Fluctuations of addition spectra of independent quantum systems

Abstract

Motivated by recent experiments on large quantum dots, we consider the energy spectrum in a system consisting of $N$ particles distributed among $K<N$ independent sub-systems, such that the energy of each sub-system is a quadratic function of the number of particles residing on it. On a large scale, the ground state energy E(N) of such a system grows quadratically with $N$, but in general there is no simple relation such as $E(N)=a N +b N^{2}$. The deviation of E(N) from exact quadratic behavior implies that its second difference (the inverse compressibility) $χ_{N} \equiv E(N+1)-2 E(N)+E(N-1)$ is a fluctuating quantity. Regarding the numbers $χ_{N}$ as values assumed by a certain random variable $χ$, we obtain a closed-form expression for its distribution $F(χ)$. Its main feature is that the corresponding density $P(χ)=\frac{dF(χ)} {dχ}$ has a maximum at the point $χ=0$. As $K \to \infty$ the density is Poissonian, namely, $P(χ) \to e^{-χ}$.