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arXiv:1611.06623 · Progress of Theoretical and Experimental Physics

Influence of the Nambu–Goldstone mode on the energy-weighted sum of excitation strengths in the random-phase approximation

Hitoshi Nakada

Published Jan 30, 2017• 4 citations
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Abstract

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................

DOI
10.1093/ptep/ptx004

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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................
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.
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.
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.
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.
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.
Abstract Symmetry breaking at the mean-field level leads to an appearance of a symmetry restoring Nambu-Goldstone (NG) mode in the linear response theory. These modes represent a special kind of collective motion of the system. However, they can interfere with the calculated intrinsic physical excitations and, hence, they are often called as spurious modes. I discuss translational and rotational NG mode and the inertia parameter associated with these modes, by using the finite amplitude method formalism. I will also discuss how to remove spurious mode from the calculated transition strength function.