Draft:Eliashberg theory

Eliashberg theory (also known as Nambu–Migdal–Eliashberg theory) is a theory of superconductivity which provides a microscopic justification for the BCS theory, and improves on it by including Coulomb interaction and effects that arise from the phonon dynamics. The theory can also be used to describe unconventional pairing mediated by exchange of bosons other than phonons.

History

The BCS theory was published in 1957 [something about the reception here]. However, there were also featured in the theory that were less than ideal. In its original form, BCS theory was based on finding the coefficients for the variational ansatz of the ground state. It soon turned out that in this form, the theory was difficult to extend to disordered and other systems. In 1959, Lev Gor'kov reformulated the theory in a quantum field theoretic approach by introducing the so-called anomalous Green's function to describe the superconducting correlations. He showed that Ginzburg–Landau theory, a theory which predated the BCS theory and explained many experiments, could be derived near the critical temperature from his reformulated BCS theory, both for clean superconductors and for superconducting alloys. Yoichiro Nambu introduced so-called Nambu spinors, which simplified the theory.

The electron-phonon interaction had been considered in the context of superconductivity since the early 1950s and lies in the heart of the BCS theory, albeit in an approximate way. Russian physicist Arkady Migdal's was the first to successfully apply the diagrammatic technique to this problem. According to the Migdal's theorem most of the diagrams can be neglected and the theory should be accurate to the order of (m/M)^1/2, where m is the mass of an electron, and M is the mass of the ion. Since the ions are much heavier than the electrons, neglecting those diagrams is often well justified.

The Eliashberg theory was first formulated by the Russian physicist Gerasim Eliashberg in two articles published in 1960. An important precursor for Eliashberg was also paper concerning electron-phonon interactions in the normal state. The same theory was formulated independently by the Japanese physicist Yoichiro Nambu, who showed that the BCS theory can be formulated in a gauge-invariant way.

The theory was soon extended by Philip W. Anderson...

Eliashberg equations

[Ummarino]

In the simplest case, we have two coupled equations: one for the superconducting order parameter and one for the quasiparticle weight:

$\Delta (i\omega _{n})Z(i\omega _{n})={\frac {\pi }{\beta }}\sum _{n'}{\frac {\Delta (i\omega _{n'})}{\sqrt {\omega _{n'}^{2}+\Delta ^{2}(i\omega _{n'})}}}\left[\lambda (i\omega _{n'}-i\omega _{n})-\mu ^{*}(\omega _{C})\right]\theta (\omega _{C}-|\omega _{n'}|)$ $Z(i\omega _{n})=1+{\frac {\pi }{\omega _{n}\beta }}\sum _{n'}{\frac {\omega _{n'}}{\sqrt {\omega _{n'}^{2}+\Delta ^{2}(i\omega _{n'})}}}\lambda (i\omega _{n'}-i\omega _{n})$

Coulomb pseudopotential

[Ummarino]

Short-range Coulomb interaction can be included.

$\mu ^{*}={\frac {\mu }{1+\mu \ln(E_{\rm {F}}/\omega _{D})}}$

Tunneling density of states

[Ummarino]

From the solution of the gap, the tunneling density of states can be solved.

$\left({\frac {dI}{dV}}\right)_{S}{\Big /}\left({\frac {dI}{dV}}\right)_{N}\propto \Re \left\{{\frac {|V|}{\sqrt {V^{2}-\Delta ^{2}(V)}}}\right\}$

Extensions

The Gor'kov-type equations of the Eliashberg theory can be subjected to quasiclassical approximation that simplifies the equations drastically. [50 years of BCS]

Although the original formulation of the theory describes pairing mediated by phonons, one can derive an analogous set of equations when the pairing is mediated by other bosonic modes. These modes are often no fully independent of the electrons that become superconducting, but are their collective excitations. There is no analog of the Migdal theorem, so unlike the phonon-mediated case, the theory is expected to be valid only in the weak-coupling limit. These interactions are also usually repulsive, and because of this the excepted pairing state is often not s-wave, but d-wave or some other state which has a lower symmetry than the s-wave.

According to one of the leading explanations for high-temperature superconductivity in cuprates, the pairing is mediated by spin fluctuations. This is usually described with an Eliashberg-type theory, in which the phonon propagator is replaced spin susceptibility.

In contrast to most other theories of superconductivity, Philip W. Anderson's resonating valence bond theory assumes that already the normal state is a non-Fermi liquid, and hence the resulting superconducting state is not described by an Eliashberg-type theory.

References

• Schrieffer, J. R. (1999). Theory of Superconductivity. CRC Press. doi:10.1201/9780429495700. ISBN 978-0-429-49570-0.
• Marsiglio, F. (2020). "Eliashberg theory: A short review". Annals of Physics. 417. Elsevier BV: 168102. doi:10.1016/j.aop.2020.168102. ISSN 0003-4916.
• Eliashberg, G. M. (1960). "Interactions between electrons and lattice vibrations in a superconductor" (PDF). Soviet Physics JETP. 11 (3): 696–702. (Zh. Eksp. Teor. Fiz., 38 (1960), p. 966)
• Eliashberg, G. M. (1961). "Temperature Green's function for electrons in a superconductor" (PDF). Soviet Physics JETP. 12 (5): 1000–1002. (Zh. Eksp. Teor. Fiz., 38 (1960), pp. 1437-1441)
• Migdal, A. B. (1958). "Interactions between electrons and lattice vibrations in a normal metal" (PDF). Soviet Physics JETP. 34 (7): 996–1001.
• Nambu, Yoichiro (1960-02-01). "Quasi-Particles and Gauge Invariance in the Theory of Superconductivity". Physical Review. 117 (3): 648–663. doi:10.1103/PhysRev.117.648. ISSN 0031-899X.
• Marsiglio, F.; Carbotte, J. P. (2008). "Electron-Phonon Superconductivity". Superconductivity. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 73–162. doi:10.1007/978-3-540-73253-2_3. ISBN 978-3-540-73252-5.
• Bogolinkov, N. N., V. V. Tolmachev, and D. V. Shirkov, 1959, A New Method in the Theory of Superconductivity (Consultants Bureau, New York)
• Morel, P.; Anderson, P. W. (1962-02-15). "Calculation of the Superconducting State Parameters with Retarded Electron-Phonon Interaction". Physical Review. 125 (4). American Physical Society (APS): 1263–1271. doi:10.1103/physrev.125.1263. ISSN 0031-899X.
• Allen, Philip B.; Mitrović, Božidar (1983). "Theory of Superconducting Tc". Solid State Physics. Elsevier. doi:10.1016/s0081-1947(08)60665-7. ISBN 978-0-12-607737-7. ISSN 0081-1947.
• Bardeen, John (1973). "Electron-Phonon Interactions and Superconductivity". Science. 181 (4106): 1209–1214. doi:10.1126/science.181.4106.1209. ISSN 0036-8075. JSTOR 1736977. PMID 17821583.
• Carbotte, J. P. (1990-10-01). "Properties of boson-exchange superconductors". Reviews of Modern Physics. 62 (4). American Physical Society (APS): 1027–1157. doi:10.1103/revmodphys.62.1027. ISSN 0034-6861.
• Scalapino, Douglas J. (1969). "The Electron-Phonon Interaction and Strong-Coupling Superconductors". In Parks, R.D. (ed.). Superconductivity. Taylor & Francis. ISBN 978-0-203-73796-5.
• Rainer, D.; Sauls, J. A. (1995). "Strong-coupling theory of superconductivity". In Butcher, P.N. (ed.). Superconductivity: From Basic Physics to the Latest Developments. World Scientific. arXiv:1809.05264. doi:10.1142/9789814503891_0002. ISBN 978-981-4503-89-1.