Abstract
The work presents a proof of convergence of the density of energy levels to a Gaussian distribution for a wide class of quadratic forms of Fermi operators. This general result applies also to quadratic operators with disorder, e.g., containing random coefficients. The spacing distribution of the unfolded spectrum is investigated numerically. For generic systems, the level spacings behave as the spacings in a Poisson process. Level clustering persists in the presence of disorder.
Highlights
In a variety of situations, one encounters quadratic forms in Fermi operators Hn = nAij ci † cj i,j=1 + 1 2 Bij(cicj − ci † cj † ), (1)where the Fermi operators ci’s obey the canonical anticommutation relations {ci, cj} = 0 and {ci, cj†} = δij, and the coefficients satisfy Aij = Aji ∈ R and Bij = −Bij ∈ R for i, j = 1, 2
Our goal here is to show that the density of energy levels of a wide class of quadratic Fermi operators converges to a Gaussian distribution in the limit of large n
III, we present our main results on the limiting density of energy levels and the rate of convergence to the limit
Summary
Hartmann, Mahler, and Hess[19] considered generic many body quantum systems with nearest neighbour interaction They proved that, provided that the energy per particle has an upper bound, the energy distribution for almost every product state becomes a Gaussian in the limit of infinite number of particles. Our goal here is to show that the density of energy levels of a wide class of quadratic Fermi operators (both deterministic and random) converges to a Gaussian distribution in the limit of large n. The proof of this universal result relies on the connection between the spectrum of Hn and the subset-sum structure arising in the normal mode decomposition.
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