Migdal's theorem in heavy fermion systems

Wojciechowski, Ryszard

doi:10.1016/s0921-4526(98)00907-7

Cited by 5 publications

(9 citation statements)

References 7 publications

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“…. For the explicit calculation it is convenient to use x = q/2p f and z = mω/2p 2 f x to get for the structure factor (10) with (15)…”

Section: Pacs Numbersmentioning

confidence: 99%

“…Here and in the following we use V (q) = v(q)e 2 /4πǫ 0 . The vertex correction (18) with (15) leads to the part…”

Section: Pacs Numbersmentioning

confidence: 99%

“…Violations of this theorem appear if the magnon frequency becomes large 12,13 or non-adiabaticity leading eventually to a polaron collapse 14 . For heavy fermion systems the Migdal theorem is not valid 15 and near the magnetic boundary the quasiparticle spectra is different from the Eliashberg theory 16 which means the applicability of RPA calculations in high-T c superconductivity 17,18 is questionable.…”

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confidence: 99%

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Conditions where random phase approximation becomes exact in the high-density limit

Morawetz

Ashokan

Bala³

et al. 2018

Phys. Rev. B

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It is shown that in d-dimensional systems, the vertex corrections beyond the random phase approximation (RPA) or GW approximation scales with the power d − β − α of the Fermi momentum if the relation between Fermi energy and Fermi momentum is ǫ f ∼ p β f and the interacting potential possesses a momentum-power-law of ∼ p −α . The condition d − β − α < 0 specifies systems where RPA is exact in the high-density limit. The one-dimensional structure factor is found to be the interaction-free one in the high-density limit for contact interaction. A cancellation of RPA and vertex corrections render this result valid up to second-order in contact interaction. For finite-range potentials of cylindrical wires a large-scale cancellation appears and found to be independent of the width parameter of the wire. The proposed high-density expansion agrees with the Quantum Monte Carlo simulations. PACS numbers:The correlation energy in electron gases has been a topic of long-time investigations. In order to avoid divergences in perturbation theory already Macke 1 summed an infinite series of diagrams (RPA). Later Gell-Man and Bruckner show that the RPA at zero temperature becomes exact in the high-density limit 2,3 confirmed up to orders of the logarithm of density 4 . The corresponding momentum distributions in RPA have been computed already in 5,6 and recently an improved parametrization has been presented by cummulant expansions 7 . It confirms that the high-density limit is indeed given by the RPA calculation. The analytic expressions of the electron gas have been found 8,9 and an approximation bridging the low and high-density expansion of the correlation energy has been provided 10 .The Migdal theorem 11 contains a similar statement that for an electron-phonon coupling, higher-order vertex corrections vanish in orders of the ratio of the phonon frequency to the Fermi energy. Violations of this theorem appear if the magnon frequency becomes large 12,13 or non-adiabaticity leading eventually to a polaron collapse 14 . For heavy fermion systems the Migdal theorem is not valid 15 and near the magnetic boundary the quasiparticle spectra is different from the Eliashberg theory 16 which means the applicability of RPA calculations in high-T c superconductivity 17,18 is questionable.It is therefore desirable to have a simple criterion when vertex corrections vanish in the high-density limit. Here we provide an argument from simple scaling for interacting Fermi systems which shows that the power exponent of the interaction and the dimensionality of the system together with the exponent of the relation between Fermi energy and momentum, determines the expansion of the vertex corrections in terms of the Fermi momentum. First we drive an exact scaling law combining the dimensionality, the form of Fermi energy, and the momentum behavior of the potential into a condition when RPA is exact. As an application, we calculate the structure factor and pair correlation function for a wire of Fermions and compare the result with recent Quantum Mo...

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“…. For the explicit calculation it is convenient to use x = q/2p f and z = mω/2p 2 f x to get for the structure factor (10) with (15)…”

Section: Pacs Numbersmentioning

confidence: 99%

“…Here and in the following we use V (q) = v(q)e 2 /4πǫ 0 . The vertex correction (18) with (15) leads to the part…”

Section: Pacs Numbersmentioning

confidence: 99%

mentioning

confidence: 99%

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Conditions where random phase approximation becomes exact in the high-density limit

Morawetz

Ashokan

Bala³

et al. 2018

Phys. Rev. B

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“…Eliashberg equations including vertex corrections and an explicit dependence on k are also given in the literature [ 65 – 67 ]. These equations were derived in the context of research on the superconducting state in fullerene systems [ 68 – 69 ], in high- T C cuprates [ 70 – 72 ], in heavy fermion compounds [ 73 ], and in superconductors under high magnetic fields [ 74 ]. Unfortunately, due to enormous mathematical difficulties, their full self-consistent solutions are still unknown (Δ n , k and Z n , k ).…”

Section: Theoretical Modelmentioning

confidence: 99%

Nonadiabatic superconductivity in a Li-intercalated hexagonal boron nitride bilayer

Szewczyk¹,

Domagalska²,

Durajski³

et al. 2020

Beilstein J. Nanotechnol.

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When considering a Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the vertex corrections of electron–phonon interaction cannot be omitted. This is evidenced by the very high value of the ratio λωD/εF ≈ 0.46, where λ is the electron–phonon coupling constant, ωD is the Debye frequency, and εF represents the Fermi energy. Due to nonadiabatic effects, the phonon–induced superconducting state in Li-hBN is characterized by much lower values of the critical temperature (T LOVC C ∈ {19.1, 15.5, 11.8} K, for μ* ∈ {0.1, 0.14, 0.2}, respectively) than would result from calculations not taking this effect into account (T ME C∈ {31.9, 26.9, 21} K). From the technological point of view, the low value of T C limits the possible applications of Li-hBN. The calculations were carried out under the classic Migdal–Eliashberg formalism (ME) and the Eliashberg theory with lowest-order vertex corrections (LOVC). We show that the vertex corrections of higher order (λ3) lower the value of T LOVC C by a few percent.

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“…[8][9][10][11], should raise the question of this scheme's universal utility [12- (1) for Ω = 50 meV and different microscopic couplings, with the bare-band ε b k , quasiparticle band ε q k , and km(ω) path shown. 16], and generally whether the limits implied by such analysis do apply to all systems being measured [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Spectral function tour of electron-phonon coupling outside the Migdal limit

et al. 2011

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We simulate spectral functions for electron-phonon coupling in a filled band system -far from the asymptotic limit often assumed where the phonon energy is very small compared to the Fermi energy in a parabolic band and the Migdal theorem predicting (1+λ) quasiparticle renormalizations is valid. These spectral functions are examined over a wide range of parameter space through techniques often used in angle-resolved photoemission spectroscopy (ARPES). Analyzing over 1200 simulations we consider variations of the microscopic coupling strength, phonon energy and dimensionality for two models: a momentum-independent Holstein model, and momentum-dependent coupling to a breathing mode phonon. In this limit we find that any 'effective coupling', λ eff , inferred from the quasiparticle renormalizations differs from the microscopic dimensionless coupling characterizing these Hamiltonians, λ, and could drastically either over-or under-estimate it depending on the particular parameters and model. In contrast, we show that perturbation theory retains good predictive power for low coupling and small momenta, and that the momentum-dependence of the self-energy can be revealed via the relationship between velocity renormalization and quasiparticle strength. Additionally we find that (although not strictly valid) it is often possible to infer the selfenergy and bare electronic structure through a self-consistent Kramers-Kronig bare-band fitting; and also that through lineshape alone, when Lorentzian, it is possible to reliably extract the shape of the imaginary part of a momentum-dependent self-energy without reference to the bare-band. PACS numbers: 71.38.-k, 79.60.-i, 74.25.Jb arXiv:1102.3716v2 [cond-mat.str-el]

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Migdal's theorem in heavy fermion systems

Cited by 5 publications

References 7 publications

Conditions where random phase approximation becomes exact in the high-density limit

Conditions where random phase approximation becomes exact in the high-density limit

Nonadiabatic superconductivity in a Li-intercalated hexagonal boron nitride bilayer

Spectral function tour of electron-phonon coupling outside the Migdal limit

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