An effective Hamiltonian approach to quantum random walk

In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative (see Chandrashekar, Sci. Rep. 3, 2829 (18)). We showed that in the case of two-step walk, the time evolution operator effectively can have multiplicative form. In the case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the graphene Hamiltonian, the walk has been studied on a graphene lattice and we conclude the preference of additive approach over the multiplicative one.     

An effective spin Hamiltonian approach to heavy electron metamagnetism

ABSTRACT

We describe a minimal model for metamagnetism, based on a quantum spin Hamiltonian with a single energy scale. Within this model, the metamagnetic critical field is proportional to the temperature where a peak in the linear susceptibility occurs which in turn is proportional to the temperatures where the nonlinear susceptibilities also peak. The thermodynamic properties are derived in a straightforward manner and bear a striking resemblance to observations in such strongly correlated systems as heavy fermion materials. We also consider extensions of the model by including effects such as a mean field and a tilt of the quantisation axis to encompass observed deviations from a minimal metamagnetic behaviour.     

A symmetry-based approach to the effective Hamiltonian for symmetric top molecules in degenerate vibronic states

The effective Hamiltonian for a symmetric top molecule in a degenerate vibronic state is obtained. Included in this Hamiltonian are the rotational, spin-rotational, spin-orbit coupling and electronic spin-spin interactions. The terms of the Hamiltonian are expressed as the product of molecular ‘constants,’ rotational angular momentum operators, and symmetry operators. A formalism is derived, and tables included, to determine whether or not a symmetry operator vanishes for a given vibronic state of a particular molecular symmetry. In this way, one can easily obtain all the non-vanishing Hamiltonian terms for a particular application.     

On the periodic orbits of perturbed Hooke Hamiltonian systems with three degrees of freedom

We study periodic orbits of Hamiltonian differential systems with three degrees of freedom using the averaging theory. We have chosen the classical integrable Hamiltonian system with the Hooke potential and we study periodic orbits which bifurcate from the periodic orbits of the integrable system perturbed with a non-autonomous potential.     

The effective Hamiltonian approach for the two-nucleon distribution function

The microscopic two-body effective Hamiltonian equation satisfied by the two-nucleon distribution function needed for two-particle transfer processes is derived. The usual approximations made in order to obtain a partial solution to this equation are discussed and compared. It is concluded that contributions to the distribution function from both core state and channel coupling terms are likely to be comparable to continuum configurations in importance. Such contributions must certainly be included if an unambiguous normalization for the distribution function is desired.     

Magnetic moments of negative parity baryons from the effective Hamiltonian approach to QCD

The magnetic moments of the S ₁₁(1535) and S ₁₁(1650) baryons are studied in the framework of the relativistic three-quark Hamiltonian derived in the Field Correlation Method. The baryon magnetic moments are expressed via the average current quark energies which are defined by the fundamental QCD parameters: the string tension σ, the quark masses, and the strong coupling constant α _s . The resulting magnetic moments for the J ^P = 1/2^? nucleons are compared both to model calculations and to those from lattice QCD.     

A Hamiltonian approach to the strong gravity limit

The Hamiltonian for general relativity is examined in the strong gravity (SG) limitG . In this limit a perfect fluid moves irrotationally along geodesics. An appropriate SG limit for the scalar field is developed such that the energy density has a limit. The solutions in this limit, which were previously known, are shown to come out simply and directly. A classification of the trace-free part of the momentum is given in the Appendix.     

A perturbed martingale approach to global optimization

A new global stochastic search, guided mainly through derivative-free directional information computable from the sample statistical moments of the design variables within a Monte Carlo setup, is proposed. The search is aided by imparting to the directional update term additional layers of random perturbations referred to as ‘coalescence’ and ‘scrambling’. A selection step, constituting yet another avenue for random perturbation, completes the global search. The direction-driven nature of the search is manifest in the local extremization and coalescence components, which are posed as martingale problems that yield gain-like update terms upon discretization. As anticipated and numerically demonstrated, to a limited extent, against the problem of parameter recovery given the chaotic response histories of a couple of nonlinear oscillators, the proposed method appears to offer a more rational, more accurate and faster alternative to most available evolutionary schemes, prominently the particle swarm optimization.     

The effective potential of the confinement order parameter in the Hamiltonian approach

The effective potential of the order parameter for confinement is calculated within the Hamiltonian approach by compactifying one spatial dimension and using a background gauge fixing. Neglecting the ghost and using the perturbative gluon energy one recovers the Weiss potential. From the full non-perturbative potential calculated within a variational approach a critical temperature of the deconfinement phase transition of 269 MeV is found for the gauge group SU(2).     

A Hamiltonian approach to fermion Casimir energy

A derivative expansion of the one-fermion-loop effective action for theO(4) σ model can be computed in a variety of ways. This can then be used to investigate the effect of fermions on skyrmion stability. Here a method of calculating the energy density itself as a derivative expansion is outlined. The method would be applicable to a wide class of models. It is illustrated by computing the four-derivative terms in the fermion Casimir energy density in theO(4) σ model. The result agrees with previous calculations.     

Logarithmic correction to the probability of capture for dissipatively perturbed Hamiltonian systems

Hamiltonian systems are analyzed with a double homoclinic orbit connecting a saddle to itself. Competing centers exist. A small dissipative perturbation causes the stable and unstable manifolds of the saddle point to break apart. The stable manifolds of the saddle point are the boundaries of the basin of attraction for the competing attractors. With small dissipation, the boundaries of the basins of attraction are known to be tightly wound and spiral-like. Small changes in the initial condition can alter the equilibrium to which the solution is attracted. Near the unperturbed homoclinic orbit, the boundary of the basin of attraction consists of a large sequence of nearly homoclinic orbits surrounded by close approaches to the saddle point. The slow passage through an unperturbed homoclinic orbit (separatrix) is determined by the change in the value of the Hamiltonian from one saddle approach to the next. The probability of capture can be asymptotically approximated using this change in the Hamiltonian. The well-known leading-order change of the Hamiltonian from one saddle approach to the next is due to the effect of the perturbation on the homoclinic orbit. A logarithmic correction to this change of the Hamiltonian is shown to be due to the effect of the perturbation on the saddle point itself. It is shown that the probability of capture can be significantly altered from the well-known leading-order probability for Hamiltonian systems with double homoclinic orbits of the twisted type, an example of which is the Hamiltonian system corresponding to primary resonance. Numerical integration of the perturbed Hamiltonian system is used to verify the accuracy of the analytic formulas for the change in the Hamiltonian from one saddle approach to the next. (c) 1995 American Institute of Physics.     

Hamiltonian approach to nonlinear oscillators

A Hamiltonian approach to nonlinear oscillators is suggested. A conservative oscillator always admits a Hamiltonian invariant, H, which keeps unchanged during oscillation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with acceptable accuracy. Two illustrating examples are given to elucidate the solution procedure.     

Hamiltonian approach to frame dragging

A Hamiltonian approach makes the phenomenon of frame dragging apparent “up front” from the appearance of the drag velocity in the Hamiltonian of a test particle in an arbitrary metric. Hamiltonian (1) uses the inhomogeneous force equation (4), which applies to non-geodesic motion as well as to geodesics. The Hamiltonian is not in manifestly covariant form, but is covariant because it is derived from Hamilton’s manifestly covariant scalar action principle. A distinction is made between manifest frame dragging such as that in the Kerr metric, and hidden frame dragging that can be made manifest by a coordinate transformation such as that applied to the Robertson–Walker metric in Sect. 2. In Sect. 3 a zone of repulsive gravity is found in the extreme Kerr metric. Section 4 treats frame dragging in special relativity as a manifestation of the equivalence principle in accelerated frames. It answers a question posed by Bell about how the Lorentz contraction can break a thread connecting two uniformly accelerated rocket ships. In Sect. 5 the form of the Hamiltonian facilitates the definition of gravitomagnetic and gravitoelectric potentials.     

A three-body approach to the effective nucleon-nucleon interaction

The problem of the effective nucleon-nucleon interaction is formulated in a three-body model. The interaction of two nucleons and a heavy core having internal excitations is described by means of Faddeev-type coupled equations. The elimination of the excited states of the core leads to a set of coupled equations with an effective kernel which incorporates the effect of the eliminated degrees of freedom. An exactly soluble model is constructed and solved. The exact and an approximate solution are compared. The comparison demonstrates the applicability of the model for the investigation of problems connected with the effective nucleon-nucleon interaction.     

A generic approach to boundary reflection in periodic media

Waves in periodic media, whose propagation is governed by nearest neighbour interaction, are investigated. The reflection and transmission coefficients are derived for a plane wave incident from medium 1 upon medium 2, without invoking common approximations. The derivation is valid for a class of waves including magneto- and electro-inductive waves, waves on short loaded dipoles, nanoparticles, coupled waveguides and acoustic waves in monatomic media. For this last case hitherto unknown microscopic reflection and transmission coefficients are derived and shown to reduce in the continuous limit to the well-known expressions in terms of acoustic impedances.     

Nonequilibrium diagram technique for tunneling problems in the effective mass approach

It is shown how the general formulas of the nonequilibrium diagram technique can be used in problems of tunnel planar structures described in the effective mass approach. The relation between such a “continual” approach and the tunneling Hamiltonian method is established, and the applicability conditions for this method are determined. The effects beyond the applicability limits of the tunneling Hamiltonian method, which can be described by the continual approach, are considered.