Random phase approximation: Difference between revisions

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[[File:Random phase approximation ring diagrams.png|thumb|400px|Ring diagrams that are summed up in order to obtain the RPA approximation. Above bold lines stand for interacting Green functions, non-bold lines stand for non-interaction Green function, and dashed lines stand for two-body interactions.]]

The '''random phase approximation''' ('''RPA''') is an approximation method in [[condensed matter physics]] and in [[nuclear physics]]. It was first introduced by [[David Bohm]] and [[David Pines]] as an important result in a series of seminal papers of 1952 and 1953.<ref name="Bohm Pines pp. 625–634">{{cite journal | last=Bohm | first=David |author-link= David Bohm| last2=Pines | first2=David |author-link2=David Pines| title=A Collective Description of Electron Interactions. I. Magnetic Interactions | journal=Physical Review | publisher=American Physical Society (APS) | volume=82 | issue=5 | date=1 May 1951 | issn=0031-899X | doi=10.1103/physrev.82.625 | pages=625–634}}</ref><ref name="Pines Bohm pp. 338–353">{{cite journal | last=Pines | first=David |author-link=David Pines| last2=Bohm | first2=David |author-link2=David Bohm| title=A Collective Description of Electron Interactions: II. CollectivevsIndividual Particle Aspects of the Interactions | journal=Physical Review | publisher=American Physical Society (APS) | volume=85 | issue=2 | date=15 January 1952 | issn=0031-899X | doi=10.1103/physrev.85.338 | pages=338–353}}</ref><ref name="Bohm Pines pp. 609–625">{{cite journal | last=Bohm | first=David |author-link=David Bohm| last2=Pines | first2=David |author-link2=David Pines| title=A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas | journal=Physical Review | publisher=American Physical Society (APS) | volume=92 | issue=3 | date=1 October 1953 | issn=0031-899X | doi=10.1103/physrev.92.609 | pages=609–625}}</ref> For decades physicists had been trying to incorporate the effect of microscopic [[Quantum mechanics|quantum mechanical]] interactions between [[Electron|electrons]] in the theory of matter. Bohm and Pines' RPA accounts for the weak screened Coulomb interaction and is commonly used for describing the dynamic linear electronic response of electron systems.

In the RPA, [[electron]]selectrons are assumed to respond only to the total [[total electric potential]] ''V''('''r''') which is the sum of the external perturbing potential ''V''_ext('''r''') and a screening potential ''V''_sc('''r'''). The external perturbing potential is assumed to oscillate at a single frequency ''ω'', so that the model yields via a [[self-consistent field]] (SCF) method <ref name="Ehrenreich Cohen pp. 786–790">{{cite journal | last=Ehrenreich | first=H. | last2=Cohen | first2=M. H. | title=Self-Consistent Field Approach to the Many-Electron Problem | journal=Physical Review | publisher=American Physical Society (APS) | volume=115 | issue=4 | date=15 August 1959 | issn=0031-899X | doi=10.1103/physrev.115.786 | pages=786–790}}</ref> a dynamic [[dielectric]] function denoted by ε_RPA('''k''', ''ω'').

dynamic [[dielectric]] function denoted by ε_RPA('''k''', ω).

The contribution to the [[dielectric function]] from the total electric potential is assumed to ''average out'', so that only the potential at wave vector '''k''' contributes. This is what is meant by the random phase approximation. The resulting dielectric function, also called the ''[[Lindhard theory|Lindhard dielectric function]]'',<ref>{{cite journal|author=J. Lindhard| journal=Kongelige Danske Videnskabernes Selskab, Matematisk-Fysiske Meddelelser|volume=28|issue=8|year=1954|url=http://gymarkiv.sdu.dk/MFM/kdvs/mfm%2020-29/mfm-28-8.pdf|title=On the Properties of a Gas of Charged Particles}}</ref><ref>N. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976)</ref> correctly predicts a number of properties of the electron gas, including [[plasmon]]s.<ref>G. D. Mahan, ''Many-Particle Physics'', 2nd ed. (Plenum Press, New York, 1990)</ref>

The RPA was criticized in the late 501950's for overcounting the degrees of freedom and the call for justification led to intense work among theoretical physicists. In a seminal paper [[Murray Gell-Mann]] and [[Keith Brueckner]] showed that the RPA can be derived from a summation of leading-order chain [[Feynman diagram]]s in a dense electron gas.<ref name="Gell-Mann Brueckner pp. 364–368">{{cite journal | last=Gell-Mann | first=Murray | last2=Brueckner | first2=Keith A. | title=Correlation Energy of an Electron Gas at High Density | journal=Physical Review | publisher=American Physical Society (APS) | volume=106 | issue=2 | date=15 April 1957 | issn=0031-899X | doi=10.1103/physrev.106.364 | pages=364–368| url=https://authors.library.caltech.edu/3713/1/GELpr57b.pdf }}</ref>

The consistency in these results became an important justification and motivated a very strong growth in theoretical physics in the late 50's and 60's.

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<math>\left|\mathrm{RPA}\right\rangle=\mathcal{N}\mathbf{e}^{Z_{ij}\mathbf{a}_{i}^{\dagger}\mathbf{a}_{j}^{\dagger}/2}\left|\mathrm{MFT}\right\rangle</math>

where ''Z'' is a symmetric matrix with <math>|Z|\leq 1</math> and

<math>\mathcal{N}= \frac{\left\langle \mathrm{MFT}\right|\left.\mathrm{RPA}\right\rangle}{\left\langle \mathrm{MFT}\right|\left.\mathrm{MFT}\right\rangle}</math>