The Crossover from the Bulk to the Few-Electron limit in Ultrasmall Metallic Grains

TL;DR

This paper investigates the transition from bulk superconductivity to fluctuation-dominated behavior in ultrasmall metallic grains, using advanced computational methods to analyze their quantum properties and crossover characteristics.

Contribution

It introduces a particle-hole DMRG approach, compares it with exact solutions, and proposes a simplified wave function model to better understand the crossover regime.

Findings

Smooth crossover between large and small grains without critical level spacings.

Complete agreement between DMRG and Richardson exact solutions.

A new recursive method for PBCS wave function calculations.

Abstract

We study the properties of ultrasmall metallic grains with sizes in the range of 20 up to 400 electrons. Using a particle-hole version of the DMRG method we compute condensation energies, spectroscopic gaps, pairing parameters and particle-hole probabilities of the ground state wave function. The results presented in this paper confirm that the bulk superconducting regime (large grains) and the fluctuation dominated regime (small grains) are qualitative different, but show that the crossover between them is very smooth with no signs of critical level spacings separating them. We compare our DMRG results with the exact ones obtained with the Richardson solution finding complete agreement. We also propose a simplified version of the DMRG wave function, called the Particle-Hole BCS ansatz, which agrees qualitatively with the DMRG solution and illustrates what is lacking in the PBCS wave function in order to describe correctly the crossover. Finally we present a new recursive method to compute norms and expectation values with the PBCS wave function.