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arXiv:1606.02216 · Progress of Theoretical and Experimental Physics

Nambu–Goldstone modes in the random phase approximation

Kai Neergård

Published Oct 1, 2016• 1 citation
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Abstract

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.

DOI
10.1093/ptep/ptw135

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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.
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 linear in canonic coordinates, with conjugate momenta of cyclic coordinates (Goldstone modes) and a subspace with a basis whose vectors are associated with pairs of conjugate canonic coordinates that do not enter the Hamiltonian at all. In a subspace complementary to the one spanned by all these coordinates including the conjugate coordinates of the Goldstone momenta, the RPA matrix behaves as in the case of a zerodimensional 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.
It is mathematically proven that, if the stability matrix S\mathsf{S}S is positive-semidefinite, solutions of the random-phase approximation (RPA) equation are all physical or belong to Nambu-Goldstone (NG) modes, and the NG-mode solutions may form Jordan blocks of N S\mathsf{N\,S}NS (N\mathsf{N}N is the norm matrix) but their dimension is not more than two. This guarantees that the NG modes in the RPA can be separated out via canonically conjugate variables.
The influence of the Nambu-Goldstone (NG) mode on the energy-weighted sum (EWS) of the excitation strengths is analyzed within the random-phase approximation (RPA).When a certain symmetry is broken at the mean-field level, an NG mode emerges in the RPA, which can be represented by canonical variables forming a two-dimensional Jordan block.A general formula is rederived which separates out the NG-mode contribution to the EWS, via the projection on the subspace directed by the NG mode.As examples, the formula is applied to the E1 excitation and the rotational excitations in nuclei, reinforcing the theoretical consistency of the RPA................
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.
In this article, we revisit the question of the validity of Hartree-Fock and random-phase approximations. We show that there is a connection between the two and while the RPA as it is known in much of the physics literature is of limited validity, there is a generalised sense in which the random phase approximation is of much wider applicability including to systems that do not possess Fermi surfaces. The main conclusion is that the Hartree-Fock approximation is a mean-field idea applied to the density operator, and the random-phase approximation is a mean-field idea applied to the number operator. The generalised RPA is used to compute single-particle properties such as momentum distribution and spectral functions. It is found that we have to go beyond the generalised RPA and include fluctuations in the momentum distribution in order to recover a nonzero imaginary part of the one-particle self energy, which is also explicitly computed, all this in any number of spatial dimensions and no bosonization is needed.