Odd Number Of Particles Research Articles

Monte Carlo calculations of the lowestenergy level of a given angular momentum in the pairing model for evenand odd Sn isotopes

We discuss how to evaluate with numerical Monte Carlo techniquesthe lowest energy levels of a given angularmomentum for both even and odd numbers of particles in the pairing modelĤ = Ĥ0-GP̂†P̂ for some Sn isotopes. Itis shown that a properselection of the chemical potential strongly suppresses the signfluctuations.

Near-integrability of periodic FPU-chains

The FPU-chain with periodic boundary conditions is studied by Birkhoff–Gustavson normalisation. In the cases of up to 6 particles and for β-chains with an odd number of particles the normal forms are integrable, which permits us to apply KAM-theory. This leads to the presence of many invariant tori on which the motion is quasi-periodic. Thus we explain the recurrence phenomena and the small size of chaos observed in experiments. Furthermore, we find a certain clustering of modes.

Exact form factors in integrable quantum field theories: the sine-Gordon model

We provide detailed arguments on how to derive properties of generalized form factors, originally proposed by one of the authors (M.K.) and Weisz twenty years ago, solely based on the assumption of ‘maximal analyticity” and the validity of the LSZ reduction formalism. These properties constitute consistency equations which allow the explicit evaluation of the n-particle form factors once the scattering matrix is known. The equations give rise to a matrix Riemann-Hilbert problem. Exploiting the “off-shell” Bethe ansatz we propose a general formula for form factors for an odd number of particles. For the sine-Gordon model alias the massive Thirring model we exemplify the general solution for several operators. In particular we calculate the three-particle form factor of the soliton field, carry out a consistency check against the Thirring model perturbation theory and thus confirm the general formalism.

Kinetic-energy density functionals based on the homogeneous response function applied to one-dimensional fermion systems

Explicit kinetic-energy density functionals for noninteracting electron systems that are based on reproducing the linear response function in the homogeneous limit are the most accurate approximations to the exact Kohn-Sham functional. In this work we study several such models for one-dimensional systems. In this case, the geometrical simplicity allows the calculation of the static linear response resulting from these functionals for inhomogeneous systems. Likewise, the discussion of the energies and density profiles obtained is easier and clearer than that for more realistic problems. Special attention is paid to the application of explicit kinetic energy functionals to systems with odd and noninteger number of particles.

Tunneling in the topological mechanism of superconductivity

We compute the order parameter and Josephson tunneling amplitude in a two-dimensional model of topological superconductivity which captures the physics of the doped Mott insulator. The hydrodynamics of topological electronic liquid consists of the compressible charge sector and the incompressible chiral topological spin liquid. We show that ground states differing by an odd number of particles are orthogonal and insertion of two extra electrons is followed by the emission of soft modes of the transversal spin current. The orthogonality catastrophe makes the physics of superconductivity drastically different from the BCS theory but similar to the physics of one-dimensional electronic liquids. The wave function of a pair is dressed by soft modes. As a result the two-particle matrix element forms a complex d-wave representation (i.e., changes sign under $90\ifmmode^\circ\else\textdegree\fi{}$ rotation), although the gap in the electronic spectrum has no nodes. In contrast to the BCS theory the tunneling amplitude has an asymmetric broad peak (much bigger than the gap) around the Fermi surface. We develop an operator algebra, that allows one to compute other correlation functions.

Tunneling and Orthogonality Catastrophe in the Topological Mechanism of Superconductivity

We compute the angular dependence of the order parameter and tunneling amplitude in a model exhibiting topological superconductivity and sketch its derivation as a model of a doped Mott insulator. We show that ground states differing by an odd number of particles are orthogonal and the order parameter is in the d-representation, although the gap in the electronic spectrum has no nodes. We also develop an operator algebra, that allowes one to compute off-diagonal correlation functions.

Parity Effect in Ground State Energies of Ultrasmall Superconducting Grains

We study the superconductivity in small grains in the regime when the quantum level spacing $\delta\varepsilon$ is comparable to the gap $\Delta$. As $\delta\varepsilon$ is increased, the system crosses over from superconducting to normal state. This crossover is studied by calculating the dependence of the ground state energy of a grain on the parity of the number of electrons. The states with odd numbers of particles carry an additional energy $\Delta_P$, which shows non-monotonic dependence on $\delta\varepsilon$. Our predictions can be tested experimentally by studying the parity-induced alternation of Coulomb blockade peak spacings in grains of different sizes.

Resonant coherent quantum tunneling of the magnetization of spin- systems: Spin-parity effects

We perform quantum dynamical calculations to study the reversal of the magnetization for systems of a few spin- particles with a general biaxial anisotropy in the presence of an external magnetic field at T=0 and with no dissipation. Collective quantum tunneling of the magnetization is demonstrated to occur only for some specific resonant values of the magnetic field proving the invalidity of semiclassical approximations for clusters of a few spin- particles. Quantum tunneling of the magnetization direction at zero field is shown to be spin-parity dependent: whereas clusters with an odd number of particles do not exhibit quantum tunneling of the magnetization due to the degeneration between the first two levels, clusters with an even number of spins present resonant coherent quantum tunneling of the magnetization. On the contrary, resonant coherent quantum tunneling of the magnetization at finite magnetic field has no detectable parity dependence. We study the dependence of such collective tunneling of the magnetization on the magnetic field, the number of spins, and the anisotropy.

Intrinsic localized modes in the bulk and at the surface of anharmonic diatomic chains.

Intrinsic localized vibrational modes in a diatomic anharmonic chain are discussed in detail. We consider a finite (even or odd) number of particles interacting with nearest-neighbor harmonic and cubic and quartic anharmonic potentials. For suitable values of the potential parameters we find two odd parity modes localized about light and heavy atoms, respectively, with frequencies above the top of the optical branch. In the gap there are localized modes arising from the top of the acoustic branch with odd parity if they are centered on a heavy atom and even parity if centered on a light atom. There are also two localized modes originating from the bottom of the optical branch. The odd parity mode is centered on a light atom, while the even parity mode is centered on a heavy atom. In addition, we have found two types of surface modes. One is the anharmonic version of the surface mode occurring in a finite diatomic harmonic chain with frequency in the gap, and the other is entirely due to the anharmonicity. The latter mode has its frequency above the top of the optical branch. A comparison with other work is given.

Exact S matrix of the deformed c=1 matrix model.

We consider the $c=1$ matrix model deformed by the operator ${1\over 2} M\Tr\Phi^{-2}$, which was conjectured by Jevicki and Yoneya to describe a two-dimensional black hole of mass $M$. We calculate the exact non-perturbative $S$-matrix and show that all the amplitudes involving an odd number of particles vanish at least to all orders of perturbation theory. We conjecture that these amplitudes vanish non-perturbatively and prove this for the $2n \to 1$ scattering. For the 2-- and 4--particle amplitudes we give some leading terms of the perturbative expansion.

Even-odd anomalous tunneling effect.

The analysis of the interacting fermion models with the SU(2) symmetry indicates the different behavior of the splittings of opposite-parity lowest-energy eigenstates for even and odd numbers of fermions. The imaginary time-dependent mean-field approach reproduces the average results only. For the odd particle numbers, an introduction of the universal logarithmic term to the action integral is needed. The even-odd effect appears in a wide class of models with the SU(2) symmetry and is connected with the crossing of the energy surfaces of opposite-parity states for an odd number of fermions.

Spontaneous fission properiies and lieftime systematics

Half-lives for spontaneous fission of nuclides with even and odd numbers of particles are compared with recent theoretical calculations. A summary of odd particle hindrance factors is given. The most recent measurements of kinetic-energy and mass distributions and neutron emission for spontaneous fission of the heaviest nuclides are summarized and discussed.

Selection rules for production and η-π decays of odd-CP exotics

Statements and conditions of validity are presented for three selection rules relevant to the analysis of data indicating the existence of a possible CP-exotic state and to the determination of whether it is a four-quark or hybrid state. SU(3) flavor symmetyry forbids the decay of an odd-CP SU(3)-octet state into two octet pseudoscalar mesons. Isospin symmetry forbids the ηπ decay of an odd-parity hybrid state via an OZI-allowed diagram. No selection rule forbids the ηπ decay of a four-quark state. All n-point functions in which there are an odd number of particles with unnatural parity, all momental lying in the same plane and all particles with spin having zero helicity, are forbidden by angular momentum and parity conservation.

Configuration mixing and electromagnetic properties of odd-even nuclei.

The electromagnetic properties of ground and excited states of nuclei with an odd number of particles and holes outside the closed shells are described in the j-j coupling shell model. The residual interaction between the valence particles and holes causes two different configuration mixing: (a) one where the valence particles and holes are scattered to higher shell model states, and (b) ones where the valence particles and holes excite particles out of the core. The electromagnetic properties of odd-even nuclei are sensitive to the correct treatment of both types of configuration mixing. In this paper the matrix elements of the nuclear Hamiltonian evaluated using the basis of configuration mixing states of types (a) and (b), and the resulting eigenvalue equations, have been exactly calculated. The matrix elements of the electromagnetic multipole operators calculated with the resulting eigenvectors of type (b) are then related to the deviation observed between the experimental and valence expectation values.

Analytic evaluation of certain zeroth order coulombic hyperangular interaction integrals

As has been presented in a previous paper, the use of hyperspherical coordinates in research on a system of electrically charged particles carries mathematical complications into the evaluation of certain kinds of hyperangular interaction integrals. These integrals contain the hyperangular interaction potential and various powers of the inverse of the total angular momentum operator on the space of hyperharmonics, from which the zeroth order hyperharmonic is excluded. In this work the simplest one of these integrals has been taken into consideration. After some intermediate steps, it has been shown that it can be expressed in terms of elementary functions of the cosine of the angle (γ) between the hyperaxes of the potential term for an odd number of particles. In the case of an even number of particles, these integrals can be given in terms of a generalized hypergeometric function of the same argument (γ) and its derivatives.

Charge Transfer in P̄P Reactions at 22.4 GeV/c

Charge transfer is investigated between c.m. hemispheres and across the largest rapidity gap in an event. The latter method is found to give a larger contribution to zero charge transfer than the former. The rapidity gap size is noted to be bigger for a separation into groups with an odd number of charged particles than into groups with an even number of charged particles. Ordering of produced particles in rapidity in high multiplicity events reveals charge asymmetry and a topology independent charge distribution inside an event.

A new boson expansion for odd-particle systems

The Beliaev-Zelevinsky boson expansion is extended to include systems with odd particle number. Closed forms, valid to all orders, are given for the particle-boson as well as boson-boson interaction.

Validity of the hartree-bogoliubov-theory in an exactly solvable model

The structure and the validity of the Hartree-Bogoliubov (HB) theory are investigated within a special, exactly solvable model of the n-body problem, which was introduced previously and which contains only pairing and monopole forces. The HB theory is applied to this model in the variational form since there are convergence difficulties for the diagonalization procedure of solving the HB equation. For including the blocking effect the problems of even and odd numbers of particles are treated separately. For our model the HB equations yield solutions of essentially different type simultaneously (Hartree-Fock, BCS, mixed HB solutions). A comparison to the exact results shows that the properties of the energetically lowest solutions are satisfactory for ranges of model parameters where one of the two interaction types is overwhelming, whereas in the transition region there are large discrepancies (especially for the structure of the spectrum for the systems of an odd number of particles). The importance of the blocking effect is small and it can be taken into good account by introducing suitable constraints on the mean number of particles.

Exactly Solvable Many-Boson Model

A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.

Validity of the BCS and RPA approximations in the pairing-plus-monopole solvable model

A solvable many-body model, which simulates qualitatively the main characteristics of the pairing-plus-quadrupole model, is presented and investigated for systems of even and odd numbers of particles. The exact solution, when the force parameters are fixed and the number of particles is varied throughout the “major shell” of the model, is compared to the approximate BCS (or HB) solution for the ground state, and to the quasi-particle RPA result for the one-phonon excited states. We conclude that the BCS (or HB) solution for the ground state energy and wave function is good and practically exact if the number-of-particles projection is performed. The RPA for even systems seems to work for the so-called “pairing vibration” (0 +), whereas, for the model “quadrupole vibration”, errors of about 20% of the exact excitation are encountered. For odd systems, we examine a particle-conserving approximation for the corresponding quadrupole vibration.