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{{Short description|Mathematical formalism used in quantum field theory}}
[[File:Random phase approximation ring diagrams.png|thumb|
The '''random phase approximation''' ('''RPA''') is an approximation method in [[condensed matter physics]] and [[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 | last1=Bohm | first1=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| bibcode=1951PhRv...82..625B }}</ref><ref name="Pines Bohm pp. 338–353">{{cite journal | last1=Pines | first1=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| bibcode=1952PhRv...85..338P }}</ref><ref name="Bohm Pines pp. 609–625">{{cite journal | last1=Bohm | first1=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| bibcode=1953PhRv...92..609B }}</ref> For decades physicists had been trying to incorporate the effect of microscopic [[Quantum mechanics|quantum mechanical]] interactions between [[electron]]s 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. It was further developed to the relativistic form (RRPA) by solving the [[Dirac equation]].<ref>{{Cite journal |last1=Deshmukh |first1=Pranawa C. |author-link=Pranawachandra Deshmukh |last2=Manson |first2=Steven T. |date=September 2022 |title=Photoionization of Atomic Systems Using the Random-Phase Approximation Including Relativistic Interactions |journal=Atoms |language=en |volume=10 |issue=3 |pages=71 |doi=10.3390/atoms10030071 |issn=2218-2004 |doi-access=free|bibcode=2022Atoms..10...71D }}</ref><ref>{{Cite journal |last1=Johnson |first1=W R |last2=Lin |first2=C D |last3=Cheng |first3=K T |last4=Lee |first4=C M |date=1980-01-01 |title=Relativistic Random-Phase Approximation |url=https://iopscience.iop.org/article/10.1088/0031-8949/21/3-4/029 |journal=Physica Scripta |volume=21 |issue=3–4 |pages=409–422 |doi=10.1088/0031-8949/21/3-4/029 |bibcode=1980PhyS...21..409J |s2cid=94058089 |issn=0031-8949|url-access=subscription }}</ref>
In the RPA,
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
▲In the RPA, [[electron]]s 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>H. Ehrenreich and M. H. Cohen, [http://dx.doi.org/10.1103/PhysRev.115.786 Phys. Rev. '''115''', 786 (1959)]</ref> a
The RPA was criticized in the late
▲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>J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. '''28''', 8 (1954)</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>
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 RPA was criticized in the late 50'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>M. Gell-Mann, K.A. Brueckner, Phys. Rev. '''106''', 364 (1957)</ref>
The consistency in these results became an important justification and motivated a very strong growth in [[theoretical physics]] in the late
==Applications==
==Application: RPA Ground State of an interacting bosonic system ==▼
The RPA vacuum <math>\left|\mathbf{RPA}\right\rangle</math> for a bosonic system can be expressed in terms of non-correlated bosonic vacuum <math>\left|\mathbf{MFT}\right\rangle</math> and original boson excitations <math>\mathbf{a}_{i}^{\dagger}</math>▼
{{off topic|date=July 2025}}
▲The RPA vacuum <math>\left|\
:<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>
The normalization can be calculated by
:<math>\langle
\mathrm{RPA}|\mathrm{RPA}\rangle=
\mathcal{N}^2 \langle \mathrm{MFT}|
Line 35 ⟶ 38:
<math>\tilde{\mathbf{q}}^{i}=(X^{\dagger})^{i}_{j}\mathbf{a}^{j}</math>
:<math>\mathcal{N}^{-2}=
\sum_{m_{i}}\sum_{n_{j}} \frac{(z_{i}/2)^{m_{i}}(z_{j}/2)^{n_{j}}}{m!n!}
\langle \mathrm{MFT}|
Line 44 ⟶ 47:
</math>
:<math>=\prod_{i}
\sum_{m_{i}} (z_{i}/2)^{2 m_{i}} \frac{(2 m_{i})!}{m_{i}!^2}=
</math>
:<math>
\prod_{i}\sum_{m_{i}} (z_{i})^{2 m_{i}} {1/2 \choose m_{i}}=\sqrt{\det(1-|Z|^2)}
</math>
Line 54 ⟶ 57:
the connection between new and old excitations is given by
:<math>\tilde{\mathbf{a}}_{i}=\left(\frac{1}{\sqrt{1-Z^2}}\right)_{ij}\mathbf{a}_{j}+
\left(\frac{1}{\sqrt{1-Z^2}}Z\right)_{ij}\mathbf{a}^{\dagger}_{j}</math>.
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