Self-consistent random phase approximation in the Schütte–Da Providencia fermion-boson model

Abstract

The self-consistent random phase approximation (SCRPA) is applied to the exactly solvable model with fermion-boson coupling proposed by Schütte and Da Providencia. Very encouraging results in comparison with the exact solution of the model for various observables are obtained. The transition from the normal phase to the phase with a spontaneously broken symmetry is carefully investigated. The strong reduction of the variance in SCRPA vs Hartree-Fock is pointed out. PACS number(s): 21.60. Jz, 24.10.Cn During the last decade the so-called self-consistent version of the random phase approximation (SCRPA) has seen very encouraging successes in a number of nontrivial model cases (see, for example, Ref.[1] for a detailed description of the method and Ref.[2] for the application of SCRPA to the many-level pairing model). In spite of these performances of the theory, there are remaining problems. In first place this concerns situations with spontaneously broken symmetries. Such situations were treated in Refs. [1,3,4]. Whereas in the Lipkin model [3] the symmetry broken ("deformed") phase caused no problem because the broken symmetry is discrete (parity), in the other two cases [1,4], with a continuous broken symmetry, problems appeared with the low-lying mode known to be exactly at zero energy in the standard HF-RPA approach (the spurious or Goldstone mode), where HF stands for Hartree-Fock. In the two cases cited [1,4] the low-lying mode does not appear at zero energy in SCRPA because the RPA operator does not contain the symmetry operator as a limit case. Indeed, e.g., the number operator in quasiparticle (BCS) representation contains a purely Hermitian piece ␣ k † ␣ k which cannot be incorporated in the RPA operator which by definition is non-Hermitian. The same situation is present in the Schütte-Da-Providenica boson-fermion model [5] where the symmetry operator contains the boson and fermion number operators. The violation of the Goldstone theorem signifies that the Ward identities and conservation laws are not respected. Though this violation seems relatively mild and to go away in macroscopic systems (the Hermitian pieces becoming of zero weight), the situation remains annoying for finite systems.In this paper which can be considered as a sequel of Ref.[4] we want again to investigate the Schütte-Da Providenica model:where the a † , a are fermion operators. In analogy to the work in Ref. ͓4͔ we will introduce the more general RPA operatorThe operators T ± , T 0 are obtained from ± , 0 , by writing the latter ones in the deformed basisThe bosons operators B † and B are obtained from the original ones by a shift transformation B → b − , where is a c number characterizing the appearance of the Bose condensate. The introduction of the boson pair operators ␤ † ␤ † is motivated by the fact that otherwise there exists a certain dissymmetry between fermions and bosons, the fermions being in any case bilinear whereas the bosons are otherwise only contained to linear order in Eq. ͑2͒. Also the symmetry operator P = b † ...

Vacuum effects on the properties of nuclear matter under an external magnetic field

“…3, is that the vacuum correction preserves the thermodynamical instabilities of the MFA. Therefore a spinodal decomposition similar to that shown in [6] must be expected, with the same range of densities but extending to higher pressures. It must be said that in order to allow an easier comparison within the same figure, the constant contribution of the magnetic field to the total energy density is not shown in Fig.…”

“…Hence I analyze the dependence on B of the difference ∆M between the full magnetization and the corresponding result without vacuum effect. To compare with previous calculations [5,25] which used the same hadronic interaction, the adimensional ratio ∆M/q 2 B as a function of the magnetic intensity is shown in Fig. 4, separately for the proton and neutron contributions within the range 10 17 < B < 10 19 G. By considering the results of [25] for B > 10 18 G, an almost general trend is that this quantity decreases by increasing the magnetic intensity and decreasing the matter density.…”

“…To compare with previous calculations [5,25] which used the same hadronic interaction, the adimensional ratio ∆M/q 2 B as a function of the magnetic intensity is shown in Fig. 4, separately for the proton and neutron contributions within the range 10 17 < B < 10 19 G. By considering the results of [25] for B > 10 18 G, an almost general trend is that this quantity decreases by increasing the magnetic intensity and decreasing the matter density. Due to the tiny results I obtained for the neutron component, one can expect that the vacuum corrections to this component could have significative effects only in the very low density regime.…”

“…The interaction between matter and strong magnetic fields is a subject of permanent research [1,2]. In particular the combination of magnetic fields and the strong interaction has been intensively debated in the low [3,4,5,6,7,8,9,10,11,12] and medium energy regimes [13,14,15,16,17]. In the first case, the use of hadronic degrees of freedom is indispensable.…”

“…Among the most used models of the hadronic interaction, the Quantum Hadro-Dynamics(QHD) has a remarkable versatility to describe a variety of phenomena and its results have a satisfactory accuracy when it is required. QHD has been used to study the interaction of hadrons and magnetics fields, for instance in the structure and composition of neutron stars [3,4,5],the liquidgas phase transition [6], the neutrino propagation in nuclear matter [7], the deconfinement phase transition [8], magnetic catalysis [9,10], the modification of nuclear structure [11], and the formation of magnetic domains [12]. Within this description there is a general agreement that the anomalous magnetic moments (AMM) of the hadrons play a significative role when the magnetic 2 Vacuum corrections to the MFA within the QHD model…”