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Journal of Physics G Nuclear Physics

Random-phase approximation and broken symmetry

E. D. D. Davis, W. D. Heiss

Published Sep 1, 1986• 26 citations
Publisher

Abstract

The validity of the random-phase approximation (RPA) in broken-symmetry bases is tested in an appropriate many-body system for which exact solutions are available. Initially the regions of stability of the self-consistent quasiparticle bases in this system are established and depicted in a 'phase' diagram. It is found that only stable bases can be used in an RPA calculation. This is particularly true for those RPA modes which are not associated with the onset of instability of the basis; the authors see that these modes do not describe any excited state when the basis is unstable, although from a formal point of view they remain acceptable. The RPA does well in a stable broken-symmetry basis provided one is not too close to a point where a phase transition occurs. This is true for both energies and matrix elements.

DOI
10.1088/0305-4616/12/9/006

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The validity of the random-phase approximation (RPA) in broken-symmetry bases is tested in an appropriate many-body system for which exact solutions are available. Initially the regions of stability of the self-consistent quasiparticle bases in this system are established and depicted in a 'phase' diagram. It is found that only stable bases can be used in an RPA calculation. This is particularly true for those RPA modes which are not associated with the onset of instability of the basis; the authors see that these modes do not describe any excited state when the basis is unstable, although from a formal point of view they remain acceptable. The RPA does well in a stable broken-symmetry basis provided one is not too close to a point where a phase transition occurs. This is true for both energies and matrix elements.
The "phase transitions" predicted within finite microscopic systems by Hartree-Fock-Bogoliubov (HFB) and aspects of the use of the associated broken-syrometry bases in the random-phase approximation (RPA) are considered using soluble models-Evidence is presented in support of the conjecture that the success of these techniques lies in the fact that they mimic singularities in the dependence on interaction strengths of the exact solution.This conjecture provides a natural explanation for why such methods fail close to a point where a phase transition occurs and indicates possible directions for improvement.Phase diagrams at both zero and finite temperature are determined, and simple analytic expressions for the way in which critical scrcn;the scale with particle number are found.It is shown that the ''phase transitions" predicted at finite temperature are relevant.A connection between the singularities referred to above and real phase transitions found in the thermodynamic limit is discussed.It is found that only stable bases can be used in an RPA calculation.This is in particular true for those RPA modes which are not associated with the onset of instability of the basis; these modes do not describe any excited state when tht. basis is unstable.Outside transitional regions certain undesirable features of HFB are unearthed, notably that the HFB ground state energy is not necessarily an upper bound to the exact ground state energy.The effectiveness in this regime of the Hartree-Fock Seniority approximation as a substitute to projection methods is ■ -■■.ated.
The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.
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We show that it is possible to restore broken symmetries associated with the Goldstone mode within the self-consistent random phase approximation (SCRPA) applied to the three-level Lipkin model. To do this, it is necessary to include the so-called scattering terms in the RPA operator. We determine one- and two-body densities as very convergent expansions in terms of the generators of the RPA basis. We show that SCRPA excitations correspond to the heads of some rotational bands in the exact spectrum.
We calculate the ground-state expectation value of scalar observables in the matrix formulation of the random phase approximation (RPA). Our expression, derived using the quasiboson approximation, is a straightforward generalization of the RPA correlation energy. We test the reliability of our expression by comparing against full diagonalization in $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces. In general the RPA values are an improvement over mean-field (Hartree-Fock) results, but are not always consistent with shell-model results. We also consider exact symmetries broken in the mean-field state and whether or not they are restored in RPA.
I show that the kernel of the random phase approximation (RPA) matrix based on a stable Hartree, Hartree–Fock, Hartree–Bogolyubov, or Hartree–Fock–Bogolyubov mean field solution is decomposed into a subspace with a basis whose vectors are associated, in the equivalent formalism of a classical Hamiltonian homogeneous of second degree in canonical coordinates, with conjugate momenta of cyclic coordinates (Nambu–Goldstone modes) and a subspace with a basis whose vectors are associated with pairs of a coordinate and its conjugate momentum neither of which enters the Hamiltonian at all. In a subspace complementary to the one spanned by all these coordinates including the conjugate coordinates of the Nambu–Goldstone momenta, the RPA matrix behaves as in the case of a zero-dimensional kernel. This result was derived very recently by Nakada as a corollary to a general analysis of RPA matrices based on both stable and unstable mean field solutions. The present proof does not rest on Nakada's general results.