Random phase approximation

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.^[1][2][3] 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 accounts for the weak screened Coulomb interaction and is commonly used for describing the dynamic linear electronic response of electron systems.

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_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 ^[4] a 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 dielectric function,^[5][6] correctly predicts a number of properties of the electron gas, including plasmons.^[7]

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 diagrams in a dense electron gas.^[8]

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.

Application: RPA Ground State of an interacting bosonic system

The RPA vacuum $\left|\mathbf{RPA}\right\rangle$ for a bosonic system can be expressed in terms of non-correlated bosonic vacuum $\left|\mathbf{MFT}\right\rangle$ and original boson excitations $\mathbf{a}_{i}^{\dagger}$

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

where Z is a symmetric matrix with $|Z|\leq 1$ and

$\mathcal{N}= \frac{\left\langle \mathrm{MFT}\right|\left.\mathrm{RPA}\right\rangle}{\left\langle \mathrm{MFT}\right|\left.\mathrm{MFT}\right\rangle}$

The normalization can be calculated by

$\langle \mathrm{RPA}|\mathrm{RPA}\rangle= \mathcal{N}^2 \langle \mathrm{MFT}| \mathbf{e}^{z_{i}(\tilde{\mathbf{q}}_{i})^2/2} \mathbf{e}^{z_{j}(\tilde{\mathbf{q}}^{\dagger}_{j})^2/2} | \mathrm{MFT}\rangle=1$

where $Z_{ij}=(X^{\mathrm{t}})_{i}^{k} z_{k} X^{k}_{j}$ is the singular value decomposition of $Z_{ij}$. $\tilde{\mathbf{q}}^{i}=(X^{\dagger})^{i}_{j}\mathbf{a}^{j}$

$\mathcal{N}^{-2}= \sum_{m_{i}}\sum_{n_{j}} \frac{(z_{i}/2)^{m_{i}}(z_{j}/2)^{n_{j}}}{m!n!} \langle \mathrm{MFT}| \prod_{i\,j} (\tilde{\mathbf{q}}_{i})^{2 m_{i}} (\tilde{\mathbf{q}}^{\dagger}_{j})^{2 n_{j}} | \mathrm{MFT}\rangle$

$=\prod_{i} \sum_{m_{i}} (z_{i}/2)^{2 m_{i}} \frac{(2 m_{i})!}{m_{i}!^2}=$

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

the connection between new and old excitations is given by

$\tilde{\mathbf{a}}_{i}=\left(\frac{1}{\sqrt{1-Z^2}}\right)_{ij}\mathbf{a}_{j}+ \left(\frac{1}{\sqrt{1-Z^2}}Z\right)_{ij}\mathbf{a}^{\dagger}_{j}$.

References

1. ↑ D. Bohm and D. Pines: A Collective Description of Electron Interactions. I. Magnetic Interactions, Phys. Rev. 82, 625–634 (1951) (abstract)
2. ↑ 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) (abstract)
3. ↑ 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) (abstract)
4. ↑ H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959)
5. ↑ J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 28, 8 (1954)
6. ↑ N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, Toronto, 1976)
7. ↑ G. D. Mahan, Many-Particle Physics, 2nd ed. (Plenum Press, New York, 1990)
8. ↑ M. Gell-Mann, K.A. Brueckner, Phys. Rev. 106, 364 (1957)