Migdal Approximation Research Articles

Development of a self-consistent approximation

A self-consistent approximation of a higher level than the standard self-consistent approximation, known in various fields of physics as the Migdal, Kraichnan or Born self-consistent approximation, is derived taking into account both the first and second terms of the series for the vertex function. In contrast to the standard approximation, the new self-consistent approximation is described by a system of two coupled nonlinear integral equations for the self-energy and the vertex function. In addition to all the diagrams with non-intersecting lines of correlation/interaction taken into account by the standard self-consistent approximation, the new approach takes into account in each term of the Green’s function expansion a significant number of diagrams with intersections of these lines. Because of this, the shape, linewidth, and amplitude of the resonance peaks of the dynamic susceptibility calculated in this approximation are much closer to the exact values of these characteristics. The advantage of the new self-consistent approach is demonstrated by the example of calculation of the dynamic susceptibility of waves in an inhomogeneous medium.

Interaction quench in the Holstein model: Thermalization crossover from electron- to phonon-dominated relaxation

We study the relaxation of the Holstein model after a sudden switch-on of the interaction by means of the nonequilibrium dynamical mean field theory, with the self-consistent Migdal approximation as an impurity solver. We show that there exists a qualitative change in the thermalization dynamics as the interaction is varied in the weak-coupling regime. On the weaker interaction side of this crossover, the phonon oscillations are damped more rapidly than the electron thermalization timescale, as determined from the relaxation of the electron momentum distribution function. On the stronger interaction side, the relaxation of the electrons becomes faster than the phonon damping. In this regime, despite long-lived phonon oscillations, a thermalized momentum distribution is realized temporarily. The origin of the thermalization crossover found here is traced back to different behaviors of the electron and phonon self-energies as a function of the electron-phonon coupling. In addition, the importance of the phonon dynamics is demonstrated by comparing the self-consistent Migdal results with those obtained with a simpler Hatree-Fock impurity solver that neglects the phonon self-energy. The latter scheme does not properly describe the evolution and thermalization of isolated electron-phonon systems.

Temperature dependence of the conductivity sum rule in the normal state due to inelastic scattering

We examine the temperature dependence of the optical sum rule in the normal state due to interactions. To be concrete we adopt a weak coupling approach which uses an electron-boson exchange model to describe inelastic scattering of the electrons with a boson, in the Migdal approximation. While a number of recent works attribute the temperature dependence in the normal state to that which arises in a Sommerfeld expansion, we show that in a wide parameter regime this contribution can be quite small. Instead, most of the temperature dependence arises from the zeroth order term in the `expansion', through the temperature dependence of the spectral function, and the interaction parameters contained therein. For low boson frequencies this circumstance causes a linear T-dependence in the sum rule. We develop some analytical expressions and understanding of the temperature dependence.

Phonon contribution to high- Tc superconductivity: Tunneling conductance and photoelectron dispersion relation

Quasi-particle dispersion relations were calculated within the Migdal approximation for the tunneling spectral function of the electron–phonon interaction obtained by us in slightly overdoped Bi 2Sr 2CaCu 2O 8. These relations were compared with the photoelectron results by Gweon et al. If a linear dispersion relation is assumed for a bare state, considerable discrepancies are observed. However, many similarities are noted with regard to local changes in the dispersion relations such as a kink, a discontinuity at the maximum phonon frequency, and an isotope shift in a single-phonon frequency range.

Self-consistent modification to the electron density of states due to electron-phonon coupling in metals

The "standard" theory of a normal metal consists of an effective electron band which interacts with phonons and impurities. The effects due to the electron-phonon interaction are often delineated within the Migdal approximation; the properties of many simple metals are reasonably well described with such a description. On the other hand, if the electron-phonon interaction is sufficiently strong, a polaron approach is more appropriate. The purpose of this paper is to examine to what degree the Migdal approximation is self-consistent, as the coupling strength increases. We find that changes in the electron density of states become significant for very large values of the coupling strength; however, there is no critical value, nor even a crossover regime where the Migdal approximation has become inconsistent. Moreover, the extent to which the electron band collapses is strongly dependent on the detailed characteristics of the phonon spectrum.

Symmetry of superconductivity inNH3K3C60superconductors: nonadiabatic effects in multiband systems

We have studied effects of nonadiabatic electron-intramolecular-vibration couplings on superconductivity in a multiband system of fullerides. The self-energy of the anomalous Green function was studied. Corrections to the Migdal approximation were taken into account for both the intraband and the interband couplings. While the nonadiabatic corrections to the intraband couplings favor anisotropic superconductivity, the corrections to the interband couplings favor isotropic superconductivity. This raises the theoretical possibility of the change of superconducting pairing symmetry close to the boundary between the superconducting and the antiferromagnetic phases of ${\mathrm{NH}}_{3}{\mathrm{K}}_{3}{\mathrm{C}}_{60}$ family of fullerides where lifting of the degeneracy of the ${t}_{1u}$ bands is expected.

Possible change of the superconducting symmetry in the vicinity of the SC–AF transition in NH3A3C60

We have studied vertex corrections to both the intraband and interband electron–phonon couplings in the doped C60. The corrections to the Migdal approximation was studied for the self-energy of the anomalous Green function. On the basis of our results, we suggest a possible change of the symmetry of the superconductivity in the vicinity of the phase boundary between the superconducting and the antiferromagnetic phases observed in the family of NH3A3C60 fullerides.

Test of final state approximations using threshold pp→ ppπ0

The Watson–Migdal approximation scheme to take into account final state interactions is shown to give the actual threshold momentum dependence of the reaction pp→ ppπ 0. However, by an explicit plane wave replacement of the final state wave function it is stressed that not too much physical significance should be given to the proportionality coefficient extracted using this procedure. The plane wave approximation is not physically reliable even after introducing the Watson–Migdal or a more sophisticated final state interaction factor, since direct production is missed. Also with short range interactions there can be discrepancies of a factor two.

Some consequences of the breakdown of Migdal's theorem in high- Tc superconductors

In order to reproduce the experimental T c and tunneling data of high- T c superconductors in the framework of the Eliashberg theory for strong electron–boson interaction, it is necessary to use large values of the coupling constant. In this work, we investigate whether this fact is a possible consequence of the breakdown of Migdal's theorem that certainly occurs in the cuprates due to their very low value of Fermi energy as compared to the characteristic phonon energy. In order to numerically quantify the consequences of the violation of the Migdal's approximation, we use the theoretical approach developed by Pietronero et al. [L. Pietronero, S. Strässler, C. Grimaldi, Phys. Rev. B 52, 10516 (1995)] and Grimaldi et al. [C. Grimaldi, L. Pietronero, S. Strässler, Phys. Rev. B 52, 10530 (1995)] on experimental data taken from the literature. The effects on the electron–phonon coupling constant λ, on the critical temperature T c and on the superconducting gap Δ are calculated also by means of a direct solution of the standard Eliashberg equations and discussed in different kinds of superconducting materials both with low and high critical temperature.

Resistivity Saturation Revisited: Results from a Dynamical Mean Field Theory

We use the dynamical mean field method to study the high-temperature resistivity of electrons strongly coupled to phonons. The results reproduce the qualitative behavior of the temperature and disorder dependence of the resistivity of ``A-15'' materials which is normally described in terms of saturation, but imply that the resistivity does not saturate. Rather, a change in temperature dependence occurs when the scattering becomes strong enough to cause a breakdown of the Migdal approximation.

Neutron - proton fsi in <sup>2</sup>H(d,pd) reaction at Energy 13,6 mev

The pd coincidence spectra of the d + d reaction are measured at deuteron beam energy E 0 = 13. 6 MeV. Particle emission ʋ angles ʋ p/ ʋ d = 39.3/39.3, 39.3/31.0, Ψ p- Ψ d =180 degrees have been taken to accent the neutron - proton (np) final state interaction (FSI) and in some degree to diminish influence of other mechanisms. The experimental results are compared to theoretical calculations using the Watson - Migdal approximation for FSI of np 3 S 1 and 1 S 0 (isospin forbidden) states. At symmetric angles the optimum fitting is obtained with contribution of a singlet state equal (8 ± 2)%, in nonsymmetrical geometry no influence of the singlet np FSI has been found.

Nonequilibrium linked cluster expansion for steady-state quantum transport

We generalize the linked cluster expansion of Matsubara Green functions into nonequilibrium situations. This allows us to compute nonequilibrium correlation functions and, consequently, physical observables for interacting quantum systems with intermediate interaction strengths. As a specific example, we study nonlinear dc transport in a resonant tunneling system with electron-phonon interaction. The first- and second-order nonequilibrium cluster approximations are explicitly evaluated. The results are compared with those obtained by the standard nonequilibrium Green functions approach within the Migdal approximation for the self-energy and in the low-density limit also with the known exact solution. For moderately strong electron-phonon interactions the first-order linked cluster approximation is more precise than the Migdal approach. The second-order approximation appears to be problematic in the presence of the Fermi sea.

Deformation, pairing and magnetic moments in rare-earth nuclei

Abstract The global behaviour of g -factors in the rare-earth region is examined. Magnetic moments of the 2 1 + states in even-even rare-earth nuclei with 60 ⩽ Z ⩽ 76 are calculated using the Migdal approximation for the moment of inertia. Attention is drawn to the implications of g -factors for the problem of identical bands, particularly to the application of the concept of F -spin multiplets. The magnetic properties of the heavy, transitional nuclei are discussed in terms of the possibility that the proton and neutron distributions in these soft nuclei might have different deformations.

Small parameter for Migdal-type description of t-t′-J model

The small parameter, validating Migdal's approximation for the spin-charge coupling vertex in the two-dimensional t-t′-J model close to half-filling, is found to be 1/ln (t/J), t ≫ J. An effective spin-charge separation, in the vicinity of the Fermilevel, reduces to zero the quasi-particle weight of electron states at half-filling. The calculated optical conductivity σ(ω) exhibits an “infrared” 1/ωtail with superimposed, multimagnon resonances.

Transition temperature and isotope effect near an extended van Hove singularity

Using Eliashberg theory and a model density for α2F the transition temperatureTcand the isotope effect α are calculated near an extended van Hove singularity. We show that, at least in the one-particle and the Migdal approximation, even the considered strong van Hove singularity cannot yield large enhancements ofTcand strong reductions of α of the kind observed in experiment around optimal doping.

From electron to small polaron: An exact cluster solution

We perform the exact diagonalization of two, four, and six vibrating molecules strongly coupled with one electron to show that a small polaron describes both adiabatic and nonadiabatic regimes. Electron spectra are derived and compared with the adiabatic Holstein approximation and with the Lang-Firsov canonical transformation in a wide range of the adiabatic parameter and the coupling strength. We show that the self-consistent Migdal approximation is violated at an intermediate value of the coupling constant \ensuremath{\lambda}\ensuremath{\simeq}1 and the adiabatic or nonadiabatic small polaron is the ground state for the intermediate and strong coupling \ensuremath{\lambda}g1. The phonon frequencies are calculated for a wide range of the coupling, including the intermediate region.

The spectral function of a one-dimensional Holstein polaron

The spectral function for a Holstein polaron in one dimension is calculated using exact diagonalization techniques, as a function of Einstein phonon frequency and electron-phonon coupling strength. We also use the self-consistent Migdal approximation to calculate the same spectrum; we find quantitative agreement for all frequencies at low coupling strengths. The ground state energy and quasiparticle residue evolve smoothly from band electron like to small polaron like. We compare with Migdal and small polaron approximations and find that neither is in quantitative agreement with the exact result at intermediate coupling strengths.

Enhancement, Reduction and Inversion of Isotope Effects Due to Anharmonic Phonons –Migdal-Eliashberg Theory for Superconductivity–

Superconducting transition temperatures of a many-electron system coupled strongly with anharmonic phonons have been studied, to clarify how drastically anharmonicity changes the isotope effect from the harmonic case. The diagonal self-energy of electron as well as the off-diagonal one is determined self-consistently within Migdal's approximation. Typical two types of anharmonicities are studied. The first is a sextic anharmonicity induced by a hard-core repulsion. Isotope shifts become almost zero in this case. When this repulsion is stronger, the shifts are reversed. The second is a small negative quartic anharmonicity in addition to the sextic one, just in the case of a mixture of a hard-core and a soft one. In this case, on the contrary, the shifts are greatly enhanced. Discussion is also given in relation to the observed anomalous isotope effects of newly discovered superconductors.

Eliashberg theory of the critical temperature and isotope effect. Dependence on bandwidth, band-filling, and direct Coulomb repulsion

We investigate the dependence of the superconducting critical temperature and the isotope coefficient on bandwidth, band-filling, and the direct Coulomb repulsion, within Eliashberg theory. The Migdal approximation is assumed throughout, and the Coulomb repulsion is modelled by the Hubbard U and treated in the simplest approximation. We assume a constant density of states with a finite bandwidth. We find that while, in principle, small isotope coefficients are possible, it is unlikely that the isotope coefficient can ever be negative within this model. Furthermore, it is difficult to achieve small isotope coefficients for realistic parameters. Finally, we discuss a possible means by which large isotope coefficients can occur at low filling.

The dilute Potts model on fractal lattices

The authors study the critical properties of the randomly bond-diluted q-state Potts ferromagnetic and antiferromagnetic models for a family of infinitely ramified exact fractals. Within Migdal's approximation they investigate the influence of bond dilution on the phase diagrams, and calculate the dependence of the critical exponents upon fractal geometries.