Liouville–Sturm problem, we apply complex-plane WKB methods to study the spectrum in the semi-classical limit for vanishing diffusivity. The spectral limit graph is found to consist of analytic curves (branches) related to Stokes graphs forming a tree-structure. Eigenvalues in the neighborhood of branches emanating from the imaginary axis are subject to various sublinear power laws with respect to diffusivity, leading to convection-enhanced rates of dissipation of the corresponding modes. The solution of the ADE converges in the limit of vanishing diffusivity to the solution of the effective diffusion equation on convective time scales that are sublinear with respect to the diffusive time scales.

Quantum integrable models of interacting bosons and classical r-matrices with spectral parameters

Using the technique of classical r -matrices with spectral parameters we construct a general form of quantum Lax operators of interacting boson systems corresponding to an arbitrary simple (or reductive) Lie algebra. We prove quantum integrability of these models in the physically important case of g = g l ( n ) and “diagonal” in the root basis classical r -matrices. We consider in detail two classes of non-skew-symmetric classical r -matrices with spectral parameters and obtain the corresponding quantum Lax operators and quantum integrable many-boson hamiltonians that generalize Bose–Hubbard dimer hamiltonians.

Variations on Quantum Ergodic Theorems

We derive some quantum ergodic theorems, related to microlocal behavior of eigenfunctions of a positive, self-adjoint, elliptic pseudodifferential operator Λ on a compact Riemannian manifold M, emphasizing results that hold without the hypothesis that the Hamiltonian flow generated by the symbol of Λ be ergodic. Cases treated include both integrable Hamiltonians and some associated with “soft chaos.”

Limit cycles bifurcating from a degenerate center

We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic polynomial differential systems we prove that at most three limit cycles can bifurcate from the degenerate center. As far as we know this is the first time that a complete study up to second order in the small parameter of the perturbation is done for studying the limit cycles which bifurcate from the periodic orbits surrounding a degenerate center (a center whose linear part is identically zero) having neither a Hamiltonian first integral nor a rational one. This study needs many computations, which have been verified with the help of the algebraic manipulator Maple.

Integrability and the conformal field theory of the Higgs branch

In the context of the AdS3/CFT2 correspondence, we investigate the Higgs branch CFT2. Witten showed that states localised near the small instanton singularity can be described in terms of vector multiplet variables. This theory has a planar, weak-coupling limit, in which anomalous dimensions of single-trace composite operators can be calculated. At one loop, the calculation reduces to finding the spectrum of a spin-chain with nearest-neighbour interactions. This CFT2 spin-chain matches precisely the one that was previously found as the weak-coupling limit of the integrable system describing the AdS3 side of the duality. We compute the one-loop dilatation operator in a non-trivial compact subsector and show that it corresponds to an integrable spin-chain Hamiltonian. This provides the first direct evidence of integrability on the CFT2 side of the correspondence.

Origin of the enhancement of tunneling probability in the nearly integrable system.

The enhancement of tunneling probability in the nearly integrable system is closely examined, focusing on tunneling splittings plotted as a function of the inverse of the Planck's constant. On the basis of the analysis using the absorber which efficiently suppresses the coupling, creating spikes in the plot, we found that the splitting curve should be viewed as the staircase-shaped skeleton accompanied by spikes. We further introduce renormalized integrable Hamiltonians and explore the origin of such a staircase structure by investigating the nature of eigenfunctions closely. It is found that the origin of the staircase structure could trace back to the anomalous structure in tunneling tail which manifests itself in the representation using renormalized action bases. This also explains the reason why the staircase does not appear in the completely integrable system.

Many-body characterization of particle-conserving topological superfluids.

What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order.

NANOMETRIC MODELISATION OF GAS STRUCTURE, A MULTIDIMENTIONNAL QUANTUM WELL

The present work is dedicated to the numeric physical modelisation of the potential to the interface of a heterostructure in GaAsAl/GaAs. Calculs, using some projective methods, permitting the integration of Hamiltonian, with Green functions in the equation of Schrödinger, for a rigorous resolution with the equation of Poisson are elaborated. A study of convergence of globally no linear system is done and it confirmed for ten bases functions and a very determined electric wall position. The different parameter influence on performances of the GaAsAl/GaAs heterostructure is put in evidence; what will permit us subsequently to consider the dynamic of the carriers in a HEMT heterostructure by rigorous and complete manner.

Persistence of Diophantine flows for quadratic nearly integrable Hamiltonians under slowly decaying aperiodic time dependence

The aim of this paper is to prove a Kolmogorov-type result for a nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists of the possibility to choose an arbitrarily small decaying coefficient, consistently with the perturbation size.

A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

A new integrable generalization to the two-dimensional (2D) sphere, , and to the hyperbolic space, , of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of motion is shown to be quadratic in the momenta. To construct such a new integrable Hamiltonian, , we will use a group theoretical approach in which the curvature, , of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. When the oscillator parameters Ω1 and Ω2 are such that , the system turns out to be the well-known superintegrable oscillator on and . Nevertheless, numerical integration of the trajectories of suggests that for other values of the parameters Ω1 and Ω2, the system is not superintegrable. In this way, we support the conjecture that for each commensurate, and thus superintegrable, Euclidean oscillator there exists a two-parametric family of curved integrable oscillators that turns out to be superintegrable only when the parameters are tuned to the commensurability condition.

Exponential-six potential scaling for the calculation of free energies in molecular simulations

An adjustable, scaled form of the exponential-six (exp-6) potential is presented. The potential form allows stable scaling from a fully interacting exp-6 system to a non-interacting reference system for the direct computation of free energy differences or efficient particle growth simulations, particularly for high-density systems. Additional scaling parameters were introduced to overcome known endpoint effects, whereby reducing the potential to an ideal gas state can produce singularities in simulation averages or prohibit the sampling of close particle distances. The scaled potential is validated in several ways, using Hamiltonian thermodynamic integration, by comparison to vapour–liquid and solid–liquid coexistence free energies reported in the literature, and by the application of the Gibbs–Helmholtz equation. Forms of the scaled exp-6 potential for its implementation into molecular simulations and the thermodynamic integration methods are also developed.

F4 Quantum Integrable, rational and trigonometric models: space-of-orbits view

Algebraic-rational nature of the four-dimensional, F4-invariant integrable quantum Hamiltonians, both rational and trigonometric, is revealed and reviewed. It was shown that being written in F4 Weyl invariants, polynomial and exponential, respectively, both similarity-transformed Hamiltonians are in algebraic form, they are quite similar the second order differential operators with polynomial coefficients; the flat metric in the Laplace-Beltrami operator has polynomial (in invariants) matrix elements. Their potentials are calculated for the first time: they are meromorphic (rational) functions with singularities at the boundaries of the configuration space. Ground state eigenfunctions are algebraic functions in a form of polynomials in some degrees. Both Hamiltonians preserve the same infinite flag of polynomial spaces with characteristic vector (1, 2, 2, 3), it manifests exact solvability. A particular integral common for both models is derived. The first polynomial eigenfunctions are presented explicitly.

Hamiltonian functional integrals representing regularized traces of higher-order differential operators

In this paper, we obtain a representation for the regularized traces (1) of evenorder differential opera� tors in terms of Hamiltonian functional integrals and the corresponding Feynman formulas. We also present an example showing how such a representation can be applied to obtain formulas not containing functional integrals. A Feynman formula (2) is a representation of a solution of an evolution differential equation or an object somehow related to this equation in terms of limit integrals over Cartesian powers of some space. If the space under consideration coincides with the spa� tial domain of the solution, then the corresponding formulas are said to be Lagrangian; if this is the prod� uct of this spatial domain and the dual of the linear space containing it, then the formulas are said to be Hamiltonian. A similar terminology is used for func� tional integrals being approximated by Feynman for� mulas. In this paper, we extend results of (3) to higher� order differential operators, which requires substan� tially modifying the technique; namely, we use Hamil� tonian (rather that Lagrangian, as in (3)) Feynman formulas and functional integrals. Moreover, in those Hamiltonian functional integrals which are used in the paper, integration is over the set of functions taking values in the product of the momentum space and a domain in the configuration space not coinciding with the entire space. Main attention is paid to the algebraic aspects of the problems under consideration; some analytic assumptions are omitted.

Limits of integrable Hamiltonians on semisimple Lie algebras

It is proved that the limit of integrable Hamiltonians on a semisimple Lie algebra is an integrable Hamiltonian. Some limits of integrable Hamiltonians obtained by the argument shift method such that these limits themselves cannot be obtained by this method are constructed.

Solution for an interaction quench in the Lieb-Liniger Bose gas

We study a quench protocol where the ground state of a free many-particle bosonic theory in one dimension is let unitarily evolve in time under the integrable Lieb-Liniger Hamiltonian of $\delta$-interacting repulsive bosons. By using a recently-proposed variational method, we here obtain the exact non-thermal steady-state of the system in the thermodynamic limit, and discuss some of its main physical properties. Besides being a rare case of a thermodynamically exact solution to a truly interacting quench situation, this interestingly represents an example where a naive implementation of the generalized Gibbs ensemble fails.