Random phase approximation: Difference between revisions

Revision as of 22:08, 1 September 2012

Revision as of 18:02, 12 October 2012

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[[File:Random_phase_approximation_ring_diagramsRandom phase approximation ring diagrams.png|thumb|400px|Ring diagrams that are summed up in order to obtain the RPA approximation]]

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>D. Bohm and D. Pines: ''A Collective Description of Electron Interactions. I. Magnetic Interactions'', Phys. Rev. '''82''', 625–634 (1951) ([http://prola.aps.org/abstract/PR/v82/i5/p625_1 abstract])</ref><ref>D. Pines and D. Bohm: ''A Collective Description of Electron Interactions: II. Collective vs Individual Particle Aspects of the Interactions'', Phys. Rev. '''85''', 338–353 (1952) ([http://prola.aps.org/abstract/PR/v85/i2/p338_1 abstract])</ref><ref>D. Bohm and D. Pines: ''A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas'', Phys. Rev. '''92''', 609–625 (1953) ([http://prola.aps.org/abstract/PR/v92/i3/p609_1 abstract])</ref> For decades physicists had been trying to incorporate the effect of microscopic quantum mechanical interactions between electrons in the theory of matter. Bohm and Pines RPA approximation accounts for the weak screened Coulomb interaction, and RPA is commonly used for describing the dynamic linear electronic response of electron systems.

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dynamic [[dielectric]] function

: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 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>.

The RPA was criticized in the late 50's for overcounting the degrees of freedom and the call for justification lead 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 50's and 60's.

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<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

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where <math>Z_{ij}=(X^{\mathrm{t}})_{i}^{k} z_{k} X^{k}_{j}</math> is the [[singular value decomposition]] of <math>Z_{ij}</math>.

<math>\tilde{\mathbf{q}}^{i}=(X^{\dagger})^{i}_{j}\mathbf{a}^{j}</math>

<math>\mathcal{N}^{-2}=

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\prod_{i}\sum_{m_{i}} (z_{i})^{2 m_{i}} {1/2 \choose m_{i}}=\sqrt{\det(1-|Z|^2)}

</math>

the connection between new and old excitations is given by

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[[Category:Condensed matter physics]]

{{condensed-matter-stub}}