Periodic Hamiltonian Research Articles

Theoretical investigation of the periodic anderson hamiltonian of samarium hexaboride

Theoretical investigation of the periodic anderson hamiltonian of samarium hexaboride

Boundary value problems associated with Hamiltonian systems coupled with positively-(p, q)-homogeneous systems

We study the multiplicity of solutions for a two-point boundary value problem of Neumann type associated with a Hamiltonian system which couples a system with periodic Hamiltonian in the space variable with a second one with positively-(p, q)-homogeneous Hamiltonian. The periodic problem is also treated.

Trace formulas for time periodic complex Hamiltonians on lattice

We consider time periodic Hamiltonians with complex potentials on the lattice and determine trace formulas. As a corollary we estimate imaginary parts of eigenvalues of the quasienergy operator in terms of the norm of potentials.

Homogenization of some periodic Hamilton-Jacobi equations with defects

We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case

We study the asymptotic behavior of the viscosity solutions of the Hamilton-Jacobi (HJ) equation as the positive discount factor λ tends to 0, where is the perturbation of a Hamiltonian –periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential The constant c(G) appearing above is defined as the infimum of values for which the HJ equation in admits bounded viscosity subsolutions. We prove that the functions locally uniformly converge, for to a specific solution of the critical equation We identify in terms of projected Mather measures for G and of the limit to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.

Fractional Floquet theory

A fractional generalization of the Floquet theorem is suggested for fractional Schrödinger equations (FTSE)s with the time-dependent periodic Hamiltonians. The obtained result, called the fractional Floquet theorem (fFT), is formulated in the form of the Mittag-Leffler function, which is considered as the eigenfunction of the Caputo fractional derivative. The suggested formula makes it possible to reduce the FTSE to the standard quantum mechanics with the time-dependent Hamiltonian, where the standard Floquet theorem is valid. Two examples related to quantum resonances are considered as well to support the obtained result.

A perspective on the relative merits/demerits of time-propagators based on Floquet theorem.

The present report examines the nuances of analytic methods employed in the derivation of evolution operators in periodically driven quantum systems based on Floquet theorem. Specifically, time-propagators of the form, U(t) = P(t)e-iH̄t defined in the Hilbert space (of finite dimension), are derived through generalized multimodal expansion of the operators involved. While Floquet methods defined in the extended Hilbert space (of infinite dimension) have remained the method of choice for the description of time-evolution at non-stroboscopic time-intervals, the expansion schemes discussed do present an attractive option for similar studies in the standard Hilbert space. Nevertheless, the convergence criteria and suitability of such methods deserve formal validation in problems of experimental relevance. Employing examples comprising periodic Hamiltonians from magnetic resonance spectroscopy, the exactness of Floquet based time-propagators in the Schrödinger and interaction representation is discussed. Through rigorous comparisons between simulations emerging from analytic and exact numerical methods, the relative merits and demerits of different formulations of Floquet based methods are also discussed.

A continuous approach to Floquet theory for pulse-sequence optimization in solid-state NMR.

We present a framework that uses a continuous frequency space to describe and design solid-state nuclear magnetic resonance (NMR) experiments. The approach is similar to the well-established Floquet treatment for NMR, but it is not restricted to periodic Hamiltonians and allows the design of experiments in a reverse fashion. The framework is based on perturbation theory on a continuous Fourier space, which leads to effective, i.e., time-independent, Hamiltonians. It allows the back-calculation of the pulse scheme from the desired effective Hamiltonian as a function of spin-system parameters. We show as an example how to back-calculate the rf irradiation in the MIRROR experiment from the desired chemical-shift offset behavior of the sequence.

Continuous-time crystal from a spontaneous many-body Floquet state

Floquet driven systems represent an extremely interesting arena to study out-of-equilibrium phenomena. For instance, they provide realizations of discrete time crystals, where the discrete time translation symmetry of the periodic Hamiltonian is spontaneously broken by a subharmonic response of the system. However, the continuous presence of an external periodic driving is required within the current Floquet paradigm. We propose here the concept of spontaneous many-body Floquet state. This is a state that, in the absence of external periodic driving, self-oscillates like in the presence of a periodic Hamiltonian, this behavior being spontaneously induced by many-body interactions. In addition, its quantum fluctuations are described by regular Floquet theory. Furthermore, it is also a time crystal, presenting long-range time-periodic order. However, this crystalline behavior is very different to that of conventional Floquet discrete time crystals: here, there is no external periodic driving, energy is conserved, and the nature of the spontaneous symmetry breaking is continuous instead of discrete. We demonstrate that spontaneous many-body Floquet states can emerge in a variety of canonical many-body problems, ranging from interacting fermions to Bose-Hubbard models. We specifically show that a spontaneous many-body Floquet state is a universal intrinsic state of a one-dimensional flowing atom condensate, both subsonic and supersonic, resulting from a dynamical phase transition and robust against external perturbations and quantum fluctuations, proposing also realistic experimental scenarios for its observation. A spontaneous many-body Floquet state not only represents a realization of a continuous time crystal, but also a novel paradigm in Floquet physics.

A polyfold proof of the Arnold conjecture

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from {{mathbb {C}}{mathbb {P}}}^1times M to {{mathbb {C}}}^+ times M and {{mathbb {C}}}^-times M, as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

Numerical recipes for faster MAS-DNP simulations

Numerical simulations of Magic Angle Spinning Dynamic Nuclear Polarization (MAS-DNP) have transformed the way the DNP process is understood in rotating samples. In 2012, two methods were concomitantly developed to simulate small spin systems (< 4 spin-1/2). The development of new polarizing agents, including those containing metal centers with S > 1/2, makes it necessary to further expand the numerical tools with minimal approximations that will help rationalize the experimental observations and build approximate models. In this paper, three strategies developed in the past five years are presented: an adaptive integration scheme, a hybrid Hilbert/Liouville formalism, and a method to truncate the Liouville space basis for periodic Hamiltonian. Each of these methods enable time savings ranging from a factor of 3 to > 100. We illustrate the code performance by reporting for the first time the MAS-DNP field profiles for “AMUPol”, in which the couplings to the nitrogen nuclei are explicitly considered, as well as Cross-Effect MAS-DNP field profiles with two electrons spin 5/2 interacting with a nuclear spin 1/2.

Variational projector-augmented wave method: A full-potential approach for electronic structure calculations in solid-state physics

In solid-state physics, energies of crystals are usually computed with a plane-wave discretization of Kohn-Sham equations. However the presence of Coulomb singularities requires the use of large plane-wave cut-offs to produce accurate numerical results. In this paper, an analysis of the plane-wave convergence of the eigenvalues of periodic linear Hamiltonians with Coulomb potentials using the variational projector-augmented wave (VPAW) method is presented. In the VPAW method, an invertible transformation is applied to the original eigenvalue problem, acting locally in balls centered at the singularities. In this setting, a generalized eigenvalue problem needs to be solved using plane-waves. We show that cusps of the eigenfunctions of the VPAW eigenvalue problem at the positions of the nuclei are significantly reduced. These eigenfunctions have however a higher-order derivative discontinuity at the spheres centered at the nuclei. By balancing both sources of error, we show that the VPAW method can drastically improve the plane-wave convergence of the eigenvalues with a minor additional computational cost. Numerical tests are provided confirming the efficiency of the method to treat Coulomb singularities.

Topological and geometrical aspects of band theory

This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. Amusingly, many of these developments have a connection to contributions in high-energy physics by Dirac. The review starts by a presentation of the Dirac magnetic monopole, goes on with the Berry phase in a two-level system and the geometrical/topological band theory for Bloch electrons in crystals. Next, specific examples of tight-binding models giving rise to lattice versions of the Dirac equation in various space dimension are presented: in 1D (Su–Schrieffer–Heeger (SSH) and Rice–Mele models), 2D (graphene, boron nitride, Haldane model) and 3D (Weyl semi-metals). The focus is on topological insulators and topological semi-metals. The latter have a Fermi surface that is characterized as a topological defect. For topological insulators, the two alternative view points of twisted fiber bundles and of topological textures are developed. The minimal mathematical background in topology (essentially on homotopy groups and fiber bundles) is provided when needed. Topics rarely reviewed include: periodic versus canonical Bloch Hamiltonian (basis I/II issue), Zak versus Berry phase, the vanishing electric polarization of the SSH model and Dirac insulators.

A simple real-space scheme for periodic Dirac operators

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation

We present an algorithm for constructing analytically approximate integrals of motion in simple time periodic Hamiltonians of the form $H=H_0+ \varepsilon H_i$, where $\varepsilon$ is a perturbation parameter. We apply our algorithm in a Hamiltonian system whose dynamics is governed by the Mathieu equation and examine in detail the orbits and their stroboscopic invariant curves for different values of $\varepsilon$. We find the values of $\varepsilon_{crit}$ beyond which the orbits escape to infinity and construct integrals which are expressed as series in the perturbation $\varepsilon$ and converge up to $\varepsilon_{crit}$. In the absence of resonances the invariant curves are concentric ellipses which are approximated very well by our integrals. Finally we construct an integral of motion which describes the hyperbolic stroboscopic invariant curve of a resonant case.

Supercanonical transformations and time-dependent Hamiltonians

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp. The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

The Study of the Ground State Properties of Heavy Famion System Using an Extended Kondo-Anderson Model in One Dimension

The Kondo interaction coupling, Heisenberg exchange coupling, and Coulomb interactions within d-sites, were introduced in a one dimensional Periodic Anderson Model Hamiltonian (PAMH) to further investigate the effects of interaction parameters on the ground state energy of systems with heavy fermions (HF) behavior. Periodic Anderson model PAM being one of the most successful model for studying the heavy fermions System (HFS) was used in an extended version (mixed Kondo-Anderson representation) on a system of three-electrons interacting on three-sites cluster. Exact Diagonalization technique (EDT) normally used to solve conventional PAM calculation was considered in this work for a very small cluster. Hamiltonian used to describe this model contains the usual term describing the kinetic energy of the system, on-site coulomb repulsion and a hopping integral. The Hamiltonian is acted on the different Hilbert states of the lattice system and results of the interactions were obtained in terms of hopping integral, coulomb repulsions, exchange couplings and the hybridization term. Graphs of ground state energy Eo plotted agains tthese interaction parameters were presented in a clear format. As these parameters were varied numerically through a finite range of values, the individual effects of these parameters on the system’s ground state energy were observed and discussed. Hence, the results obtained from this work shows theoretically how the tuning of the Columbic interaction within the conduction band provides information that sheds light on the underlying physics of the heavy fermions systems models. Results obtained from this work further demonstrate the reliability of the model Hamiltonians that we have harnessed and the importance of considering electron-lattice interactions as well as interactions that account for magnetic impurities for the proper description of heavy fermions material.

Approximate Research of Problems on Perturbation of Periodic and Autonomous Hamiltonian Systems in Critical Cases

The paper studies the problem of approximate construction of eigenvalues and multipliers of linear (autonomous and periodic) Hamiltonian systems depending on a small parameter in the main critical cases. New formulas for the asymptotic (in powers of a small parameter) representation of the eigenvalues and multipliers of the task are proposed. The obtained formulas allow us to effectively study the problems of stability and hyperbolicity of linear systems, equilibrium points and periodic solutions of nonlinear Hamiltonian systems, the problem of constructing the boundaries of the regions of stability and hyperbolicity, problems of local bifurcations of nonlinear dynamical systems, etc.

On the Reducibility of a Class of Nonlinear Periodic Hamiltonian Systems with Degenerate Equilibrium

In this paper, we prove the existence of periodic solutions of a class of Hamiltonian systems with degenerate equilibriums under small nonlinear periodic perturbations. Actually we prove that the periodic Hamiltonian systems with small perturbations can be reducible to a periodic Hamiltonian system with an equilibrium by a periodic symplectic mapping. This result is a reformulation of the result in Lu and Xu (Nonlinear Differ Equ Appl 21:361–370, 2014) in the case of Hamiltonian systems.

Normalization of a Periodic Hamiltonian System

First, the normal form in the vicinity of the stationary solution of the autonomous Hamiltonian system is recalled. Next, linear periodic Hamiltonian systems are considered. For them, normal forms of the Hamiltonians in the complex and real cases are found. A feature of the real case in the situation of parametric resonance is discovered. Next, normal forms of Hamiltonians of nonlinear periodic systems are found. Using an additional canonical transformation, such a normal form can be always reduced to an autonomous Hamiltonian system that preserves all small parameters and symmetries of the original system. The local families of fixed points of this autonomous system are associated with families of periodic solutions to the original system. A similar theory is constructed in the neighborhood of the periodic solution to the autonomous system. All transformations are algorithmic and can be implemented in a computer algebra system.