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arXiv:2404.11820 · Physical review. B./Physical review. B

Applicability of Eliashberg theory for systems with electron-phonon and electron-electron interaction: A comparative analysis

Shang-Shun Zhang, Zachary M. Raines, Andrey V. Chubukov

Published Jun 24, 2024• 10 citations
arXivHTMLPDF

Abstract

The applicability of Eliashberg theory, for fermions interacting with fluctuations of a soft collective boson near a quantum-critical point (QCP) in two dimensions, remains a subject of debate. The velocity of the soft boson is comparable to the Fermi velocity. Hence, Migdal's justification of Eliashberg theory is formally not applicable. The authors argue here that some elements of Migdal's reasoning remain valid because the collective boson is overdamped, yet vertex corrections to Eliashberg theory are $O$(1). At the same time, they show that these corrections are numerically small, at least near an Ising-nematic QCP.

DOI
10.1103/physrevb.109.245132

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The applicability of Eliashberg theory, for fermions interacting with fluctuations of a soft collective boson near a quantum-critical point (QCP) in two dimensions, remains a subject of debate. The velocity of the soft boson is comparable to the Fermi velocity. Hence, Migdal's justification of Eliashberg theory is formally not applicable. The authors argue here that some elements of Migdal's reasoning remain valid because the collective boson is overdamped, yet vertex corrections to Eliashberg theory are OOO(1). At the same time, they show that these corrections are numerically small, at least near an Ising-nematic QCP.
We consider the problem of fermions interacting with gapless long-wavelength collective bosonic modes. The theory describes, among other cases, a ferromagnetic quantum-critical point (QCP) and a QCP towards nematic ordering. We construct a controllable expansion at the QCP in two steps: we first create a non-Fermi-liquid ``zero-order'' Eliashberg-type theory, and then demonstrate that the residual interaction effects are small. We prove that this approach is justified under two conditions: the interaction should be smaller than the fermionic bandwidth, and either the band mass mB{m}_{B}mB​ should be much smaller than m=kF∕vFm={k}_{F}∕{v}_{F}m=kF​∕vF​, or the number of fermionic flavors NNN should be large. For an SU(2) symmetric ferromagnetic QCP, we find that the Eliashberg theory itself includes a set of singular renormalizations which can be understood as a consequence of an effective long-range dynamic interaction between quasiparticles, generated by the Landau damping term. These singular renormalizations give rise to a negative nonanalytic q3∕2{q}^{3∕2}q3∕2 correction to the static spin susceptibility, and destroy a ferromagnetic QCP. We demonstrate that this effect can be understood in the framework of the ${\ensuremath{\phi}}^{4}$ theory of quantum criticality. We also show that the nonanalytic q3∕2{q}^{3∕2}q3∕2 correction to the bosonic propagator is specific to the SU(2) symmetric case. For systems with a scalar order parameter, the q3∕2{q}^{3∕2}q3∕2 contributions from individual diagrams cancel out in the full expression of the susceptibility, and the QCP remains stable.
We diagnose the stability of the Migdal-Eliashberg theory for a Fermi surface coupled to a gapless boson in 2+1 dimensions. We provide a scheme for diagonalizing the Bethe-Salpeter ladder when small-angle scattering mediated by the boson plays a dominant role. We found a large number of soft modes which correspond to shape fluctuations of the Fermi surface, and these shape deformations follow a diffusion-like dynamics on the Fermi surface. Surprisingly, the odd-parity deformations of a convex Fermi surface becomes unstable near the non-Fermi liquid regime of the Ising-Nematic quantum critical point and our finding calls for revisit of the Migdal-Eliashberg framework. The implication of the Bethe-Salpeter eigenvalues in transport will be discussed in the companion paper [H.Guo,arXiv:2311.03458].
It is widely believed that in the adiabatic limit the Fermi liquid state of an electron-phonon system, described by Migdal-Eliashberg theory, remains stable until the dressed phonon softens. Our variational and analytic analysis of the prototypical Holstein model shows that, in a wide range of fillings both in 3D and 2D, a polaronic/bipolaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bipolaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions retain Fermi liquid behavior, yet Luttinger's theorem is violated. At even larger couplings the density of states gradually approaches its form in the atomic limit.
It is widely believed that in the adiabatic limit the Fermi liquid state of an electron-phonon system, described by Migdal-Eliashberg theory, remains stable until the dressed phonon softens. Our variational and analytic analysis of the prototypical Holstein model shows that, in a wide range of fillings both in 3D and 2D, a polaronic/bipolaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bipolaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions retain Fermi liquid behavior, yet Luttinger's theorem is violated. At even larger couplings the density of states gradually approaches its form in the atomic limit.
We study the effects of finite temperature on normal state properties of a metal near a quantum critical point to an antiferromagnetic or Ising-nematic state.At T ¼ 0, bosonic and fermionic self-energies are traditionally computed within Eliashberg theory, and they obey scaling relations with characteristic power laws.Corrections to Eliashberg theory break these power laws but only at very small frequencies.Quantum Monte Carlo (QMC) simulations have shown that, already at much larger frequencies, there are strong systematic deviations from these predictions, casting doubt on the validity of the theoretical analysis.We extend Eliashberg theory to finite T and argue that in the T range accessible in the QMC simulations above the superconducting transition, the scaling forms for both fermionic and bosonic self-energies are quite different from those at T ¼ 0. We compare finite T results with QMC data and find good agreement for both systems.We argue that this agreement resolves the key apparent contradiction between the theory and the QMC simulations.
Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\ensuremath{\lambda}}$ divergence in the critical temperature Tc{T}_{c}Tc​ for strong electron-boson coupling $\ensuremath{\lambda}$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that ${\ensuremath{\lambda}}_{E}=\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\ll}1$, where ${\ensuremath{\omega}}_{D}$ is the Debye frequency and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy. Here we go beyond this restriction in the context of the two-dimensional Yukawa-SYK (Y-SYK) model, which is solvable for arbitrary values of ${\ensuremath{\lambda}}_{E}$. We find that ${T}_{c}\ensuremath{\approx}0.18\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}\sqrt{\ensuremath{\lambda}}$ for large $\ensuremath{\lambda}$, provided ${\ensuremath{\lambda}}_{E}$ remains small, and crosses over to a universal value of ${T}_{c}\ensuremath{\approx}0.04\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{F}$ for large ${\ensuremath{\lambda}}_{E}$. The saturation of Tc{T}_{c}Tc​ is due to a self-consistent account of the boson dynamics for large ${\ensuremath{\lambda}}_{E}$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\ensuremath{\lambda}$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\ensuremath{\lambda}}$ growth of Tc{T}_{c}Tc​ in Migdal-Eliashberg theory saturates to a universal value independent of $\ensuremath{\lambda}$ and ${\ensuremath{\omega}}_{D}$, providing an upper bound on the critical temperature at strong electron-boson coupling.