Equation of Motion Method for strongly correlated Fermi systems and Extended RPA approaches

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In this review are summarized about 20 years of theoretical research with applications in the field of many-body physics for strongly correlated fermions with Rowe's equation of motion (R-EOM) method and extended RPA equations. One major goal is to set up, via EOM, RPA equations with a correlated ground state. Since the correlations depend on the RPA amplitudes, it follows that RPA becomes a selfconsistency problem which is called Self-Consistent RPA (SCRPA). This then also improves very much the Pauli principle violated with standard RPA. The method was successfully applied to several non trivial problems, like the nuclear pairing Hamiltonian in the particle–particle channel (pp-RPA) and the Hubbard model of condensed matter. The SCRPA has several nice properties, as for instance, it can be formulated in such a way that all very appreciated qualities of standard RPA as, e.g., appearance of zero (Goldstone) modes in the case of broken symmetries, conservation laws, Ward identities, etc. are maintained. For the Goldstone mode an explicit example of a model case is presented. The formalism has its sound theoretical basis in the fact that an extension of the usual RPA operator has been found which exactly annihilates the Coupled Cluster Doubles (CCD) ground state wave function. This has been a longstanding problem for all RPA practitioners from the beginning. There exists a rather simplified version of SCRPA which is the so-called renormalized RPA (r-RPA) where only the correlated occupation numbers are involved in the selfconsistent cycle. Because its numerical solution is rather similar to standard RPA, it has known quite a number of applications, like beta and double beta decays, which are reviewed in this article. In this review also an extended version of second RPA (ERPA)is described. This ERPA maintains all appreciable properties of standard RPA. Several realistic applications for, e.g., the damping of giant resonances are presented. Another important aspect of the extended RPA-theories is that it can be formulated symmetry conserving replacing and eventually improving symmetry (e.g., number) projected mean field approaches. A practical application is given in the review. The EOM formalism allows to treat alpha clustering and alpha particle (quartet) condensation in nuclear matter. This EOM approach succeeded where earlier attempts have failed. It is, for instance, shown how the critical temperature for alpha particle condensation in symmetric and asymmetric nuclear matter can efficiently be calculated.
D.S.D. was supported by the grant PN-18090101/2019–2021 of the Romanian Ministry of Education and Research, J.D. by the Spanish Ministerio de Ciencia, Innovacion y Universidades and the European regional development fund (FEDER), Project No. PGC2018-094180-B-I00, G.R. by the German Research Foundation (DFG), Grant No. RO905/38-1, and E.L. by the US-NSF Career Grant PHY-1654379.