Figure 4 - uploaded by Roger Balian
Content may be subject to copyright.
Specific heat of an odd-number system for w F ∆ = 10, w F Λ = 200 andˆNandˆ andˆN = 201. Symbols and abbreviations are the same as in Fig. 3. The solid curve reaches the classical value for one degree of freedom near T = 0.1∆ while the BCS value tends to zero, as exhibited by the inset. The projected and unprojected curves with no interaction are identical and have the linear behaviour of the Fermi gas.
Source publication
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used a...
Citations
We investigate fluctuations and even-odd effects in small superfluid systems at finite temperature by means of the static path approximation (SPA) plus random-phase approximation (RPA) treatment, where exact number parity projection is introduced. The RPA correction to the SPA is evaluated exactly. Results are given first for a schematic 20-level pairing model, where an excellent agreement with the exact canonical results is obtained both for even and odd systems, and then for a heavy nucleus, where the smoothing of the BCS transition and the even-odd effects in the pairing energy and specific heat are examined.
Fluctuations and odd-even effects in small superfluid systems at finite temperature are investigated by means of the static path plus RPA approximation. A general derivation of this method is presented, which allows a straightforward implementation in statistical ensembles with fixed number parity (NP). A significant smoothing of the superconducting to normal transition is obtained in systems where the gap is comparable to the level spacing, as well as a decrease of pairing correlations in the odd case. Results are shown for a schematic model, where an excellent agreement with exact canonical results is obtained with the present method, and then for a heavy nucleus. Comparison with NP projected BCS and BCS+RPA results is made. An effective BCS+RPA approach which remains smooth in transitional regions is also derived.
The thermodynamic functions of a Fermi gas with spin population imbalance are studied in the temperature-asymmetry plane in the BCS limit. The low-temperature domain is characterized by an anomalous enhancement of the entropy and the specific heat above their values in the unpaired state, decrease of the gap and eventual unpairing phase transition as the temperature is lowered. The unpairing phase transition induces a second jump in the specific heat, which can be measured in calorimetric experiments. While the superfluid is unstable against a supercurrent carrying state, it may sustain a metastable state if cooled adiabatically down from the stable high-temperature domain. In the latter domain the temperature dependence of the gap and related functions is analogous to the predictions of the BCS theory.
We develop a theory of superconductivity in ultrasmall (nm-scale) metallic grains having a discrete electronic eigenspectrum with a mean level spacing of order of the bulk gap. The theory is based on calculating the eigenspectrum using a generalized BCS variational approach, whose applicability has been extensively demonstrated in studies of pairing correlations in nuclear physics. We discuss how conventional mean field theory breaks down with decreasing sample size, how the so-called blocking effect weakens pairing correlations in states with non-zero total spin, and how this affects the discrete eigenspectrum's behavior in a magnetic field, which favors non-zero total spin. In ultrasmall grains, spin magnetism dominates orbital magnetism, just as in thin films in a parallel field; but whereas in the latter the magnetic-field induced transition to a normal state is known to be first-order, we show that in ultrasmall grains it is softened by finite size effects. Our calculations qualitatively reproduce the magnetic-field dependent tunneling spectra for individual aluminum grains measured recently by Ralph, Black and Tinkham. We argue that previously-discussed parity effects for the odd-even ground state energy difference are presently not observable for experimental reasons, and propose an analogous parity effect for the pair-breaking energy that should be observable provided that the grain size can be controlled sufficiently well. Finally, experimental evidence is pointed out that the dominant role played by time-reversed pairs of states, well-established in bulk and in dirty superconductors, persists also in ultrasmall grains. Comment: 21 pages RevTeX, 12 EPS figures included, uses epsf.sty
