Small Superconducting Grain in the Canonical Ensemble

Abstract

By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing δ. Both quantities turn out to be parity dependent and universal functions of the ratio δ/∆ (∆ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.

Spectroscopy of discrete energy levels in ultrasmall metallic grains

“…BCS approach breaks down; (iv) show that the PBCS crossover [35] is qualitatively correct, but not quantitatively, being somewhat too abrupt; (v) are reproduced remarkably well by the approaches of MFF [33] and BH [34]; (vi) are fully reproduced by the DMRG of [36,37] with a relative error of < 10 −4 for n ≤ 400; our figures don't show DMRG curves, since they are indistinghuishable from the exact ones and are discussed in detail in [36,37].…”

“…A series of more sophisticated canonical approaches [33][34][35][36][37][38][39][40][41] (summarized in Sec. 5.2) confirmed the parity dependence of pairing correlations, but established that the abrupt vanishing of pairing correlations at d BCS p/2 is an artifact of g.c.…”

“…Describing this crossover constituted a conceptual challenge, since the standard grand-canonical mean-field BCS treatment of pairing correlations [21][22][23][24][25][26][27][28] breaks down for d ∆ . This challenge elicited a series of increasingly sophisticated canonical treatments of pairing correlations [33][34][35][36][37][38][39][40][41], based on a simple reduced BCS-Hamiltonian for discrete energy levels, which showed that the crossover is completely smooth, but, interestingly, depends on the parity of the number of electrons on the grain, as pointed out by von Delft et al [21]. Very recently, the main conclusions of these works were confirmed [39] using an exact solution of the reduced BCS model, discovered by Richardson in the context of nuclear physics in the 1960s [46][47][48][49][50][51][52][53][54].…”

Superconductivity in Ultrasmall Grains: Introduction to Richardson’s Exact Solution

“…Most previous works [2,3,4,6,7,8,9, 10] studied a half-filled band with fixed width 2ω D of uniformly-spaced levels (i.e. ε j = j d), containing N = 2n + b electrons.…”

“…studies [2]- [6], the first canonical study was that of Mastellone, Falci and Fazio (MFF) [7], who used Lanczos exact diagonalization (with n ≤ 12) and a scaling argument to probe the crossover regime.…”