Class Of Hamiltonians Research Articles

High-order Foldy-Wouthuysen transformations of the Dirac and Dirac-Pauli Hamiltonians in the weak-field limit

The low-energy and weak-field limit of the Dirac equation can be obtained by an order-by-order block diagonalization approach to any desired order in the parameter $\ensuremath{\pi}/mc$ $(\ensuremath{\pi}$ is the kinetic momentum and $m$ is the mass of the particle). In previous work, it has been shown that, up to the order of ${(\ensuremath{\pi}/mc)}^{8}$, the Dirac-Pauli Hamiltonian in the Foldy-Wouthuysen (FW) representation may be expressed as a closed form and consistent with the classical Hamiltonian, which is the sum of the classical relativistic Hamiltonian for orbital motion and the Thomas-Bargmann-Michel-Telegdi Hamiltonian for spin precession. In order to investigate the exact validity of the correspondence between classical and Dirac-Pauli spinors, it is necessary to proceed to higher orders. In this paper, we investigate the FW representation of the Dirac and Dirac-Pauli Hamiltonians using Kutzelnigg's diagonalization method. We show that the Kutzelnigg diagonalization method can be further simplified if nonlinear effects of static and homogeneous electromagnetic fields are neglected (in the weak-field limit). Up to the order of ${(\ensuremath{\pi}/mc)}^{14}$, we find that the FW transformation for both Dirac and Dirac-Pauli Hamiltonians is in agreement with the classical Hamiltonian with the gyromagnetic ratio given by $g=2$ and $g\ensuremath{\ne}2$, respectively. Furthermore, with higher-order terms at hand, it is demonstrated that the unitary FW transformation admits a closed form in the low-energy and weak-field limit.

Diagonal quantum circuits: Their computational power and applications

Diagonal quantum circuits are quantum circuits comprising only diagonal gates in the computational basis. In spite of a classical feature of diagonal quantum circuits in the sense of commutativity of all gates, their computational power is highly likely to outperform classical one and they are exploited for applications in quantum informational tasks. We review computational power of diagonal quantum circuits and their applications. We focus on the computational power of instantaneous quantum polynomial-time (IQP) circuits, which are a special type of diagonal quantum circuits. We then review an approximate generation of random states as an application of diagonal quantum circuits, where random states are an ensemble of pure states uniformly distributed in a Hilbert space. We also present a thermalizing algorithm of classical Hamiltonians by using diagonal quantum circuits. These applications are feasible to be experimentally implemented by current technology due to a simple and robust structure of diagonal gates.

Ensemble inequivalence in systems with wave-particle interaction.

The classical wave-particle Hamiltonian is considered in its generalized version, where two modes are assumed to interact with the coevolving charged particles. The equilibrium statistical mechanics solution of the model is worked out analytically, both in the canonical and the microcanonical ensembles. The competition between the two modes is shown to yield ensemble inequivalence, at variance with the standard scenario where just one wave is allowed to develop. As a consequence, both temperature jumps and negative specific heat can show up in the microcanonical ensemble. The relevance of these findings for both plasma physics and free electron laser applications is discussed.

Correspondence between classical and Dirac-Pauli spinors in view of the Foldy-Wouthuysen transformation

The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-1/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of $1/m^{14}$ ($m$ is the particle's mass) in the low-energy/weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g., Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.

Spectrum of quantum transfer matrices via classical many-body systems

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous ${\mathfrak g}{\mathfrak l}_n$-invariant XXX spin chain on $N$ sites with twisted boundary conditions can be found in terms of velocities of particles in the rational $N$-body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all $N$ particles and the other one is an $N$-dimensional Lagrangian submanifold obtained by fixing levels of $N$ classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the ${\mathfrak g}{\mathfrak l}_n$ Gaudin model with $N$ marked points (on the quantum side) and the Calogero-Moser system with $N$ particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

Finite size effects in L1o-FePt nanoparticles

Finite size effects on the temperature dependence of the uniaxial magnetic anisotropy, longitudinal and transverse susceptibilities and specific heat are examined for L1o-ordered FePt nanoparticles using an atomistic model based on an effective classical spin Hamiltonian. At low temperatures below criticality, we study the intrinsic uniaxial magnetic anisotropy energy (MAE) K1 and its scaling with magnetization K1(T)∼Ms(T)δ and using Langevin dynamics simulations we show that the dependence of the exponent δ on the size L and aspect ratio of the grain arises from decomposition of the MAE into bulk and surface dependent terms. Monte Carlo simulations in the critical regime near the Curie temperature Tc, show that the temperature variation of the specific heat and longitudinal susceptibility is given by finite size scaling relations c=Lα/νc̃(L1/νϵ) and χ=Lγ/νχ̃(L1/νϵ), respectively, where ϵ=(T−Tc)/Tc is the reduced temperature, and the susceptibility scaling function χ̃ can be approximated by a Lorentzian. Our estimates of the critical exponents α,γ, and ν appear to be in agreement with the universality class of the 3D Ising model.

Information Processing and the Second Law of Thermodynamics: An Inclusive, Hamiltonian Approach

We obtain generalizations of the Kelvin-Planck, Clausius, and Carnot statements of the second law of thermodynamics, for situations involving information processing. To this end, we consider an information reservoir (representing, e.g. a memory device) alongside the heat and work reservoirs that appear in traditional thermodynamic analyses. We derive our results within an inclusive framework in which all participating elements -- the system or device of interest, together with the heat, work and information reservoirs -- are modeled explicitly by a time-independent, classical Hamiltonian. We place particular emphasis on the limits and assumptions under which cyclic motion of the device of interest emerges from its interactions with work, heat, and information reservoirs.

Quantum quench in ap+ipsuperfluid: Winding numbers and topological states far from equilibrium

We study the non-adiabatic dynamics of a 2D p+ip superfluid following a quantum quench of the BCS coupling constant. The model describes a topological superconductor with a non-trivial BCS (trivial BEC) phase appearing at weak (strong) coupling strengths. We extract the exact long-time asymptotics of the order parameter \Delta(t) by exploiting the integrability of the classical p-wave Hamiltonian, which we establish via a Lax construction. Three different types of behavior can occur depending upon the strength and direction of the quench. In phase I, the order parameter asymptotes to zero. In phase II, \Delta(t) goes to a non-zero constant. Phase III is characterized by persistent oscillations of \Delta(t). For quenches within I and II, we determine the topological character of the asymptotic states. We show that two different formulations of the bulk topological winding number, although equivalent in the ground state, must be regarded as independent out of equilibrium. The first number Q characterizes the Anderson pseudospin texture of the initial state; we show that it is conserved. For non-zero Q, this leads to the prediction of a "gapless topological" state when \Delta(t) goes to zero. The presence or absence of Majorana edge modes in a sample with a boundary is encoded in the second winding number W, formulated in terms of the retarded Green's function. We show that W can change following a quench across the quantum critical point. We discuss the implications for the (dis)appearance of Majorana edge modes. Finally, we show that the parity of zeros in the bulk out-of-equilibrium Cooper pair distribution function constitutes a Z2-valued quantum number, which is non-zero whenever W differs from Q. The pair distribution can in principle be measured using RF spectroscopy in an ultracold atom realization, allowing direct experimental detection of the bulk Z2 number.

On the Josephson effect in a Bose–Einstein condensate subject to a density-dependent gauge potential

We investigate the coherent dynamics of a Bose–Einstein condensate in a double well, subject to a density-dependent gauge potential. Further, we derive the nonlinear Josephson equations that allow us to understand the many-body system in terms of a classical Hamiltonian that describes the motion of a nonrigid pendulum with an initial angular offset. Finally we analyse the phase-space trajectories of the system, and describe how the self-trapping is affected by the presence of an interacting gauge potential.

Symmetries of N = 4 supersymmetric mechanics

We explicitly constructed the generators of the SU(n + 1) group that commute with the supercharges of N = 4 supersymmetric mechanics in the background U(n) gauge fields. The corresponding classical Hamiltonian can be represented as a direct sum of two Casimir operators: one Casimir operator on the SU(n + 1) group contains our bosonic and fermionic coordinates and momenta, while the second one, on the SU(1, n) group, is constructed from isospin degrees of freedom only.

Theory of half-metallic double perovskites. II. Effective spin Hamiltonian and disorder effects

Double perovskites like Sr$_2$FeMoO$_6$ are materials with half-metallic ground states and ferrimagnetic T$_{\rm{c}}$'s well above room temperature. This paper is the second of our comprehensive theory for half metallic double perovskites. Here we derive an effective Hamiltonian for the Fe core spins by "integrating out" the itinerant Mo electrons and obtain an unusual double square-root form of the spin-spin interaction. We validate the classical spin Hamiltonian by comparing its results with those of the full quantum treatment presented in the companion paper "Theory of Half-Metallic Double Perovskites I: Double Exchange Mechanism". We then use the effective Hamiltonian to compute magnetic properties as a function of temperature and disorder and discuss the effect of excess Mo, excess Fe, and anti-site disorder on the magnetization and T$_{\rm{c}}$. We conclude with a proposal to increase T$_{\rm{c}}$ without sacrificing carrier polarization.

Theory of half-metallic double perovskites. I. Double exchange mechanism

The double perovskite material \SFMO has the rare and desirable combination of a half-metallic ground state with 100% spin polarization and ferrimagnetic \Tc$\simeq 420$K, well above room temperature. In this two-part paper, we present a comprehensive theoretical study of the magnetic and electronic properties of half metallic double perovskites. In this paper we present exact diagonalization calculations of the "fast" Mo electronic degrees coupled to "slow" Fe core spin fluctuations treated by classical Monte Carlo techniques. From the temperature dependence of the spin-resolved density of states, we show that the electronic polarization at the chemical potential is proportional to magnetization as a function of temperature. We also consider the effects of disorder and show that excess Fe leaves the ground state half-metallic while anti-site disorder greatly reduces the polarization. In a companion paper titled "Theory of Half-Metallic Double Perovskites II: Effective Spin Hamiltonian and Disorder Effects", we derive an effective classical spin Hamiltonian that provides a new framework for understanding the magnetic properties of half-metallic double perovskites including the effects of disorder. Our results on the dependence of the spin polarization on temperature and disorder has important implications for spintronics.

FIRST-ORDER QUANTUM-GRAVITATIONAL CORRECTION TO FRIEDMANNIAN COSMOLOGY FROM COVARIANT, HOLOMORPHIC SPINFOAM COSMOLOGY

The first-order loop quantum gravity correction of the simplest, classical general relativistic Friedmann Hamiltonian constraint, emerging from a holomorphic spinfoam cosmological model peaked on homogeneous, isotropic geometries, is studied. A quantum Hamiltonian constraint satisfied by the EPRL transition amplitude between the boundary cosmological coherent states includes a contribution of the order of the Planck constant ℏ that also appears in the corresponding semiclassical, symplectic model. The analysis of this term gives a quantum-gravitational correction to the classical Friedmann dynamics of the scale factor yielding an insignificantly small deceleration contribution in the expansion of the universe. The robustness of the physical interpretation is established for arbitrary refinements of the boundary graphs. Also, mathematical equivalences between the semiclassical cosmological model and certain classical fluid and scalar field theories are explored.

Perturbation method to calculate the density of states

Monte Carlo switching moves ("perturbations") are defined between two or more classical Hamiltonians sharing a common ground-state energy. The ratio of the density of states (DOS) of one system to that of another is related to the ensemble averages of the microcanonical acceptance probabilities of switching between these Hamiltonians, analogously to the case of Bennett's acceptance ratio method for the canonical ensemble [C. H. Bennett, J. Comput. Phys. 22, 245 (1976)]. Thus, if the DOS of one of the systems is known, one obtains those of the others and, hence, the partition functions. As a simple test case, the vapor pressure of an anharmonic Einstein crystal is computed, using the harmonic Einstein crystal as the reference system in one dimension; an auxiliary calculation is also performed in three dimensions. As a further example of the algorithm, the energy dependence of the ratio of the DOS of the square-well and hard-sphere tetradecamers is determined, from which the temperature dependence of the constant-volume heat capacity of the square-well system is calculated and compared with canonical Metropolis Monte Carlo estimates. For these cases and reference systems, the perturbation calculations exhibit a higher degree of convergence per Monte Carlo cycle than Wang-Landau (WL) sampling, although for the one-dimensional oscillator the WL sampling is ultimately more efficient for long runs. Last, we calculate the vapor pressure of liquid gold using an empirical Sutton-Chen many-body potential and the ideal gas as the reference state. Although this proves the general applicability of the method, by its inherent perturbation approach the algorithm is suitable for those particular cases where the properties of a related system are well known.

Generalized U(N) gauge transformations in the realm of the extended covariant Hamilton formalism of field theory

The Lagrangians and Hamiltonians of classical field theory require to comprise gauge fields in order to be form-invariant under local gauge transformations. These gauge fields have turned out to correctly describe pertaining elementary particle interactions. In this paper, this principle is extended to require additionally the form-invariance of a classical field theory Hamiltonian under variations of the space-time curvature emerging from the gauge fields. This approach is devised on the basis of the extended canonical transformation formalism of classical field theory which allows for transformations of the space-time metric in addition to transformations of the fields. Working out the Hamiltonian that is form-invariant under extended local gauge transformations, we can dismiss the conventional requirement for gauge bosons to be massless in order for them to preserve the local gauge invariance. The emerging equation of motion for the curvature scalar R turns out to be compatible with that from a Proca system in the case of a static gauge field.Communicated by H Stöcker

Noncommutative quantum mechanics of a test particle under linearly polarized gravitational waves

We consider the quantum dynamics of a test particle first with no potential and then with hermonic oscillator potential in noncommutative space under the influence of a passing linearized gravitational wave in the long wave-length and low-velocity limit. A prescription for quantizing the classical Hamiltonian for these systems is proposed. While both the solutions show noncommutative signatures, for the hermonic oscillator case a resonance scenario is found. We expect this noncommutative signature to show up as some noise source in the GW detection experiments since the recent phenomenological upper-bounds set on spatial noncommutative parameter implies a length-scale comparable to the length-variations due to the passage of gravitational waves, detectable in the present day GW detectors.

THE EFFECTS OF THE BEYOND MEAN FIELD CORRECTIONS OF FERMI SUPERFLUID GAS IN A DOUBLE-WELL POTENTIAL

By considering the contribution of the higher-order term representing the lowest approximation of beyond mean field corrections, we investigate a superfluid Fermi gas confined in a double-well potential in Bose–Einstein Condensation (BEC) side of the Bardeen–Cooper–Schrieffer (BCS) to BEC crossover. Two limited cases of deep BEC regime and BEC regime of BCS–BEC crossover, corresponding to the two-body scattering length a sc is small enough and large enough, respectively. We derive a simple two-mode model that could depict the dynamics effectively. With making thorough analysis on the two-mode model and its corresponding classical Hamiltonian, we find that the Josephson oscillation or self-trapping phenomenon could emerge at certain parameters. We find three kinds of the phase states: Josephson oscillation (JO), oscillating-phase-type self-trapping (OPTST) and running-phase-type self-trapping (RPTST). The dependence of these three phase states on the dimensionless interaction parameter y = 1/(kFa sc ) and the initial system energy are given in this paper.

The invariant representation of generalized momentum

The relativistic-invariance representation of generalized momentum of a particle in the external field is proposed. In this representation the dependence of potentials of the particle-field interaction on the particle velocity is taken into account. An exact correspondence of expressions for the energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions of problems of mechanics with relativistic and classical approaches. The invariance of the generalized momentum representation allows one to describe equivalently a physical system in geometrically conjugated spaces of kinematical and dynamical variables.

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation

A new classical model for the general second-quantized many-electron Hamiltonian in Cartesian coordinates and momenta is presented; this makes semiclassical (SC) calculations using an initial value representation (IVR) more useful than the classical Hamiltonian in action-angle variables given earlier by Miller and White [J. Chem. Phys. 84, 5059-5066 (1986)]. If only 1-electron terms are included in this Hamiltonian, the classical equations of motion for the Cartesian variables are linear, and the SC-IVR gives exact results for the propagator (and thus for transition probabilities, the energy spectrum, etc.), as confirmed by analytic proof and numerical calculations. Though this new Hamiltonian is not exact when 2-electron interactions are included, we observe good results for the SC-IVR transition probabilities for times that are not too long. Test calculations, for example, show that the SC-IVR is accurate for times long enough to obtain good result for the eigenvalue spectrum (i.e., the energy levels of the electronic system).

KZ Characteristic Variety as the Zero Set of Classical Calogero-Moser Hamiltonians

We discuss a relation between the characteristic variety of the KZ equations and the zero set of the classical Calogero-Moser Hamiltonians.