On the distribution of eigenvalues of grand canonical density matrices

Using physical arguments and partition theoretic methods, we demonstrate under general conditions, that the eigenvalues w(m) of the grand canonical density matrix decay rapidly with their index m, like w(m)similar toexp[-betaB(-1)(ln m)(1+1/alpha)], where B and alpha are positive constants, O(1), which may be computed from the spectrum of the Hamiltonian. We compute values of B and alpha for several physical models, and confirm our theoretical predictions with numerical experiments. Our results have implications in a variety of questions, including the behaviour of fluctuations in ensembles, and the convergence of numerical density matrix renormalization group techniques.