Niels R. Walet

Pairing in many-fermion systems: an exact renormalisation group treatment

We study the application of the exact renormalisation group to a many-fermion system with a short-range attractive force. We introduce a boson field to describe pairing effects, and take a simple ansatz for the effective action. We derive a set of approximate flow equations for the effective coupling including boson and fermionic fluctuations. The initial conditions are obtained by renormalising the interaction to fit the scattering length in vacuum. At some critical value of the running scale, the numerical solutions show a phase transition to a gapped phase. Standard results are recovered if we omit the boson loops. When boson fluctuations are included, we find that their contributions are significant only in the small-gap regime.

Reaction paths and generalized valley approximation

The generalized valley approximation has been developed as a method of approximately decoupling one or a few low‐frequency nonlinear modes from the remaining higher frequency modes of a multiparticle system. This decoupling will be best when the difference in frequencies is large; this is the case of adiabatic motion. We describe the application of this method to chemical reactions, relying in some measure on our earlier work, and contrast it with reaction‐path theories. We give an algorithm for the incorporation of our method in a chemical calculation of the Born–Oppenheimer type. Detailed calculations are reported for several standard models that couple a double well to a harmonic oscillator. The decoupling procedure leads to an effective or renormalized one‐dimensional double‐well problem. The energy splitting of the lowest doublet in this well is contrasted with the exact splitting obtained by numerical integration of the two‐dimensional Schrodinger equation. Results are good when the adiabatic condit...

Quantising the B = 2 and B = 3 skyrmion systems

Abstract We examine the quantisation of a collective Hamiltonian for the two-baryon system derived by us in a previous paper. We show that by increasing the sophistication of the approximations we can obtain a bound state — or a resonance — not too far removed from the threshold with the quantum numbers of the deuteron. The energy of this state is shown to depend very sensitively on the parameters of the model. Subsequently we construct part of a collective Hamiltonian for the three-baryon system. Large-amplitude quantum fluctuations play an important role in the intrinsic wave function of the ground state, changing its symmetry from tetrahedral to cubic. Apart from the tetrahedron describing the minimum of the potential, we identify a “doughnut” and a “pretzel” as the most important saddle points in the potential energy surface. We show that it is likely that inclusion of fluctuations through these saddle points leads to an energy close to the tritons value.

Translationally invariant treatment of pair correlations in nuclei: I. Spin and isospin dependent correlations

Abstract We study the extension of our translationally invariant treatment of few-body nuclear systems to heavier nuclei. At the same time we also introduce state-dependent correlation operators. Our techniques are tailored to those nuclei that can be dealt with in LS coupling, which includes all nuclei up to the shell closure at A = 40. We study mainly p-shell nuclei in this paper. A detailed comparison with other microscopic many-body approaches is made, using a variety of schematic nuclear interactions. It is shown that our methodology produces very good energies, and presumably also wave functions, for medium mass nuclei.

Colour superconductivity in finite systems

We have studied the effects of finite size on the two flavor colour superconducting state. Since the baryon number in the BCS state is only fixed on average, we have projected the state onto a fixed baryon number. The resulting state has been then projected onto a colour-singlet state, by integrating onto the colour group manifold. The effects of both projections have been evaluated numerically.

Splitting the gluon

In the strongly correlated environment of high-temperature cuprate superconductors, the spin and charge degrees of freedom of an electron seem to separate from each other. A similar phenomenon may be present in the strong coupling phase of Yang-Mills theories, where a separation between the color charge and the spin of a gluon could play a role in a mass-gap formation. Here we study the phase structure of a decomposed SU(2) Yang-Mills theory in a mean-field approximation, by inspecting quantum fluctuations in the condensate which is formed by the color-charge component of the gluon field. Our results suggest that the decomposed theory has an involved phase structure. In particular, there appears to be a phase which is quite reminiscent of the superconducting phase in cuprates. We also find evidence that this phase is separated from the asymptotically-free theory by an intermediate pseudogap phase.

Gender differences in conceptual understanding of Newtonian mechanics: a UK cross-institution comparison

We present the results of a combined study from three UK universities where we investigate the existence and persistence of a performance gender gap in conceptual understanding of Newtonian mechanics. Using the Force Concept Inventory, we find that students at all three universities exhibit a statistically significant gender gap, with males outperforming females. This gap is narrowed but not eliminated after instruction, using a variety of instructional approaches. Furthermore, we find that before instruction the quartile with the lowest performance on the diagnostic instrument comprises a disproportionately high fraction (~50%) of the total female cohort. The majority of these students remain in the lowest-performing quartile post-instruction. Analysis of responses to individual items shows that male students outperform female students on practically all items on the instrument. Comparing the performance of the same group of students on end-of-course examinations, we find no statistically significant gender gaps.

The kinetic energy and the geometric structure in the B = 2 sector of the Skyrme model: A study using the Atiyah-Manton ansatz

Abstract We study the construction of the collective-coordinate manifold in the baryon number two sector of the Skyrme model. To that end we use techniques of adiabatic large amplitude collective motion, which treat potential and kinetic energy on an equal footing. In this paper the starting point is the ansatz proposed by Atiyah and Manton (Phys. Lett. B 438 (1989) 222), which allows a study of the dynamics using a finite and small number of variables. From these variables we choose a subset of collective ones. We then study the behavior of inertial parameters along parts of the collective manifold, and study the dynamical parts of the interaction.

Classical theory of collective motion in the large amplitude, small velocity regime

Abstract A classical theory of collective motion is developed for the large amplitude, small velocity limit, i.e., for a hamiltonian that is at most quadratic in the momenta, allowance being made for a mass tensor that is a general function of the coordinates. It is based on the identification of decoupled motions that are confined to submanifolds of the full configuration space. Conditions for decoupling are derived and then transformed into several different sets of equivalent conditions, more useful for practical applications. Algorithms are given for constructing manifolds that are exactly decoupled if a given dynamical system admits such motions and that can be utilized as well when there is approximate decoupling, as evidenced by criteria that are established. Some examples are worked out. The connection to previous research on this problem is described.

GROUND-STATE CORRELATIONS AND RESTORATION OF BROKEN SYMMETRY TO NUCLEAR MEAN FIELD-THEORY

Abstract We reconsider the long-standing problem that the ground-state correlations predicted by the standard quasiboson (random phase) approximation are too large by a factor of two in the limit of weak residual interaction, and are thus inconsistent with the Pauli exclusion principle. We solve this problem by noting that for the derivation of the equations of the random phase approximation (RPA), which determine a set of matrix elements and excitation energies, it is unnecessary to assume that the ground state is the vacuum of a set of boson excitations, whereas in the faculty calculation of the ground-state correlation energy, this assumption is made. For the study of excitation energies and the restoration of symmetries broken by an initial mean-field solution, we apply a symmetry-preserving form of the equations of motion for fermion particle-hole excitation operators rather than for bosons. We illustrate this for the case of translations. We then describe a separate calculation of the ground-state energy that is consistent with the Pauli principle, but can nevertheless be evaluated in terms of the solutions of the RPA equations of motion. In contrast to some earlier work, we treat direct and exchange terms on an equal footing throughout.