Geometry of random interactions

Abstract

It is argued that spectral features of quantal systems with random interactions can be given a geometric interpretation. This conjecture is investigated in the context of two simple models: a system of randomly interacting d bosons and one of randomly interacting fermions in a j=7/2 shell. In both examples the probability for a given state to become the ground state is shown to be related to a geometric property of a polygon or polyhedron which is entirely determined by particle number, shell size, and symmetry character of the states. Extensions to more general situations are discussed

Bose–Einstein condensates of atoms with arbitrary spin

“…The Casimir operators appearing in Eq. (11) are known in closed form, (12) in terms of the operatorsT…”

Eigenvalue correlations and the distribution of ground state angular momenta for random many-body quantum systems

“…Chau et al showed that all energies are confined to a convex polytope (i.e., a convex polyhedron in d dimensions) and that only the states located at the vertices of this polyhedron can become the ground state [19]. The probability for a state at vertex j to become the ground state depends on the angle f j θ jf , where the sum is over all faces that contain the vertex j , and θ jf is the angle subtended at vertex j in the face f .…”

“…[19], spectroscopic properties of quantum systems with random interactions were given a geometric interpretation. In particular, it was shown that diagonal Hamiltonians, i.e., those whose energy eigenvalues depend linearly on the two-body matrix elements, can be associated with a geometric shape (convex polyhedron) defined in terms of coefficients of fractional parentage and/or generalized coupling coefficients.…”

“…In this respect, the approach of Chau et al [19] for diagonal Hamiltonians stands out, since it makes use of the geometry of the model space, is valid for both fermions and bosons, and allows the calculation of the ground state probabilities exactly.…”

“…[19] can be interpreted in terms of correlations (covariances) between energy eigenvalues to provide an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting many-body quantum systems. In addition, we show that the method can be extended to nondiagonal Hamiltonians by using perturbation theory.…”