Well-Defined Quasiparticles in Interacting Metallic Grains

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.

Figure 1

The matrix element $⟨c_{i\sigma }{c}_{i\sigma }^{†}⟩$ from Eq. (6) with $\widehat{O}=c_{i\sigma }^{†}$ (dashed line) and the contribution from the No-Gaudino state (solid line) as a function of $\lambda$. Here $i=10$, i.e., the tenth out of a total of $N=40$ energy levels above $E_{\mathrm{F}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}}$. The lost weight $w_L$, i.e., the difference between both, is plotted in the inset. It shows a maximum in the intermediate regime around $\lambda \sim 1/\ln N$ (indicated by dotted line), but even there, $w_L$ is less than 0.2% of the total spectral weight.

Figure 2

The tunneling density of states $\nu (E)$ for ${\Delta }_{\mathrm{B}\mathrm{C}\mathrm{S}}=0.07{\omega }_{\mathrm{c}\mathrm{o}}$ and $N=5,30,60,200$ energy levels above $E_{\mathrm{F}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}}$ in the No-Gaudino approximation (solid line; for the sake of better visibility, the delta peaks in Eq. (2) have been replaced by Gaussians of width $0.005{\omega }_{\mathrm{c}\mathrm{o}}$). The familiar gap ${\Delta }_{\mathrm{B}\mathrm{C}\mathrm{S}}$ emerges during the crossover from the few-electron ($d\gg {\Delta }_{\mathrm{B}\mathrm{C}\mathrm{S}}$) to the bulk limit ($d\ll {\Delta }_{\mathrm{B}\mathrm{C}\mathrm{S}}$). In the latter limit, we observe agreement with the BCS result (dashed line).