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Weakly interacting homogeneous Bose gas: The role of residual interactions

Christoph Weiß, Svend‐Age Biehs, André Eckardt, Martin Holthaus

Published Mar 29, 2005• 0 citations
OpenAlex

Abstract

The calculation of the ground-state energy of a homogeneous Bose gas with soft interparticle interaction results in an expression originally stated by Brueckner and Sawada, instead of the more accurate Lee-Yang formula. We show that the source of this discrepancy lies in the residual terms of the full Hamiltonian which are neglected in the standard Bogoliubov scheme. Incorporating these residual interactions, we derive an improved upper bound on the ground-state energy. An important link in our chain of arguments consists in the fact that the Born series of the scattering length is a convergent rather than an asymptotic expansion at least for su ciently soft potentials. We provide an exactly solvable model which demonstrates that the Born series of the scattering length reproduces, within its radius of convergence, the exact scattering length.

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The calculation of the ground-state energy of a homogeneous Bose gas with soft interparticle interaction results in an expression originally stated by Brueckner and Sawada, instead of the more accurate Lee-Yang formula. We show that the source of this discrepancy lies in the residual terms of the full Hamiltonian which are neglected in the standard Bogoliubov scheme. Incorporating these residual interactions, we derive an improved upper bound on the ground-state energy. An important link in our chain of arguments consists in the fact that the Born series of the scattering length is a convergent rather than an asymptotic expansion at least for su ciently soft potentials. We provide an exactly solvable model which demonstrates that the Born series of the scattering length reproduces, within its radius of convergence, the exact scattering length.
The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.
We argue that the energy of a strongly interacting gas in the unitary limit is parametrized by a universal constant β. For fermions, this parameter is known, but not for bosons. Finding β for bosons is the goal of this thesis. We present the basics of scattering theory, including the s-wave scattering length, which is the characteristic length scale for the interactions at low energies and which goes to infinity at unitarity. Shape and Feshbach resonances are explained. The conventional theory for Bose gases, Bogoliubov theory, is presented, which takes into account quantum fluctuations. It is used to derive identities for the groundstate energy, the condensate fraction and the contact. These are given as expansions in the scattering length, which therefore only hold true for weak interactions. To improve on this, we present an effective action from an atom-molecule theory, which also allows bound states. By a mean-field approach we derive universal identities for the associated energy scales, and in the unitary limit an analytical expression is found for the chemical potential, from which the value β = −0.54 is found. However, this mean-field result is qualitatively not reliable, since quantum depletion is neglected. A new approach is introduced, starting with a Jastrow Ansatz, from which the energy is expressed in the Jastrow function f and radial distribution function g. A cluster expansion is made for g, where a diagrammatic analogy is found. The Hyper-Netted Chain approximation is introduced to calculate g systematically by summing infinitely many diagrams. In HNC/0, we minimize the energy by a variational method, which is based on a well-constructed Ansatz for g. Numerical results using the HNC/0 approximation are given not only for the ground-state energy, but also for the condensate density and the contact. For large scattering lengths the calculations became increasingly less accurate, but the results from Bogoliubov theory are correctly included for small a. An extrapolation for the ground-state energy is made to find β ' −0.20, which is expected to be an upper bound. Moreover, quantum depletion is found to be higher when calculated in HNC/0 for large scattering lengths. Also, it is found that the contact at this value for unitarity remains finite and acquires the value of C ' 10.3n4/3, which is remarkably close to the value for fermions, C ' 11n4/3. Cover image courtesy of [18].
The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction; in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.
We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.