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{{Short description|Mathematical formalism used in quantum field theory}}
[[File:Random phase approximation ring diagrams.png|thumb|Bubble diagrams, which result in the RPA when summed up. Solid lines stand for interacting or non-interacting [[Green's function (many-body theory)|Green's functions]], dashed lines for two-particle interactions.]]
The '''random phase approximation''' ('''RPA''') is an approximation method in [[condensed matter physics]] and
In the RPA, electrons are assumed to respond only to the total [[electric potential]] ''V''('''r''') which is the sum of the external perturbing potential ''V''<sub>ext</sub>('''r''') and a screening potential ''V''<sub>sc</sub>('''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 | last1=Ehrenreich | first1=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| bibcode=1959PhRv..115..786E }}</ref> a dynamic [[dielectric]] function denoted by ε<sub>RPA</sub>('''k''', ''ω'').
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The RPA was criticized in the late 1950s 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 | last1=Gell-Mann | first1=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| bibcode=1957PhRv..106..364G | s2cid=120701027 | url=https://authors.library.caltech.edu/3713/1/GELpr57b.pdf }}</ref>
In [[superconductivity]], RPA can be used to derive [[Bardeen–Pines interaction]] between [[Phonon|phonons]] and electrons that leads to [[Cooper pair|Cooper pairing]].<ref>{{Cite book |last=Coleman |first=Piers |url=https://books.google.com/books?id=ESB0CwAAQBAJ&dq=bardeen+pines+interaction&pg=PA225 |title=Introduction to Many-Body Physics |date=2015-11-26 |publisher=Cambridge University Press |isbn=978-1-316-43202-0 |language=en}}</ref>
The consistency in these results became an important justification and motivated a very strong growth in theoretical physics in the late 50s and 60s.▼
▲The consistency in these results became an important justification and motivated a very strong growth in [[theoretical physics]] in the late 50s and 60s.
==Applications ==▼
=== Ground state of an interacting bosonic system===▼
{{off topic|date=July 2025}}
The RPA vacuum <math>\left|\mathrm{RPA}\right\rangle</math> for a bosonic system can be expressed in terms of non-correlated bosonic vacuum <math>\left|\mathrm{MFT}\right\rangle</math> and original boson excitations <math>\mathbf{a}_{i}^{\dagger}</math>
:<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>
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