The pair correlations in mesoscopic systems such as nm-size superconducting clusters and nuclei are studied at finite temperature for the canonical ensemble of fermions in model spaces with a fixed particle number: i) a degenerate spherical shell (strong coupling limit), ii) an equidistantly spaced deformed shell (weak coupling limit). It is shown that after the destruction of the pair correlations at T = 0 by a strong magnetic field or rapid rotation, heating can bring them back. This phenomenon is a consequence of the fixed number of fermions in the canonical ensemble.PACS numbers: 71.10. Li, 74.20.Fg The pair correlations in a macroscopic superconductor are destroyed by increasing the temperature or the external magnetic field. The critical field which marks the boundary between the superconducting and normal phases, is a decreasing function of the temperature T . The BCS theory, which is the mean field approximation based on the grand canonical ensemble, describes very accurately this regime. Applying the grand canonical mean field approach to rotating nuclei [1], where the angular velocity plays the role of the magnetic field, gives an analogous result: The angular velocity where the pair correlations disappear decreases with the temperature. Nuclei and atomic nano-size clusters are composed of a small and fixed number of particles, the single-particle spectrum is discrete and the level spacing is comparable with the pair gap. Due to these facts, the fluctuations of the order parameter become important, which smear out the transition from superconducting to normal phase [1,2,3,4,5] and lead to pronounced differences between a system with even and odd particle number [6].In order to properly take into account these fluctuations one has to use the canonical ensemble. The most direct way is to calculate the partition function from the exact eigenvalues of the Hamiltonian, which is possible for some models. In this Letter we study the simple model of fermions occupying an isolated shell of single particle states and interacting by a pairing force. We shall demonstrate that at zero temperature the magnetic field or the angular velocity attenuate the paring in a step-wise manner until it disappears completely above a critical value. For a mesoscopic system in the strong magnetic field heating may bring back the pair correlations. This surprising effect is a consequence of the fixed number of fermions in such a small system. The reduction of the fluctuations in particle number leads to a strong increase of the fluctuations of the order parameter, which constitute the pair correlations above the critical field. A re-entrance of pair correlations has first been discussed by Balian, Flocard and Veneroni [6], who studied ensembles with either only even numbers of particles or only odd numbers of particles. We shall demonstrate that the more stringent restriction to a fixed number of particles, which is realized in small systems, leads to qualitatively different results.First we consider fermions in a spherical pote...
A practical version of the polynomial canonical formalism is developed for normal mesoscopic systems consisting of N independent electrons. Drastic simplification of calculations is attained by means of proper ordering excited states of the system. In consequence the exact canonical partition function can be represented as a series in which the first term corresponds to the ground state whereas successive groups of terms belong to many particle-hole excitations ( one particle-hole two particle-hole and so on). At small temperatures (T < 10 inter-level spacings near the Fermi level) the number of terms which should be taken into account is weakly dependent on N and remains < 10 even if N ∼ 10 5 . The elaborated method makes canonical calculations to be not more complicated than the grand canonical ones and is free from any limitations on N and T .
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Oscillating behaviour of the susceptibility χ and heat capacity C is considered for normal and superconducting mesoscopic systems (nanoclusters and quantum dots). It is proved that at low temperature an increasing magnetic field applied to a mesoscopic system generates local extrema of χ and C. A maximum for χ and a minimum for C simultaneously arise in those points of the field where crossings of quantum levels of the normal and superconducting mesoscopic systems take place.
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