Collective Coordinate Method Research Articles

Canonical quantization of SU(3) Skyrme model in a general representation

A complete canonical quantization of the SU(3) Skyrme model performed in the collective coordinate formalism in general irreducible representations. In the case of SU(3) the model differs qualitatively in different representations. The Wess-Zumino-Witten term vanishes in all self-adjoint representations in the collective coordinate method for separation of space and time variables. The canonical quantization generates representation dependent quantum mass corrections, which can stabilize the soliton solution. The standard symmetry breaking mass term, which in general leads to representation mixing, degenerates to the SU(2) form in all self-adjoint representations.

Collective path connecting the oblate and prolate local minima in proton-rich N = Z nuclei around 68Se

By means of the adiabatic self-consistent collective coordinate method and the pairing-plus-quadrupole interaction, we have for the first time obtained a self-consistent collective path connecting the oblate and prolate local minima in 68Se and 72Kr. This self-consistent collective path is found to run approximately along the valley connecting the oblate and prolate local minima in the collective potential energy landscape. The result of this calculation clearly indicates the importance of triaxial deformation dynamics in oblate-prolate shape coexistence phenomena.

Collective Paths Connecting the Oblate and Prolate Shapes in 68Se and 72Kr Suggested by the Adiabatic Self-Consistent Collective Coordinate Method

By means of the adiabatic self-consistent collective coordinate method and the pairing-plus-quadrupole interaction, we have obtained the self-consistent collective path connecting the oblate and prolate local minima in 68Se and 72Kr for the first time. The self-consistent collective path is found to run approximately along the valley connecting the oblate and prolate local minima in the collective potential energy landscape. This result of calculation clearly indicates the importance of triaxial deformation dynamics in oblate-prolate shape coexistence phenomena.

Collective Path Connecting the Oblate and Prolate Local Minima in 68Se

By means of the adiabatic self-consistent collective coordinate method and the pairing plus quadrupole interaction, we have obtained the selfconsistent collective path connecting the oblate and prolate local minima in 68Se for the first time. Along the collective path, the triaxial deformation parameter (gamma) changes between 0 and 60 (degree) keeping the axially symmetric deformation parameter (beta) approximately constant, indicating the importance of triaxial deformation dynamics in the oblate-prolate shape coexistence phenomena.

Dynamics of topological defects in a spiral: a scenario for the spin-glass phase of cuprates.

We propose that the dissipative dynamics of topological defects in a spiral state is responsible for the transport properties in the spin-glass phase of cuprates. Using the collective-coordinate method, we show that topological defects are coupled to a bath of magnetic excitations. By integrating out the bath degrees of freedom, we find that the dynamical properties of the topological defects are dissipative. The calculated damping matrix is related to the in-plane resistivity, which exhibits an anisotropy and linear temperature dependence in agreement with experimental data.

Strong NLS soliton–defect interactions

We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. [Phys. Rev. E 49 (1994) 3400], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.

Self-similar solutions and collective coordinate methods for nonlinear Schrödinger equations

In this paper we study the phase of self-similar solutions to general nonlinear Schrödinger equations. From this analysis we gain insight on the dynamics of non-trivial solutions and a deeper understanding of the way collective coordinate methods work. We also find general evolution equations for the most relevant dynamical parameter w( t) corresponding to the width of the solution. These equations are exact for self-similar solutions and provide a shortcut to find approximate evolution equations for the width of non-self-similar solutions similar to those of collective coordinate methods.

Application of the Adiabatic Self-Consistent Collective Coordinate Method to a Solvable Model of Prolate-Oblate Shape Coexistence

The adiabatic self-consistent collective coordinate method is applied to an exactly solvable multi-O(4) model that is designed to describe nuclear shape coexistence phenomena. The collective mass and dynamics of large amplitude collective motion in this model system are analyzed, and it is shown that the method yields a faithful description of tunneling motion through a barrier between the prolate and oblate local minima in the collective potential. The emergence of the doublet pattern is clearly described.

Lectures on the functional renormalization group method

AbstractThese introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

Representation of Quantum Brownian Motion in the Collective Coordinate Method

We consider two explicitly solvable models of quantum random processes described by the Langevin equation, namely, those for a “free” quantum Brownian particle and for a quantum Brownian harmonic oscillator. The Hamiltonian (string) realization of the models reveals a soliton-like structure of “classical” solutions. Accordingly, the zero-mode collective coordinate method turns out to be an adequate means for describing the quantum dynamics of the models.

Correlations in the sine-Gordon model with finite soliton density

We study the sine-Gordon (SG) model at finite densities of the topological charge and small SG interaction constant, related to the one-dimensional Hubbard model near half-filling. Using the modified WKB approach, we find that the spectrum of the Gaussian fluctuations around the classical solution reproduces the results of the Bethe ansatz studies. The modification of the collective coordinate method allows us to write down the action, free from infra-red divergencies. The behaviour of the density-type correlation functions is non-trivial and we demonstrate the existence of leading and sub-leading asymptotes. A consistent definition of the charge-raising operator is discussed. The superconducting-type correlations are shown to decrease slowly at small soliton densities, while the spectral weight of right (left) moving fermions is spread over neighboring "4k_F" harmonics.

Semiclassical quantization of Hopf solitons

The gauge equivalent formulation of the Faddeev-Skyrme model is used for the study of the quantum theory. The rotational quantum excitations around the soliton solution of Hopf number unity are investigated by the method of collective coordinates. The quantum Hamiltonian of the system is found to coincide with the Hamiltonian of a symmetrical top rotating in SU(2). Thus, the irreducible representations of physical observables can be constructed.

Covariant collective coordinates method in path integral formalism: application to quantum gravity on classical background

In this paper the problem of quantization of the free gravitational field on classical background (the exact solutions of Einstein equations) is considered. The Bogoliubov group coordinates method in path integral formalism is developed. This approach makes it possible to take carefully into account the conservation laws alongside with the perturbation theory expansion.

Macroscopic quantum tunneling of two-component Bose-Einstein condensates

We show theoretically the existence of a metastable state and the possibility of decay to the ground state through macroscopic quantum tunneling in two-component Bose-Einstein condensates with repulsive interactions. Numerical analysis of the coupled Gross-Pitaevskii equations clarifies the metastable states whose configuration preserves or breaks the symmetry of the trapping potential, depending on the interspecies interaction and the particle number. We calculate the tunneling decay rate of the metastable state by using the collective coordinate method under the WKB approximation. Then the height of the energy barrier is estimated by the saddle point solution. It is found that macroscopic quantum tunneling is observable in a wide range of particle numbers. Macroscopic quantum coherence between two distinct states is discussed; this might give an additional coherent property of two-component Bose condensed systems. Thermal effects on the decay rate are estimated.

Symmetries of fermions in the lowest Landau level

In the first paper we presented a general theory of conformal fields on a complex plane, and in the second paper we wrote down, for the first time, a supersymmetric transformation of the conformal fields. I think I was expected to present a historical talk at the conference. But, since I had already written an essay on the reminiscences of my entire research activities in [3] and furthermore since a part of the article has been printed again in [4], I chose not to talk about my recollections on this matter. Instead, under the title Symmetries of Fermions in the Lowest Landau Level, I discussed the subjects, which may or may not be suited to the theme of this conference, namely (a) the dynamic symmetries of edge excitations and (b) the method of collective coordinates for planar fermions and the bosonization. Since I believe the viewpoint I took on the latter subject is unconventional, in this written report I expanded the discussion and have written it in detail.

Resonances in the dynamics of straight phi(4) kinks perturbed by ac forces

We study the dynamics of straight phi(4) kinks perturbed by an ac force, both with and without damping. We address this issue by using a collective coordinate theory, which allows us to reduce the problem to the dynamics of the kink center and width. We carry out a careful analysis of the corresponding ordinary differential equations, of Mathieu type in the undamped case, finding and characterizing the resonant frequencies and the regions of existence of resonant solutions. We verify the accuracy of our predictions by numerical simulation of the full partial differential equation, showing that the collective coordinate prediction is very accurate. Numerical simulations for the damped case establish that the strongest resonance is the one at half the frequency of the internal mode of the kink. In conclusion, we discuss the possible relevance of our results for other systems, especially the sine-Gordon equation. We also obtain additional results regarding the equivalence between different collective coordinate methods applied to this problem.

Diabatic Mean-Field Description of Rotational Bands in Terms of the Selfconsistent Collective Coordinate Method

Diabatic description of rotational bands provides a clear-cut picture for understanding the back-bending phenomena, where the internal structure of the yrast band changes dramatically as a function of angular momentum. A microscopic framework to obtain the diabatic bands within the mean-field approximation is presented by making use of the selfconsistent collective coordinate method. Applying the framework, both the ground state rotational bands and the Stockholm bands are studied systematically for the rare-earth deformed nuclei. An overall agreement has been achieved between the calculated and observed rotational spectra. It is also shown that the inclusion of the double-stretched quadrupole-pairing interaction is crucial to obtain an overall agreement for the even-odd mass differences and the rotational spectra simultaneously.

Mobility of Bloch walls via the collective coordinate method

We have studied the problem of the dissipative motion of Bloch walls considering a totally anisotropic one dimensional spin chain in the presence of a magnetic field. Using the so-called "collective coordinate method" we construct an effective Hamiltonian for the Bloch wall coupled to the magnetic excitations of the system. It allows us to analyze the Brownian motion of the wall in terms of the reflection coefficient of the effective potential felt by the excitations due to the existence of the wall. We find that for finite values of the external field the wall mobility is also finite. The spectrum of the potential at large fields is investigated and the dependence of the damping constant on temperature is evaluated. As a result we find the temperature and magnetic field dependence of the wall mobility.

Adiabatic Selfconsistent Collective Coordinate Method for Large Amplitude Collective Motion in Nuclei with Pairing Correlations

An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in superconducting nuclei on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a similar way as the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.

Skyrmion dynamics in quantum Hall ferromagnets

Exploring a classical solution of the nonlinear $\ensuremath{\sigma}$ model for a quantum Hall ferromagnet, a Skyrmion-magnon effective Hamiltonian is obtained via the collective coordinates method. Using the Feynman-Vernon functional-integral formalism for this model we find the temperature-dependent transport coefficients which characterize a single Skyrmion dynamics.