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arXiv:cond-mat/9906166 · Physical review. B, Condensed matter

Crossover from bulk to few-electron limit in ultrasmall metallic grains

J. Dukelsky, Germán Sierra

Published May 1, 2000• 68 citations
arXivHTMLPDF

Abstract

We study the properties of ultrasmall metallic grains with sizes in the range 20--400 electrons. Using a particle-hole version of the density-matrix renormalization-group (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 qualitatively 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 projected BCS (PBCS) wave function in order to describe correctly the crossover. Finally we present a recursive method to compute norms and expectation values with the PBCS wave function.

DOI
10.1103/physrevb.61.12302

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We study the properties of ultrasmall metallic grains with sizes in the range 20--400 electrons. Using a particle-hole version of the density-matrix renormalization-group (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 qualitatively 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 projected BCS (PBCS) wave function in order to describe correctly the crossover. Finally we present a recursive method to compute norms and expectation values with the PBCS wave function.
We analyze spectral functions of mesoscopic systems with large dimensionless conductance, which can be described by a universal Hamiltonian. We show that an important class of spectral functions are dominated by one single state only, which implies the existence of well-defined (i.e., infinite-lifetime) quasiparticles. Furthermore, the dominance of a single state enables us to calculate zero-temperature spectral functions with high accuracy using the density-matrix renormalization group. We illustrate the use of this method by calculating the tunneling density of states of metallic grains, of which we discuss the crossover from the few-electron to the bulk regime.
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 $\ensuremath{\delta}$. Both quantities turn out to be parity dependent and universal functions of the ratio $\ensuremath{\delta}/\ensuremath{\Delta}$ ( $\ensuremath{\Delta}$ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.
In this thesis, I analyze properties of mesoscopic systems that exhibit strong quantum correlations. To this purpose, I adapt and use the density-matrix renormalization group (DMRG), a numerical method that has been developed specifically for analyzing such systems. The following systems are discussed: - Josephson effect between superconducting nanograins: I investigate the Josephson effect, i.e. the dependence of the energy of two weakly coupled superconductors on the difference between their superconducting phase, in the regime that the level spacing d is comparable to the bulk superconducting gap Delta. Because the BCS solution is inapplicable in this regime, I use the DMRG to calculate the ground state of the two coupled superconductors and extract the Josephson energy E_J. The standard BCS result for E_J is reproduced only in the continuum limit of vanishing d. As d is increased, however, E_J turns out to display a nonmonotonic behaviour. A tight-binding approximation for weak Josephson coupling explains the physical mechanism underlying this reentrance in a transparent way. - Well-defined quasiparticles in small metallic grains: I analyze zero-temperature spectral functions of mesoscopic systems such as quantum dots and metallic grains, in the limit of large conductance, where they can be described by a "universal Hamiltonian" model. I show that within this model, an important class of spectral functions is dominated by one single energy eigenstate only. For an interacting system this is a very peculiar property, which implies an infinite lifetime of the quasiparticles. Moreover, I show that the dominating eigenstate contains only a limited subclass of all excitations, which I characterize as the "No-Gaudino" excitations. Hence, these are sufficient to explain many of the properties of the systems under consideration. Besides its own physical significance, the dominance of the "No-Gaudino" states has also high practical value, because it permits the calculation of zero-temperature spectral functions with high accuracy using the DMRG. I illustrate the use of this method by calculating the tunneling density of states of metallic grains and the magnetic response of mesoscopic rings. - Real-time dynamics in spin-1/2 chains I study the nonequilibrium transport properties in spin-1/2 chains by solving the time evolution of a non-stationary initial state, and investigate the influence of different interaction strength and dimerization on the magnetization transport. To this purpose, I use the "adaptive time-dependent DMRG", a DMRG variant that was recently developed to solve the many-body Schroedinger equation with high accuracy. I find that the magnetization possesses a well-defined long-time limit, whose nature does not depend on the dimerization, but only on the strength J_z of the S^z-S^z-interaction: For |J_z| < 1 I find ballistic magnetization transport, and for J_z >1 almost no transport, with a sharp crossover at |J^z|=1. I explain this crossover as a subtle consequence of a quantum phase transition which occurs at the precise value J_z| = 1. - Many-body scattering states: I present a general method for calculating many-body scattering states that does not rely on the assumptions of perturbation theory or near-equilibrium. Due to the complexity of the problem, I limit the discussion to a description of the algorithm itself and a few proof-of-principle calculations only. The strategy is to calculate the many-body scattering state that results when a scatterer (e.g. a quantum dot) is connected to two leads, to which a voltage bias V is applied. This state is obtained by solving the many-body Lippmann-Schwinger equation using the DMRG.
Superconducting properties of small metallic grains modelled by highly degenerate two-level spectrum have been studied.We have solved numerically Richardson's exact equations for the system of 2N electrons in two levels.Characterising the size of the grain by the level degeneracy we study the finite size corrections to the thermodynamic limit of the ground and lowest excited state energy.The interparticle distance ∝ N -1/3 seems to be the expansion parameter.The obtained results have been compared with those of other authors.
We develop a quantitative analytic theory that accurately describes the odd-even effect observed experimentally in a one-dimensional, trapped Fermi gas with a small number of particles [G. Z\urn et al., Phys. Rev. Lett. 111, 175302 (2013)]. We find that the underlying physics is similar to the parity effect known to exist in ultrasmall mesoscopic superconducting grains and atomic nuclei. However, in contrast to superconducting nanograins, the density fluctuation effect (Hartree correction) dominates over the superconducting pairing fluctuations and leads to a much more pronounced odd-even effect in the mesoscopic, trapped Fermi gas. We calculate the corresponding parity parameter and separation energy using both perturbation theory and a path integral framework, generalized to account for the effects of the trap, pairing fluctuations, and Hartree corrections. Our results are in an excellent quantitative agreement with experimental data and exact diagonalization. Finally, we describe a few-to-many particle crossover between the perturbative mesoscopic regime and non-perturbative many-body physics that the system approaches in the thermodynamic limit.