On interacting fermions and bosons with definite total momentum

Any exact eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a symmetry-broken function Φ. For two particles, we solve the problem exactly for all energy levels and any interparticle interaction. Especially for the ground-state, Φ is given by the simple Hartree-Fock/Hartree ansatz for fermions/bosons. Implications for several and many particles as well as a numerical example are provided. PACS numbers: 03.65.-w, 03.75.Hh, 05.30.Fk Typeset using REVTEX ∗E-mail: [email protected] 1 The homogeneous gas of interacting fermions and bosons is a fundamental concept in the physics of many-particle systems, see e.g., Refs. [1,2]. Expect for specific cases, such a manyor several-body problem cannot be solved exactly, i.e. the wavefunctions and energies of the ground and excited states are not known. In fact, for more than a few particles it becomes already impossible to numerically compute the exact ground-state energy and wavefunction. Consequently, approximations are a must. In this Letter, we introduce an exact continuous configuration-interaction ansatz for the many-body wavefunction Ψ of interacting particles in a volume with periodic boundary conditions. Examples for realizations of this case are a ring in one dimension (1D), a torus, a long, thin pipe (tube) or a sphere in 2D, and the text-book example of a “big box” in 3D. Specifically, we employ a many-body function Φ as a basis function for a continuous expansion of Ψ. The shape of Φ is to be optimized by employing the variational principle. For two particles, we solve the problem exactly for all energy levels and any inter-particle interaction. Especially for the exact ground-state, Φ is given by the simple Hartree-Fock/Hartree ansatz for fermions/bosons. For more particles as will be explained below, our ansatz with any specific choice for Φ is better than solving the Schrödinger equation by optimizing Ψ = Φ itself. For instance, taking for Φ a Hartree-Fock/Hartree ansatz, the resulting equations would lead to lower energies than the corresponding Hartree-Fock/Hartree equations. These properties make our ansatz a particularly attractive approximation for the few as well as the many-body problem. Consider the generic many-body Hamiltonian describing N fermions or bosons in a 3D box of volume V = LxLyLz: Ĥ(r1, r2, . . .) = − N