Arbitrary Two-body Potentials Research Articles

On the (anisotropic) uniform metallic ground states of fermions interacting through arbitrary two-body potentials in d dimensions

We demonstrate that the skeleton of the Fermi surface pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree–Fock contribution to the dynamic self-energy . That is to say, in order for , it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied: where stands for the underlying non-interacting energy dispersion and ϵF for the exact interacting Fermi energy. The Fermi surface consists of the set of k points which in addition to satisfying the above equation fulfil where1 The set of k points which satisfy the first of the above two equations but fail to satisfy the second constitute the pseudogap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum distribution function for k in the vicinity of and show that, whereas for the uniform metallic ground states of the conventional single-band Hubbard Hamiltonian, described in terms of an on-site interaction, and (here and denote vectors infinitesimally close to , located respectively inside and outside the underlying Fermi sea), for interactions of non-zero range these inequalities can be violated (without thereby contravening the stability condition ). This aspect is borne out by the pertaining to the normal states of for instance liquid 3He (corresponding to a range of applied pressure) as determined by means of quantum Monte Carlo calculations. We further demonstrate that for Fermi-liquid metallic states of fermions interacting through interaction potentials of non-zero range (e.g. the Coulomb potential), the zero-temperature limit of does not need to be equal to ½ for ; this in strict contrast with the uniform Fermi-liquid metallic states of the single-band Hubbard Hamiltonian (if such states at all exist). This aspect should be taken into account while analysing the deduced from the angle-resolved photoemission data concerning real materials. We discuss, in the light of the findings of the present work, the growing experimental evidence with regard to the ‘frustration’ of the kinetic energy of the charge carriers in the normal states of the copper-oxide-based high-temperature superconducting compounds.

LARGE-N COLLECTIVE FIELD THEORY APPLIED TO ANYONS IN MAGNETIC FIELDS

We present a large-N collective field formalism for anyons in external magnetic fields interacting with an arbitrary two-body potential. We discuss how the Landau levels are reproduced in our framework. We apply it to the soluble model for anyons proposed by Girvin et al., and obtain the dispersion relation of the collective modes for arbitrary statistical parameters.

Matrix elements between vector hyperspherical states.

Three-body symmetrized vector hyperspherical harmonic functions, realizing irreducible representations of the permutation group of three particles, are constructed and expressed in terms of Wigner D functions. Matrix elements of arbitrary two-body potentials between these hyperspherical states, including their velocity-dependent parts, are calculated analytically and expressed through the sum of products of the Wigner 3-j symbols and explicitly written functions of the radial variable.

Matrix elements of potentials for L=1 hyperspherical states.

Three-body L=1 symmetrized hyperspherical harmonic functions, realizing irreducible representations of the permutation group of three particles, are expressed in terms of Wigner D functions. Matrix elements of arbitrary two-body potentials between these hyperspherical states, including their velocity-dependent parts, are calculated analytically and expressed through the sum of products of the Wigner 3-j symbols and explicitly written functions of the radial variable.

Three-Body Correlations in Nuclear Matter

The possibility of computing the three-body correlation energy in nuclear matter by direct solution of integral equations in momentum space is studied. An important feature of the procedure is that the off-energy-shell reaction matrix, which comes from an arbitrary two-body potential, is approximated by a sum of separable terms. Detailed numerical calculations are made for an $S$-wave force consisting of the Reid $^{1}S_{0}$ potential acting in both singlet and triplet states. The main results are as follows: (1) The Reid $^{1}S_{0}$ reaction matrix can be accurately represented by two or three separable terms for all relative momenta $k$ in the region $0<k<8$ ${\mathrm{fm}}^{\ensuremath{-}1}$. This is the relevant region for the calculations. (2) Integrals over momenta can be adequately approximated by finite sums over only 6 or 8 Gauss points. (3) Many three-body states can be adequately treated in lowest order and therefore do not appear in the integral equations. On the basis of these results, it is estimated that a calculation using the full nuclear force would require the solution of systems of 100-200 linear equations. Solving such systems is perfectly feasible with present-day computers. The three-body correlation energy has been calculated at three different densities for the assumed two-body force.

Collective Representation of the Many-Particle System Interacting through Arbitrary Two-Body Potential

A collective description is developed for a many-particle system interacting through an arbitrary two-body potential. The description is given both in terms of the second quantizad form and in terms of an auxiliary field analogous to the longitudinal electromagnetic field. Since the BohmPines theory of the collective description has some unsatisfuactory features concerning the subsidiary condition imposed on the auxiliary field, a strict method is proposed for the treatment of tbe subsidiary condition. To obtain the useful result of tbe collective representation. the three steps of the canonical transformations must be taken. Although the first two transformations are the same as those of BohmPines, the third transformation, which is not applied by Bohm-Pines and is characteristic of tbe present theory, is necessary for obtaining a meaningful result. By using both the strict subsidiary condition and the three canonical transformations, the collective field coordinate and the panticle coordinate are separated not only in the total Hamiltonian but also in the subsidiary condition. Moreover, clarity is obtained for tbe correct meaning of the subsidiary condition. (auth)