The interaction effects on the moment of inertia of a large many-body fermion system moving under periodic boundary conditions have been explicitly derived in the second order of particle-particle coupling for the case of nonsingular repulsive interparticle forces. The derivation is facilitated by the use of a new Hamiltonian constructed by canonical transformation. This transformation is determined by requiring that it remove the compensating effects of pair excitation and that the new interaction operators of first and second order in the cranking field be also of first order in particle-particle coupling. A graphical analysis is presented in which those classes of diagrams which yield corrections vanishing relative to the rigid moment in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$ are distinguished. Some remarks on the related problem of diamagnetism are made. The possible application of the method of canonical transformation to related many-body problems is indicated.
It is shown that in the random-phase approximation the effect of particle-particle interaction on the moment of inertia of a many-fermion system moving under periodic boundary conditions vanishes when calculated on the "cranking" model of Inglis. This result is obtained by performing a unitary transformation on the equivalent Hamiltonian which reproduces the sequence of diagrams given by the random-phase approximation. The close resemblance of the general problem to the soluble model problem of a rotating dense electron gas is exploited in the calculation. This work extends that of Amado and Brueckner who have demonstrated only that interaction effects cancel in lowest order. The nature of their approximations and the connection to the problem of collective excitations are discussed.
The effects of possible superfluidity on the cranking moment of a large many-fermion system moving under periodic boundary conditions are investigated within the framework of the theory of superconductivity recently formulated by Bogolyubov. The Hamiltonian is initially subjected to Bogolyubov's general unitary transformation. The collective excitations of the fermions are then considered in the usual pair approximation; the appropriate cranking terms are linear in the boson pair operators. On performing a unitary transformation which transforms away these linear terms, one obtains an expression for the moment of inertia of the system which includes both the effects of possible superfluidity and collective excitation. This expression, by virtue of its being stationary with respect to arbitrary variations in the amplitude associated with the latter unitary transformation, is then utilized as a variational principle for the moment of inertia. For the normal state, the result previously obtained by the author, that the moment of inertia has the rigid value, is rederived in more compact form. For the superfluid state, one finds that collective excitations effect a marked increase in the superfluid moment at intermediate coupling strengths although the resulting moment is still quite small compared to the rigid value. In contrast to the normal state case, where particle-hole pairs play a major role, this increase is almost entirely due to excitations consisting of particle-pairs or hole-pairs. The precise magnitude of the apparent resonance in the moment produced by the
d-wave part of the cranking interaction is dependent to some extent on the features of the particle-particle potential which leads to the superfluid state. Variational expressions for the moment are exhibited for both Yukawa and delta-function shell potentials. These results are identical in charged and neutral Fermi systems. A calculation of the cranking moment at finite temperatures is presented in an Appendix along with an interpretation of it in terms of Bardeen's two-fluid model of superconductivity.
The response of a many-fermion system to dilatation is studied through the introduction, in close analogy with the "cranking" and "pushing" models, of a "squashing" model characterized by the Hamiltonian, $H={H}_{0}\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\epsilon}}{D}_{x}$, where
Dx is the generator of dilatations in the
x direction. The evaluation of the "rigid" dilatational moment of such a noninteracting system is carried out for both standing wave and periodic boundary conditions in a cubic box of side
L. As in the pushing model, the dilatation moment can be evaluated formally using the relation between
Dx and the commutator [
x2,
H0]. For independent fermions moving in a harmonic oscillator one-body potential, this relation is shown to lead to further simplification after the introduction of perturbed one-body wave functions. In the case of periodic boundary conditions, it is shown that the rigid dilatational moment is not altered by an interparticle interaction to first order in the interaction strength; by comparison with the analogous cranking moment problem, this result extends to all orders in the interaction strength. The response to a symmetric dilatation,
D=Dx+Dy+Dz, is studied in a soluble Hartree model of "nuclear" correlations wherein a separable monopole-monopole interaction acts between "nucleons." In this model the dilatation moment in the presence of pair correlations is found to be the rigid moment, with the kinetic energy of dilatation $\ensuremath{\propto}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\epsilon}}}^{2}$, there being no higher order corrections.NUCLEAR STRUCTURE Approximate many-body methods applied to solvable models.
The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.