# 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  School of Physics and
Astronomy and William I. Fine Theoretical Physics Institute, University of
Minnesota, Minneapolis, MN 55455, USA

###### Abstract

We present a comparative analysis of the validity of Eliashberg theory for the
cases of fermions interacting with an Einstein phonon and with soft nematic
fluctuations near an Ising-nematic/Ising-ferromagnetic quantum-critical point
(QCP). In both cases, Eliashberg theory is obtained by neglecting vertex
corrections. For the phonon case, the reasoning to neglect vertex corrections
is the Migdal “fast electron/slow boson” argument because the phonon velocity
is much smaller than the Fermi velocity, $v_{F}$. The same argument allows one
to compute the fermionic self-energy within Eliashberg theory perturbatively
rather than self-consistently. For the nematic case, the velocity of a
collective boson is comparable to $v_{F}$ and this argument apparently does
not work. Nonetheless, we argue that while two-loop vertex corrections near a
nematic QCP are not small parametrically, they are small numerically. At the
same time, perturbative calculation of the fermionic self-energy can be
rigorously justified when the fermion-boson coupling is small compared to the
Fermi energy. Furthermore, we argue that for the electron-phonon case
Eliashberg theory breaks down at some distance from where the dressed Debye
frequency would vanish, while for the nematic case it holds all the way to a
QCP. From this perspective, Eliashberg theory for the nematic case actually
works better than for the electron-phonon case.

##  I Introduction

Migdal-Eliashberg (ME) theory has been developed to describe electron-phonon
interaction in the normal state and phonon-induced superconductivity [1, 2].
Over the years, it has been successfully applied to numerous electron-phonon
systems [3, 4, 5, 6, 7, 8]. The theory, as formulated by Eliashberg [2],
consists of a set of three coupled self-consistent one-loop equations for the
normal and anomalous fermionic self-energies and the phonon polarization.
These equations can be derived either diagrammatically or from the variational
Luttinger-Ward functional [9], assuming that vertex corrections can be
neglected. ME argued that vertex corrections are small in the ratio of the
Debye frequency $\omega_{D}$ to the Fermi energy $E_{F}$. The physical
reasoning, due to Migdal [1], is that in processes leading to vertex
corrections, a fermion is forced to vibrate at the phonon frequency, which for
small $\omega_{D}/E_{F}$ is far from its own resonance. In mathematical terms,
the strength of vertex corrections is governed by the dimensionless Eliashberg
parameter $\lambda_{E}=g^{2}/(\omega_{D}E_{F})$. This parameter is different
from the dimensionless coupling $\lambda=g^{2}/\omega^{2}_{D}$, which
determines mass renormalization:
$\lambda_{E}=\lambda(\omega_{D}/E_{F})\ll\lambda$. By this reasoning, the one-
loop theory remains under control not only at weak coupling, where
$\lambda<1$, but also at strong coupling, where $\lambda>1$ but
$\lambda_{E}<1$ (i.e., at $\omega_{D}<g<(\omega_{D}E_{F})^{1/2}$). Still, when
the Debye frequency softens, $\lambda_{E}$ eventually becomes large and the
one-loop description breaks down. At larger $\lambda_{E}$, higher loop
processes become relevant and eventually lead to polaronic description.

In recent years, a similar description has been applied to metals in which
electron-electron interactions (screened Coulomb repulsion) give rise to long-
range particle-hole order either in the spin or charge channel [10, 11, 12,
13, *finger_2001, 15, *Abanov2000, *acs2, *acn, 19, 20, 21, 22, 23, 24, 25,
26, *efetov2, *efetov3, 29, 30, 31, *ital2, *ital3, 34, 35, 36, *triplet2,
*triplet3, 39, 40, 41, *Monien1993, *Nayak1994, *Altshuler1994, *Kim_1994, 46,
47, *Metzner2003, 49, 50, 51, 52, 53, *Klein2019, *Klein2020, *Klein2022, 57,
58, 59, *Nosov2024]. Near the onset of such an order, i.e., near a quantum-
critical point (QCP), it is natural to assume that the low-energy physics is
described by an effective model with fermion-fermion interactions mediated by
soft fluctuations of the bosonic order parameter that condenses at the QCP.
The theory near a QCP is based on the same set of coupled equations that
Eliashberg obtained for the electron-phonon case and bears his name.

The validity of Eliashberg theory for a system near a QCP has been questioned,
however, on the grounds that the “fast fermion/slow boson” argument, used to
justify the neglect of vertex corrections, is not applicable anymore because
collective excitations are made out of fermions, and their velocity is of
order $v_{F}$ (see e.g., Ref. 61). The counter-argument [22] is that soft
collective excitations are Landau overdamped and for this reason do behave as
slow modes compared to fermions, i.e., Migdal’s reasoning is still valid,
albeit for a different reason.

The aim of this communication is to settle the issue of applicability of the
Eliashberg (one-loop) theory for a quantum-critical metal near a QCP. We
compare the criteria for applicability of the Eliashberg theory for fermions
interacting with a soft Einstein phonon with $\omega_{D}\ll E_{F}$, and with
soft nematic fluctuations near an Ising-nematic/Ising-ferromagnetic QCP. We
argue that the criteria for these two cases are somewhat similar, but not
identical.

Below we restrict our analysis to the normal state at $T=0$. This is a
putative ground state as the true one is a superconductor in both cases.
However, for spin-singlet pairing, the criterion for validity of the
Eliashberg theory for a superconductor is actually weaker than that for the
normal state as singular quantum corrections to the self-energy cancel in the
gap equation like contributions from non-magnetic impurities [18, 62, 63]. For
this reason, if ME theory is applicable in the putative normal state at $T=0$,
it is also applicable to a superconductor. Restriction to $T=0$ is essential
because thermal fluctuations at a finite $T$ do not fit into ME reasoning and
may destroy ME theory faster than quantum fluctuations (see Refs. 8, 64 for
more detail).

We also do not discuss here logarithmic singularities in the fermionic self-
energy in the Ising-nematic/Ising-ferromagnetic case that emerge at three-loop
and higher-orders [65, 66, 67, 68, 69]. Most likely, these corrections
generate an anomalous dimension for the fermionic propagator but the branch
cut in the Green’s function remains at the same place as in the Eliashberg
theory.

Finally, for the electron-phonon case, we do not discuss here the proposals
[70, 71, 72, 73] that the transition to a ground state with polarons may be
first order and happen already at $\lambda=O(1)$ due to singular behavior of
$n$-loop contributions to the self-energy with $n\gg 1$ (Ref. [73]). In this
communication we restrict with the analysis of the self-energy at the two-loop
order.

###  I.1 Summary of the results

The summary of our results is the following. As said above, for the electron-
phonon case, the smallness of two-loop vertex corrections is controlled by the
Eliashberg parameter
$\lambda_{E}=g^{2}/(\omega_{D}E_{F})=\lambda\omega_{D}/E_{F}$. This parameter
is small at weak coupling, when $\lambda<1$, but remains small also at
$\lambda>1$, as long as $\lambda<E_{F}/\omega_{D}$. At larger $\lambda$ vertex
corrections become parametrically large and Eliashberg theory breaks down.
This happens at a finite $\omega_{D}$, unless one takes the double limit
$E_{F}\to\infty$ and $\omega_{D}\to 0$ while maintaining
$g^{2}\ll\omega_{D}E_{F}$ [63, 74]. If $E_{F}$ is kept finite, as we do in
this paper, $\lambda_{E}=O(1)$ is reached at a finite $\omega_{D}$.

For the Ising-nematic/Ising-ferromagnetic case, vertex corrections are small
at weak coupling, when the corresponding $\lambda^{*}$, defined via
$\lambda^{*}=\lim_{\omega_{m}\to 0}\partial\Sigma/\partial\omega_{m}$, is
small. In the strong coupling regime $\lambda^{*}>1$, vertex corrections are
not small parametrically, however, they remain of order one and are small
numerically (of order $10^{-2}$). This behavior holds all the way up to the
QCP, where vertex corrections remain numerically small.

There is more. The two Eliashberg equations for the dynamical fermionic self-
energy and bosonic polarization, obtained from the Luttinger-Ward functional,
are coupled self-consistent one-loop equations. The one for the bosonic
polarization accounts for Landau damping of the bosons. For the electron-
phonon case the _same_ parameter $\lambda_{E}<1$ that justifies the neglect of
vertex corrections also allows one to simplify these equations in the
following way: neglect the Landau damping and replace the self-consistent one-
loop equation for the fermionic self-energy by the perturbative one-loop
formula, in which the self-energy is expressed in terms of the bare fermionic
propagator. This simplification holds because, even at strong coupling
($\lambda>1$), typical fermionic energies $\epsilon_{k}$ are of order
$\lambda_{E}E_{F}\ll E_{F}$, in which case integration over $\epsilon_{k}$ can
be extended to infinite limits, and $\int
d\epsilon_{k}/(i(\omega_{m}+\Sigma(\omega_{m}))-\epsilon_{k})=-i\pi\operatorname{sgn}\omega_{m}$,
like for free fermions. The ability to replace the self-consistent equation
for the self-energy by a perturbative one can be also expressed as a
consequence of the smallness of typical fermionic momenta transverse to the
Fermi surface compared to typical momenta along the Fermi surface
($\lambda_{E}k_{F}$ vs. $k_{F}$).

For the Ising-nematic/Ising-ferromagnetic case Landau damping is relevant but
the self-consistent one-loop Eliashberg equations can be still reduced to the
perturbative ones because, like in the phonon case, typical momenta transverse
to the Fermi surface are parametrically smaller than typical momenta along the
Fermi surface. The smallness holds in $\lambda^{*}_{E}$, which is the ratio of
the fermion-boson coupling and the Fermi energy. This ratio must be small, as
otherwise the low-energy description would not be valid. The value of
$\lambda^{*}_{E}$ does not depend on the distance to the QCP, and hence for
the Ising-nematic/Ising-ferromagnetic case the one-loop perturbation theory
remains valid even at the QCP.

The outcome of this analysis is that Eliashberg theory for the electron-phonon
case is rigorously justified for $\lambda_{E}<1$, even when the coupling
$\lambda>1$. By the same condition $\lambda_{E}<1$, the fermionic self-energy
can be computed in one-loop perturbation theory rather than self-consistently.
Because $\lambda_{E}$ is inversely proportional to $\omega_{D}$, the theory is
valid only at $\omega_{D}$ above some critical value.

For the Ising-nematic/Ising-ferromagnetic case, Eliashberg theory is not
rigorously justified at strong coupling, but vertex corrections remain $O(1)$
and are small numerically. At the same time, there exists another small
parameter $\lambda^{*}_{E}$ (the ratio of the interaction to the Fermi
energy), which justifies the computation of the fermionic self-energy in a
perturbative one-loop analysis rather than self-consistently. This parameter
remains small even at a QCP, i.e., Eliashberg theory can be applied to a
system at a QCP, if one is willing to neglect numerically small vertex
corrections.

From this last perspective, Eliashberg theory near a nematic QCP works better
than for fermions interacting with a soft Einstein phonon, despite that the
velocity of collective nematic fluctuations is of order $v_{F}$.

The structure of the paper is the following. In the next section we briefly
review Eliashberg theory for electron-phonon interaction [2, 7, 75] and
discuss the strength of one-loop vertex corrections. In Section III we discuss
fermions near a nematic QCP. In Section IV we compare the two cases and
present our conclusions.

##  II Eliashberg theory for electron-phonon system

For definiteness we consider interaction with an Einstein phonon. Eliashberg
theory for the normal state consists of the set of two equations for the
fermionic self-energy $\Sigma(k)$ and phonon polarization $\Pi(q)$, where
$k\equiv(\mathbf{k},\omega_{m})$ and $q\equiv(\mathbf{q},\Omega_{n})$. The
equations look simplest along the Matsubara axis, when
$\omega=\omega_{m}=(2m+1)\pi T$ and $\Omega=\Omega_{n}=2n\pi T$. The full
fermionic Green’s function and the full phonon propagator are related to
$\Sigma$ and $\Pi$ via

$G^{-1}(k)=G^{-1}_{0}(k)+i\Sigma(k),\ \chi^{-1}(q)=\chi^{-1}_{0}(q)+\Pi(q),$
(1)

where $G_{0}$ and $\chi_{0}$ are electron and phonon propagators in the
absence of interaction

$G_{0}(k)=\frac{1}{i\omega_{m}-\epsilon_{\mathbf{k}}},\quad\chi_{0}(q)=\frac{\chi_{0}}{\Omega_{n}^{2}+\omega^{2}_{D}}.$
(2)

To simplify calculations, we assume a parabolic dispersion,
$\epsilon_{\mathbf{k}}=(\mathbf{k}^{2}-k^{2}_{F})/(2m)$. Note that we define
the Matsubara self-energy as $i\Sigma(\omega_{m})$ (the ${\bf k}$-dependent
term in $\Sigma(k)$ is proportional to $\lambda_{E}$ and thus negligible at
small $\lambda_{E}$, see below), so that the inverse electron propagator can
be written
$G^{-1}_{0}=i(\omega_{m}+\Sigma(\omega_{m}))-\epsilon_{\mathbf{k}}$. At $T=0$,
which we consider here, both $\omega_{m}$ and $\Omega_{n}$ are continuous
variables.

The self-consistent equations for $\Sigma$ and $\Pi$ can be either derived
diagrammatically, see Fig. 1, or obtained as stationary solutions of the
Luttinger-Ward functional. In analytic form, they are

$\displaystyle\Sigma(k)=i\tilde{g}^{2}\int\frac{d^{2}\mathbf{q}\,d\Omega_{n}}{(2\pi)^{3}}G(k+q)\chi(q),$
(3)
$\displaystyle\Pi(q)=2\tilde{g}^{2}\int\frac{d^{2}\mathbf{k}\,d\omega_{m}}{(2\pi)^{3}}G(k+q)G(k)$
(4)

where $\tilde{g}$ is the electron-phonon coupling.

Figure 1: Self-consistent one-loop electron self-energy (left panel) and
polarization bubble for Einstein phonon (right panel). Solid lines indicate
the renormalized electronic Green’s function $G(k)$ and wavy lines the
renormalized Einstein phonon propagator $\chi(q)$.

There are three energy scales in these equations: the Debye frequency
$\omega_{D}$, the Fermi energy $E_{F}=k^{2}_{F}/(2m)$, and the dimension-full
effective interaction $g=(\tilde{g}^{2}N_{F}\chi_{0})^{1/2}$, where
$N_{F}=k_{F}/(2\pi v_{F})$ is the density of states at the Fermi level per
spin component. Out of these one can introduce two dimensionless ratios

$\lambda=\frac{g^{2}}{\omega^{2}_{D}},\quad\lambda_{E}=\frac{g^{2}}{E_{F}\omega_{D}}=\lambda\frac{\omega_{D}}{E_{F}}.$
(5)

Eliashberg theory is constructed under the assumptions that $E_{F}$ is the
largest energy scale and $\lambda_{E}\ll\lambda$. The ME argument is that the
strength of vertex corrections is determined by the smaller $\lambda_{E}$,
while fermionic mass renormalization is determined by the larger $\lambda$.
When both $\lambda$ and $\lambda_{E}$ are small, the theory falls into the
weak coupling limit with $G$ and $\chi$ close to their bare expressions. In
the regime $\lambda>1$, $\lambda_{E}<1$, mass renormalization is large and the
self-energy is larger than bare $\omega_{m}$ over a wide range of frequencies,
yet vertex corrections are still small.

We will be chiefly interested in the strong coupling regime $\lambda>1$,
$\lambda_{E}<1$. We go beyond previous work [8] and analyze vertex corrections
in 2D for all phonon momenta. We also compute the two loop self-energy with
vertex correction included. We show that the latter is small in $\lambda_{E}$,
and becomes $O(1)$ when $\lambda_{E}\sim 1$.

ME found [1, 2] that the same small parameter $\lambda_{E}$ allows one to
simplify calculations within Eliashberg theory and obtain a simple expression
for the self-energy,
$\Sigma^{(E)}(\omega_{m})=\lambda\omega_{D}\arctan\left(\omega_{m}/\omega_{D}\right)$
(Eq. 12 below), which depends only on frequency. Here we analyze the
corrections to this expression, both analytically and numerically. We show
that Eq. 12 can be rigorously justified only for frequencies $\omega_{m}\ll
E_{F}$. At $\omega_{m}\sim E_{F}$, the expression is more complex (Eq. 13
below). In particular, $\Sigma(k_{F},\omega_{m})$ becomes complex on the
Matsubara axis 111This does not violate Kramers-Kronig relations, but when
$\Sigma(k_{F},\omega_{m})$ is complex, both $\Sigma^{\prime}(k_{F},\omega)$
and $\Sigma^{\prime\prime}(k_{F},\omega)$ on the real axis have even and odd
components in $\omega$., similar to the self-energy in SYK-type models [76,
77].

We also show that the momentum-dependent part of the self-energy remains small
as long at $\lambda_{E}<1$.

We discuss the solution of the Eliashberg equations first and then use it to
analyze the strength of vertex corrections.

###  II.1 Solution of Eliashberg equations

The equations for $\Sigma(k)$ and $\Pi(q)$ are coupled and in principle have
to be solved together. We argue, however, that for $\lambda_{E}\ll 1$, the two
equations can be solved independently. To demonstrate this, we make two
assumptions and verify both a posteriori. First, we assume that
$\partial\Sigma/\partial\omega_{m}$ scales as $\lambda$ and is large at strong
coupling, while $\partial\Sigma(k)/\partial\epsilon_{k}$scales as
$\lambda_{E}$ and is small when $\lambda_{E}<1$. Accordingly, we approximate
$\Sigma(\mathbf{k},\omega_{m})$ by $\Sigma(\omega_{m})$. Second, we assume
that $\Sigma(\omega_{m})$ is parametrically smaller than $E_{F}$.

Consider Eq. 4 for $\Pi(q)$ first. We express
$\Pi(q)=\Pi(\mathbf{q},\Omega_{n})$ as a sum of static and dynamic pieces
$\Pi(\mathbf{q},0)+\delta\Pi(\mathbf{q},\Omega_{n})$. The static part
$\Pi(\mathbf{q},0)$ generally comes from fermions with energies of order
$E_{F}$. It renormalizes $\omega_{D}$ and $\chi_{0}$, and may also give rise
to a momentum dependence of the phonon propagator. For the purposes of this
study we assume that this renormalization is already included into
$\chi_{0}(q)$ and that $\chi_{0}(q)$ is given by Eq. 2 (for more discussions
on this issue see Refs. [78, 8]). The second, dynamical, term comes from
fermions with low energies. For this term we obtain, after integrating over
the angle between phonon $\mathbf{q}$ and fermionic $\mathbf{k}$:

$\delta\Pi(q)=-2i\tilde{g}^{2}\int\frac{d\omega_{m}}{2\pi}\int\frac{kdk}{2\pi}\frac{1}{i\tilde{\Sigma}(\omega_{m})-\epsilon_{k}}\left[\frac{\operatorname{sgn}(\omega_{m}+\Omega_{n})}{\sqrt{\frac{(kq)^{2}}{m^{2}}+\left({\tilde{\Sigma}}(\omega_{m}+\Omega_{n})+i\left(\epsilon_{k}+\frac{q^{2}}{2m}\right)\right)^{2}}}-{\text{the
same at}}~{}\Omega_{n}=0\right],$ (6)

where for brevity we have defined
$\tilde{\Sigma}(\omega_{m})=\omega_{m}+\Sigma(\omega_{m})$. For
$\omega_{m},\Omega_{n}\ll E_{F}$ and a generic $q\sim k_{F}$, the terms
$\tilde{\Sigma}(\omega_{m}+\Omega_{n})$ and $\epsilon_{k}$ under the square-
root in Eq. 6 are much smaller than $E_{F}$ and can be neglected compared to
the two $q-$dependent terms, which are of order $E_{F}$. Equation 6 then
simplifies to

$\delta\Pi(q)=-2i\tilde{g}^{2}N_{F}\int\frac{d\omega_{m}}{2\pi}\int\frac{d\epsilon_{k}}{i\tilde{\Sigma}(\omega_{m})-\epsilon_{k}}\frac{\operatorname{sgn}(\omega_{m}+\Omega_{n})-\operatorname{sgn}(\omega_{m})}{v_{F}q\sqrt{1-\left(\frac{q}{2k_{F}}\right)^{2}}}$
(7)

where $N_{F}$ is the density of states at the Fermi level. Integration over
$\epsilon_{k}$ can now be extended to infinite limits (up to terms of order
$\tilde{\Sigma}(\omega_{m})/E_{F}\ll 1$) and yields $\int
d\epsilon_{k}/(i\tilde{\Sigma}(\omega_{m})-\epsilon_{k})=-i\pi\operatorname{sgn}(\omega_{m})$.
Integrating then over frequency, we obtain

$\delta\Pi(q)=2\tilde{g}^{2}N_{F}\frac{\Omega_{n}}{v_{F}q\sqrt{1-\left(\frac{q}{2k_{F}}\right)^{2}}}.$
(8)

This expression is valid for $q<2k_{F}$, which is the largest momentum
transfer on the Fermi surface. At small $q$, it reduces to conventional Landau
damping. Substituting $\delta\Pi$ into Eq. 1 and using Eq. 2, we obtain

$\chi(\mathbf{q},\Omega_{n})=\frac{\chi_{0}}{\Omega^{2}_{m}+\omega^{2}_{D}+\gamma\frac{|\Omega_{n}|}{q\sqrt{1-\left(\frac{q}{2k_{F}}\right)^{2}}}},$
(9)

where $\gamma=2g^{2}/v_{F}$ and $g^{2}=\tilde{g}^{2}N_{F}\chi_{0}$.

Figure 2: (Color online) One-loop perturbative Eliashberg self-energy
$\Sigma^{(E)}(\omega_{m})$ at $\lambda\gg 1$, but $\lambda_{E}\ll 1$. The
self-energy is linear in $\omega_{m}$ for $\omega_{m}<\omega_{D}$ and
saturates at larger $\omega_{m}$. It is larger than the bare $\omega_{m}$ for
$\omega_{m}<\omega_{c}=(2/\pi)E_{F}\lambda_{E}$. For $\omega_{m}<\omega_{D}$,
the system is in the Fermi liquid regime. In between $\omega_{D}$ and
$\omega_{C}$, it displays strong coupling, non-Fermi liquid behavior. At
larger $\omega_{m}$ the self-energy is smaller than $\omega_{m}$ and the
system behaves as a Fermi gas.

It is convenient to measure $q$ in units of $2k_{F}$ and $\Omega_{n}$ in units
of $\omega_{D}$. Introducing $\bar{q}=q/(2k_{F})$ and
${\bar{\Omega}}_{m}=\Omega_{n}/\omega_{D}$, we re-express Eq. 9 as

$\chi(\mathbf{q},\Omega_{n})=\frac{\chi_{0}}{\omega^{2}_{D}}\frac{1}{1+{\bar{\Omega}}^{2}_{m}+\frac{\lambda_{E}}{2}\frac{|{\bar{\Omega}}_{m}|}{\bar{q}\sqrt{1-\bar{q}^{2}}}}$
(10)

We see that the Landau damping term contains $\lambda_{E}$ in the prefactor
and is therefore small compared to one of the two other terms for all values
of ${\bar{\Omega}}_{m}$ provided $\bar{q}$ is not close to either zero or one.

We now substitute $\chi(\mathbf{q},\Omega_{n})$ from Eq. 10 into Eq. 3 for the
self-energy. We compute separately $\Sigma(\mathbf{k}_{F},\omega_{m})$ and
$\Sigma(\mathbf{k},0)$. The calculation of $\Sigma(\mathbf{k}_{F},\omega_{m})$
parallels the one for $\delta\Pi$: we first integrate over the angle between
$\mathbf{k}_{F}$ and $\mathbf{q}$ and obtain

$\Sigma(\omega_{m})=\frac{g^{2}}{N_{F}}\int\frac{d\Omega_{n}}{2\pi}\int_{0}^{2k_{F}}\frac{qdq}{2\pi}\frac{1}{\Omega^{2}_{m}+\omega^{2}_{D}+\gamma\frac{|\Omega_{n}|}{q\sqrt{1-(q/2k_{F})^{2}}}}~{}\frac{\operatorname{sgn}(\omega_{m}+\Omega_{n})}{\sqrt{(v_{F}q)^{2}+\left(\tilde{\Sigma}(\omega_{m}+\Omega_{n})+i\frac{q^{2}}{2m}\right)^{2}}}.$
(11)

We set $\omega_{m}\ll E_{F}$ and assume and verify a posteriori that (i)
typical $q$ are of order $k_{F}$ and typical $\Omega_{n}$ are of order
$\omega_{m}$ and (ii) $\tilde{\Sigma}(\omega_{m}+\Omega_{n})$ is
parametrically smaller than $E_{F}$. We then evaluate the integrals over $q$
and $\Omega_{q}$ neglecting the Landau damping term and
${\tilde{\Sigma}}(\omega_{m}+\Omega_{n})$ under the square root. The
calculation is elementary, and we obtain

$\Sigma^{(E)}(\omega_{m})=\lambda\omega_{D}\arctan\left(\frac{\omega_{m}}{\omega_{D}}\right),$
(12)

where the index $E$ denotes that this is the leading self-energy in Eliashberg
theory. The self-energy behaves as $\lambda\omega_{m}$ at small frequencies
and saturates at $(\pi/2)\lambda\omega_{D}$ at higher frequencies. We
emphasize that the right-hand side of Eq. 12 does not depend on the self-
energy and has the same form as if we used the Green’s function for free
fermions. The same holds for $\delta\Pi(q)$. In other words, to this accuracy,
the one-loop self-consistent theory becomes equivalent to the one-loop
perturbation theory. We plot $\Sigma^{(E)}(\omega_{m})$ in Fig. 2.

We next check the accuracy of approximations used to obtain Eq. 12. The
typical $q$ for the momentum integral are of order $k_{F}$ and typical
$\Omega_{n}$ for the frequency integral are of order $\omega_{m}$, as we
assumed. Further, the prefactor $\lambda\omega_{D}$ in Eq. 12 can be
equivalently expressed as $\lambda_{E}E_{F}$. We see that for small
$\lambda_{E}$, $\tilde{\Sigma}(\omega_{m}+\Omega_{n})\ll E_{F}$. This
justifies neglecting the self-energy term in Eq. 11 and in the calculation of
$\delta\Pi(q)$. Also, the self-energy $\Sigma^{(E)}(\omega_{m})$ exceeds the
bare $\omega_{m}$ at small $\omega_{m}$, up to
$\omega_{c}=(\pi/2)\lambda\omega_{D}=(\pi/2)\lambda_{E}E_{F}$. As long as
$\lambda_{E}$ is small, $\omega_{c}$ is parametrically smaller than $E_{F}$.

###  II.2 Subleading terms in the self-energy

We now go beyond the leading term and estimate the subleading terms in the
self-energy. They are the corrections to the dynamic piece of the self-energy,
arising from (i) keeping either the Landau damping term or (ii) keeping
$\tilde{\Sigma}$ in right-hand-side of Eq. 11, and (iii) the correction from
the static piece of the self-energy $\Sigma(\mathbf{k},0)$.

Keeping the Landau damping term in Eq. 11 leads to a relative correction
$(1+X(\omega_{m}))$, which is the largest at $\omega_{m}\sim\omega_{D}$, where
$X\sim\lambda_{E}\log^{2}\lambda_{E}$. This is a small correction when
$\lambda_{E}\ll 1$. The leading correction from keeping
$\tilde{\Sigma}(\omega_{m}+\Omega_{n})$ in the integrand in Eq. 11 comes from
$q\approx 2k_{F}$. This correction, which we label as
$\delta\Sigma(\omega_{m})$, is a bit tricky. First, there is a non-zero
$\delta\Sigma(0)$. This contribution is purely imaginary in our notation of
Matsubara self-energy as $i\Sigma(\omega_{m})$, i.e., it is the real part of
the actual self-energy. This term is often absorbed into the renormalization
of the bare chemical potential. This is acceptable for the low-energy
description, but we emphasize that
$\operatorname{Im}[\delta\Sigma(\omega_{m})]$ generally depends on frequency.
For this reason we don’t subtract $\delta\Sigma(0)$ from
$\delta\Sigma(\omega_{m})$. Second, at $|\omega_{m}|<\omega_{D}$, the real
part of $\delta\Sigma(\omega_{m})$ is linear in $\omega_{m}$, like
$\Sigma^{(E)}(\omega_{m})$, but with the prefactor
$\lambda\sqrt{\lambda_{E}}$, which is smaller than that in Eq. 12 by
$\sqrt{\lambda_{E}}$. For $\omega_{D}<|\omega_{m}|<E_{F}$, this correction
becomes

$\delta\Sigma(\omega_{m})=-\omega_{c}\left(\frac{|\tilde{\Sigma}^{(E)}(\omega_{m})|}{E_{F}}\right)^{1/2}\frac{1+i\operatorname{sgn}\omega_{m}}{\pi}.$
(13)

Substituting $|\tilde{\Sigma}^{(E)}(\omega_{m})|=|\omega_{m}|+\omega_{c}$, we
find that for $|\omega_{m}|<\omega_{c}$ $\delta\Sigma(\omega_{m})$ is smaller
than $\Sigma^{(E)}$ by $(\omega_{c}/E_{F})^{1/2}\sim(\lambda_{E})^{1/2}$. At
larger frequencies, $\delta\Sigma$ increases and at $\omega_{m}\sim E_{F}$
becomes of the same order as $\Sigma^{(E)}(\omega_{m})$. We emphasize that
this holds even when $\lambda_{E}$ is small. At even larger
$|\omega_{m}|>E_{F}$, the full self-energy has to be re-evaluated. We will not
analyze this frequency range here.

We note that $\delta\Sigma(\omega_{m})$ is a complex function of Matsubara
frequency. Similar behavior of $\Sigma(\omega_{m})$ has been previously
reported in SYK-type systems [76, 77]. In our case the appearance of a complex
$\Sigma(\omega_{m})$ reflects the absence of particle-hole symmetry for a
parabolic dispersion.

The computation of the static piece $\Sigma(\mathbf{k},0)$ proceeds in the
same way. After angular integration, the left-hand side of Eq. 3 becomes

$\Sigma(\mathbf{k},0)=\frac{g^{2}}{N_{F}}\int\frac{d\Omega_{n}}{2\pi}\int_{0}^{2k_{F}}\frac{qdq}{2\pi}\frac{1}{\Omega^{2}_{m}+\omega^{2}_{D}+\gamma\frac{|\Omega_{n}|}{q\sqrt{1-(q/2k_{F})^{2}}}}\frac{\operatorname{sgn}(\Omega_{n})}{\sqrt{(v_{F}q)^{2}+\left(\tilde{\Sigma}(\Omega_{n})+i\left(\epsilon_{k}+\frac{q^{2}}{2m}\right)\right)^{2}}}.$
(14)

Expanding in $\epsilon_{k}$, we obtain

$i\Sigma(\mathbf{k},0)=\epsilon_{k}\frac{2g^{2}}{N_{F}}\int\frac{d\Omega_{n}}{2\pi}\int_{0}^{2k_{F}}\frac{qdq}{2\pi}\frac{1}{\Omega^{2}_{m}+\omega^{2}_{D}+\gamma\frac{|\Omega_{n}|}{q\sqrt{1-(q/2k_{F})^{2}}}}\frac{|\tilde{\Sigma}(\Omega_{n})|}{\left({\tilde{\Sigma}}^{2}(\Omega_{n})+(v_{F}q)^{2}(1-(q/(2k_{F})^{2}))\right)^{3/2}}.$
(15)

The largest contributions to the momentum integral comes from small $q$ and
from $q\approx 2k_{F}$. Evaluating the contributions from these two regions
and performing the subsequent integration over frequency, we obtain

$i\Sigma(\mathbf{k},0)=\lambda_{E}\epsilon_{k}.$ (16)

As a consequence, the $\epsilon_{k}$ term in the bare Green’s function gets
multiplied by the factor $1-\lambda_{E}$. This renormalization is small when
$\lambda_{E}$ is small.

Summarizing, we find that the condition $\lambda_{E}\ll 1$ allows us to (i)
neglect the Landau damping term in the bosonic propagator, (ii) neglect the
momentum dependence of the self-energy, and (iii) approximate
$\Sigma(\omega_{m})$ by the one-loop perturbative result Eq. 12. For the
latter, the corrections, which make $\Sigma(\omega_{m})$ complex, are
parametrically small as long as $\omega_{m}$ remains below
$\omega_{c}\sim\lambda_{E}E_{F}$. At larger frequencies, the corrections get
stronger, and at $\omega_{m}\sim E_{F}$ become of the same order as
$\Sigma^{(E)}(\omega_{m})$. To the best of our knowledge, this last result has
not been presented earlier.

The smallness of corrections to the one-loop perturbative
$\Sigma^{(E)}(\omega_{m})$ can be understood by comparing relative energy
scales and invoking the argument about slow phonons and fast electrons, which
typically is reserved for vertex corrections. Indeed, the argument implies
that for the same frequency, fermionic momenta are far smaller than phonon
momenta. In our case, typical phonon momenta are of order $k_{F}$, while
typical fermionic momenta are of order $\tilde{\Sigma}(\omega_{m})/v_{F}$. For
$|\omega_{m}|<\omega_{c}$, where Eq. 12 for $\Sigma^{(E)}(\omega_{m})$ is
rigorously justified, typical fermionic momenta are smaller by $\lambda_{E}$.
In practical terms, this separation allows one to approximate $\int d^{2}k$ in
the Eliashberg formula for the self-energy for a generic $\chi(q)$ by $2\pi
N_{F}\int d\epsilon_{k^{\prime}}d\theta$, where
$\mathbf{k}^{\prime}=\mathbf{k}+\mathbf{q}$ and $\theta$ is the angle between
$\mathbf{k}$ and $\mathbf{k}^{\prime}$, with both set on the Fermi surface.
Eq. 3 then reduces to

$\Sigma(\mathbf{k},\omega_{m})=\Sigma(\omega_{m})=i\tilde{g}^{2}N_{F}\int{d\Omega_{n}\over
2\pi}\chi_{L}(\Omega_{n})\int{d\epsilon_{\mathbf{k}\prime}}G(k^{\prime},\omega_{m}+\Omega_{n})=g^{2}\int_{0}^{\omega_{m}}d\Omega_{n}\chi_{L}(\Omega_{n}),$
(17)

where we have defined $\chi_{L}(\Omega_{m})=(1/2\pi)\int
d\theta\chi(\Omega_{n},\theta)$. For our choice of phonon propagator $\chi(q)$
this reduces to the Eliashberg result Eq. 12. On the real frequency axis, the
same consideration yields
$\Sigma^{{}^{\prime\prime}}(\omega)=-g^{2}\int_{0}^{\omega}\chi^{{}^{\prime\prime}}_{L}(\Omega)d\Omega$.
In the literature, $g^{2}\chi^{{}^{\prime\prime}}_{L}(\Omega)$ is often
denoted $\alpha^{2}F(\Omega)$ [7, 79].

This analysis is supported by numerical solution of the Eliashberg equations.
The numerical results are obtained by solving the self-consistent equations
iteratively, where the momentum integrations are carried out without making
extra approximations. We found Fermi liquid behavior of $\Sigma(\omega_{m})$
at frequencies below $\omega_{D}$. The self-energy then saturates for an
intermediate frequency range up to $\omega_{c}$, and eventually decreases at
larger frequencies. We see that the imaginary part of $\Sigma(\omega_{m})$
remains weakly frequency dependent up to $\omega_{m}\sim E_{F}$. The
implication is that in the whole range $|\omega_{m}|\leq E_{F}$ it can be
absorbed into the renormalization of the chemical potential. The difference
between $\operatorname{Re}[\Sigma(\omega_{m})]$ and the
$\Sigma^{(E)}(\omega_{m})$ is shown in Fig. 3 (b). It matches
$\delta\Sigma(\omega_{m})$ from Eq. Eq. 13. In particular, we verified in Fig.
3 (c) that for $\omega_{m}>\omega_{D}$,
$\operatorname{Re}[\Sigma(\omega_{m})]-\Sigma^{(E)}(\omega_{m})$ scales as
$\lambda^{1/2}_{E}$, in agreement with Eq. 13.

Figure 3: (Color online) One-loop electron self-energy. We set
$E_{F}=500\omega_{D}$, $\lambda_{E}\simeq 0.06$ and $\lambda\simeq 30$. (a)
Numerical solution of the self-consistent equation Eq. 3 v.s. the perturbative
expression, $\Sigma^{(E)}(\omega_{m})$, Eq. 12. The numerical results obtained
with and without Landau damping are red and blue curves, respectively. The
right panel in (a) displays the self-energy at larger frequencies, comparable
to $E_{F}$. Both $\operatorname{Re}[\Sigma(\omega_{m})]$ and
$\operatorname{Im}[\Sigma(\omega_{m})]$ evolve with frequency
($\operatorname{Im}[\Sigma(\omega_{m})$ is a constant at smaller $\omega_{m}$
and is not shown in the left panel). (b) A comparison between
$\Sigma^{(E)}(\omega_{m})-\Sigma(\omega_{m})$ and $\delta\Sigma(\omega_{m})$
from Eq. 13. (c) The verification of the scaling relation
$(\Sigma^{(E)}(\omega_{m})-\Sigma(\omega_{m}))/\omega_{D}\propto\sqrt{\lambda_{E}}$
for a given $\omega_{m}/\omega_{D}\simeq 9$.

###  II.3 Vertex corrections

Figure 4: Relevant diagrams: (a) one-loop vertex correction, (b) two-loop
electron’s self-energy with vertex correction included, (c) ladder series for
the vertex.

The one-loop vertex correction is shown graphically in Fig. 4 (a) and the
associated two-loop self-energy with vertex renormalization is shown in Fig. 4
(b). The relative strength of the two-loop self energy compared to the one-
loop one determines the validity of the Eliashberg theory. The one-loop vertex
correction and the two-loop self-energy in 2D have been discussed previously
in Ref.[ 8]. Here we reproduce these earlier results and add additional
details.

For external fermionic $k=(\mathbf{k}_{F},\omega_{m})$ and phononic
$q=(\mathbf{q},\Omega_{n})$, the analytical expression for the one-loop
$\Delta g/g$ is

$\frac{\Delta
g(k,q)}{g}=-i\frac{g^{2}}{2\pi}\int\frac{d\omega^{\prime}_{m}}{(\omega_{m}-\omega^{\prime}_{m})^{2}+\omega_{D}^{2}}\int\frac{d\epsilon_{k^{\prime}}}{i\tilde{\Sigma}(\omega^{\prime}_{m})-\epsilon_{k^{\prime}}}\\\
\times\frac{\operatorname{sgn}(\omega^{\prime}_{m}+\Omega_{n})}{\sqrt{\left(v_{F}\left|\mathbf{q}\right|\right)^{2}+\left(\tilde{\Sigma}(\omega^{\prime}_{m}+\Omega_{n})+i\left(\epsilon_{k^{\prime}}+\mathbf{q}^{2}/2m\right)\right)^{2}}}.$
(18)

For $\mathbf{q}=0$, the vertex correction _is not small_ for any $\lambda_{E}$
as it must satisfy the Ward identity associated with the conservation of the
total density:

$\frac{g_{\text{full}}(k,q)}{g}=\frac{g_{\text{full}}(\omega_{m},\Omega_{m},\mathbf{q}=0)}{g}=\frac{\tilde{\Sigma}(\omega_{m}+\Omega_{n})-\tilde{\Sigma}(\omega_{m})}{\Omega_{n}}.$
(19)

At small $\omega_{m}$,
$\tilde{\Sigma}(\omega_{m})\approx\omega_{m}(1+\lambda)$, hence by the Ward
identity $g_{\text{full}}(k,\Omega_{m},\mathbf{q}=0)/g=\lambda\gg 1$.

Putting $\mathbf{q}=0$ in Eq. 18 and setting $\Omega_{n}>0$ for definiteness,
we perform the $\epsilon_{k^{\prime}}$ integration to obtain the one-loop
vertex correction

$\frac{\Delta
g(\omega_{m},\Omega_{m},\mathbf{q}=0)}{g}=g^{2}\int_{-\Omega_{n}}^{0}\frac{d\omega^{\prime}_{m}}{(\omega_{m}-\omega^{\prime}_{m})^{2}+\omega_{D}^{2}}\\\
\times\frac{1}{\tilde{\Sigma}(\omega^{\prime}_{m}+\Omega_{n})-\tilde{\Sigma}(\omega^{\prime}_{m})}.$
(20)

For small $\omega_{m},\Omega_{n}<\omega_{D}$, this gives $\Delta
g(k,\Omega_{m},\mathbf{q}=0)/g=\lambda/(1+\lambda)$, which approaches $1$ in
the strong coupling limit $\lambda\gg 1$. Because $\Delta
g(k,\Omega_{m},\mathbf{q}=0)/g\approx 1$, we need to include higher-order
vertex corrections. A simple experimentation shows that at large $\lambda$,
relevant diagrams for the full vertex
$g_{\text{full}}(k,\Omega_{m},\mathbf{q}=0)$ form ladder series, shown in Fig.
4. The corresponding integral equation at arbitrary $\omega_{m}$ and
$\Omega_{n}$ has the form

$\frac{g_{\text{full}}(\omega_{m},\Omega_{m},\mathbf{q}=0)}{g}=1+g^{2}\int_{-\Omega_{n}}^{0}\frac{d\omega^{\prime}_{m}}{(\omega_{m}-\omega^{\prime}_{m})^{2}+\omega_{D}^{2}}\\\
\times\frac{g_{\text{full}}(\omega^{\prime}_{m},\omega_{m},\mathbf{q}=0)/g}{\tilde{\Sigma}(\omega^{\prime}_{m}+\Omega_{n})-\tilde{\Sigma}(\omega^{\prime}_{m})}.$
(21)

One can straightforwardly check that the solution of this equation is given by
Eq. 19, in agreement with the Ward identity associated with the conservation
of the total density.

For $q\sim k_{F}$ the result is different. Integrating over
$\epsilon_{k^{\prime}}$ in Eq. 18 in infinite limits we now obtain

$\frac{\Delta g(k,q)}{g}=\frac{g^{2}}{2}\int
d\omega^{\prime}_{m}\frac{1-\operatorname{sgn}(\omega^{\prime}_{m}+\Omega_{n})\operatorname{sgn}(\omega^{\prime}_{m})}{(\omega_{m}-\omega^{\prime}_{m})^{2}+\omega_{D}^{2}}\frac{1}{\sqrt{(v_{F}q)^{2}+\left(\tilde{\Sigma}(\omega^{\prime}_{m}+\Omega_{n})-\tilde{\Sigma}(\omega^{\prime}_{m})+iq^{2}/(2m)\right)^{2}}}\\\
\approx\frac{g^{2}}{4E_{F}{\bar{q}}\sqrt{1-{\bar{q}}^{2}}}\int
d\omega^{\prime}_{m}\frac{1-\operatorname{sgn}(\omega^{\prime}_{m}+\Omega_{n})\operatorname{sgn}(\omega^{\prime}_{m})}{(\omega_{m}-\omega^{\prime}_{m})^{2}+\omega_{D}^{2}}$
(22)

where ${\bar{q}}=|\mathbf{q}|/(2k_{F})$. Evaluating the remaining frequency
integral, we find

$\frac{\Delta
g(k,q)}{g}=\frac{\lambda_{E}}{4{\bar{q}}}\frac{1}{\sqrt{1-{\bar{q}}^{2}}}\left(\arctan\left({\omega_{m}+\Omega_{n}\over\omega_{D}}\right)-\arctan\left({\omega_{m}\over\omega_{D}}\right)\right).$
(23)

We see that for generic $\omega_{m}\sim\Omega_{n}\sim\omega_{D}$, $\Delta
g(k,q)/g\sim\lambda_{E}$, as long as ${\bar{q}}$ is not too close to either
$0$ or $1$.

There is a caveat here: the vertex correction in Eq. 23 decreases at
$\omega_{m},\Omega_{n}>\omega_{D}$. On a more careful look, we found that this
is an artifact of integrating over $\epsilon_{k^{\prime}}$ in Eq. 18 in
infinite limits instead of introducing the lower cutoff at $-E_{F}$. Repeating
the integration with this lower cutoff we find that even at
$\omega_{m},\Omega_{n}>\omega_{D}$, the vertex correction remains of the form

$\frac{\Delta
g(k,q)}{g}=C(q)\frac{\lambda_{E}}{4}\frac{1}{{\bar{q}}\sqrt{1-{\bar{q}}^{2}}},$
(24)

where $C(q)=O(1)$.

We plot the one-loop vertex correction as a function of $q$ and
$\Omega_{n}\sim\omega_{D}$ in Fig. 5. We show the results obtained using both
an infinite energy cutoff ($-\infty<\epsilon_{\mathbf{k}}<\infty$) and a
finite energy cutoff ($-E_{F}<\epsilon_{\mathbf{k}}<\infty$). The two results
for $\Delta g/g$ do differ, most notably at $|\mathbf{q}|\sim k_{F}$, but both
remain of order $\lambda_{E}$ for a generic $q$.

Figure 5: (Color online) Numerical results for $\Delta g/g$ as a function of
bosonic frequency $\Omega_{n}$ and momentum $q$ for $\lambda=30$,
$\lambda_{E}=0.06$, and external fermionic frequency
$\omega_{m}=0.1\omega_{D}$. Panels (a) and (b): the results of the
calculations using infinite cutoff in the integration over
dispersion,$-\infty<\epsilon_{\mathbf{k}}<\infty$, and a finite cutoff
$-E_{F}<\epsilon_{\mathbf{k}}<\infty$, respectively.

We next substitute $\Delta g(k,q)$ into the two-loop diagram for the self-
energy, Fig. 4 (b). We label this contribution as $\Sigma^{(2)}(\omega_{m})$.
Analyzing the integral over phononic $q$, we find that it is singular at $q$
close to $0$ and $2k_{F}$, where the vertex correction is enhanced, but the
singularity is only logarithmic. Evaluating the full integral, we find, in
agreement with Ref. [ 8],

$\Sigma^{(2)}(\omega_{m})\sim\left(\lambda_{E}|\log{\lambda_{E}}|\right)\Sigma^{(E)}(\omega_{m})$
(25)

We see that as long as $\lambda_{E}$ is small, the two-loop
$\Sigma^{(2)}(\omega_{m})$ is parametrically smaller than the Eliashberg self-
energy, despite that the integral comes from $q$ where the vertex correction
is singular. This singularity only accounts for $|\log{\lambda_{E}}|$ in the
prefactor.

In Fig. 6 we present the result of numerical evaluation of
$\Sigma^{(2)}(\omega_{m})$ using an infinite cutoff in the integration over
$\epsilon_{k^{\prime}}$. We see that the ratio $\Sigma^{(2)}/\Sigma^{(E)}$ is
approximately constant at $\omega_{m}\leq\omega_{D}$ and as a function of
$\lambda_{E}$ does scale as $\lambda_{E}|\log{\lambda_{E}}|$. The drop of the
ratio $\Sigma^{(2)}/\Sigma^{(E)}$ at larger frequencies is likely an artifact
of an infinite cutoff. In any case, the ratio $\Sigma^{(2)}/\Sigma^{(E)}$ is
small at small $\lambda_{E}$. For completeness we computed this ratio at
$\lambda_{E}=1$ and found that it still remains numerically small over the
wide range of $\omega_{D}<\omega_{m}<E_{F}$ ($\Sigma^{(2)}/\Sigma^{(E)}\simeq
0.062$ at $\omega_{m}=\omega_{D}$).

For comparison with the Ising-nematic/Ising-ferromagnetic case below, we
analyze how $\Delta g/g$ depends on characteristic momenta and frequencies.
For definiteness we set $\mathbf{k}=\mathbf{k}_{F}$,
$\Omega_{n}\sim\omega_{D}$ and vary $\omega_{m}$ and $q_{\perp}$ and
$q_{\parallel}$, which are typical momentum components transverse and along
the Fermi surface. At
$\omega_{D}<\omega_{m}<\omega_{c}=(\pi/2)\lambda_{E}E_{F}$, where the self-
energy $\Sigma(\omega_{m})$ exceeds bare $\omega_{m}$, which are typical
momentum components transverse and along the Fermi surface. After a simple
experimentation we found after a simple experimentation

$\frac{\Delta
g(k,q)}{g}\sim\frac{\omega_{c}}{\omega_{c}+v_{F}q_{\perp}+\frac{q^{2}_{\parallel}}{2m}}.$
(26)

Typical $v_{F}q_{\perp}$ are of order $\omega_{c}$, typical
$q^{2}_{\parallel}/(2m)$ are of order $E_{F}$. For these momenta,

$\frac{\Delta
g(k,q)}{g}\sim\frac{v_{F}q^{\text{typ}}_{\perp}}{v_{F}q^{\text{typ}}_{\perp}+\frac{(q^{\text{typ}}_{\parallel})^{2}}{2m}}.$
(27)

The “fast electrons/slow bosons” criterion requires typical $q_{\perp}$ to be
much smaller than typical $q_{\parallel}$. This holds for $\lambda_{E}\ll 1$
because typical $q^{\text{typ}}_{\perp}\sim\lambda_{E}k_{F}$, while
$q^{\text{typ}}_{\parallel}\sim k_{F}$. The strength of the vertex correction
in Eq. 27 is, however, determined by the ratio of typical energies
$v_{F}q^{\text{typ}}_{\perp}/((q^{\text{typ}}_{\parallel})^{2}/2m)$ rather
than typical momenta. In general, the ratios of typical energies and typical
momenta are not the same, but in the electron-phonon case,
$v_{F}q^{\text{typ}}_{\perp}/((q^{\text{typ}}_{\parallel})^{2}/2m)\sim\lambda_{E}$
is the same as $q^{\text{typ}}_{\perp}/q^{\text{typ}}_{\parallel}$ because
$q^{\text{typ}}_{\parallel}\sim k_{F}$. As a consequence, the single parameter
$\lambda_{E}$ allows one to simplify the Eliashberg equations and keep vertex
corrections small.

For larger $\omega_{m}$, we found

$\frac{\Delta
g(k,q)}{g}\sim\frac{\omega_{c}}{\omega_{m}+v_{F}q^{\text{typ}}_{\perp}+\frac{(q^{\text{typ}}_{\parallel})^{2}}{2m}}.$
(28)

Now typical $v_{F}q_{\perp}$ are of order $\omega_{m}$, while typical
$q^{2}_{\parallel}/(2m)$ are still of order $E_{F}$. For these momenta,

$\frac{\Delta
g(k,q)}{g}\sim\frac{\omega_{c}}{\omega_{m}+E_{F}}\sim\lambda_{E}\frac{1}{1+\frac{\omega_{m}}{E_{F}}}$
(29)

We see that the vertex correction remains of order $\lambda_{E}$.

Figure 6: (a) Blue line – the two-loop electron self-energy, obtained using
infinite cutoff for integration over fermionic dispersion. The parameters are
the same as in Fig. 3: $\lambda=30$ and $\lambda_{E}=0.06$. Brown line – the
ratio $\Sigma^{(2)}/(\lambda_{E}\Sigma^{(E)})$. The ratio approaches a
constant at $\omega_{m}\to 0$ and evolves at $\omega_{m}\sim\omega_{D}$. The
drop of the ratio at larger $\omega_{m}$ is likely an artifact of using
infinite cutoff for the integration over the dispersion. (b) Numerical
verification of the scaling of
$\Sigma^{(2)}(\omega_{m})/\Sigma^{(E)}(\omega_{m})\sim\lambda_{E}|\log\lambda_{E}|$
at small $\omega_{m}$.

###  II.4 Summary of Section II

There are three energy scales in the electron-phonon problem: the bosonic
energy $\omega_{D}$, the dimension-full electron-phonon coupling $g$ and the
Fermi energy $E_{F}$. This allows one to introduce two dimensionless ratios
$\lambda=g^{2}/\omega^{2}_{D}$ and
$\lambda_{E}=g^{2}/(E_{F}\omega_{D})=\lambda\omega_{D}/E_{F}$. The latter is a
small parameter for the Eliasberg theory. The strong coupling regime occurs at
$\lambda\gg 1$, $\lambda_{E}\ll 1$. In this regime, the system displays Fermi
liquid behavior at $\omega<\omega_{D}$, non-Fermi liquid behavior with
$\Sigma(\omega_{m})\approx(\pi/2)\lambda\omega_{D}=(\pi/2)\lambda_{E}E_{F}$ at
$\omega_{D}<\omega_{m}<\omega_{c}$, where $\omega_{c}\sim\lambda\omega_{D}\sim
E_{F}\lambda_{E}$, and Fermi-gas behavior at larger frequencies.

The three key results for electron-phonon system, which we will later use in
comparison with the behavior near a nematic QCP, are the following.

First, Eliashberg theory is rigorously justified even at strong coupling
$\lambda\gg 1$, as long as the Eliashberg parameter $\lambda_{E}\ll 1$. Two-
loop vertex corrections to self-consistent one-loop Eliashberg theory are
small in $\lambda_{E}$ for all frequencies $\omega_{m}$.

Second, the same small parameter $\lambda_{E}$ also simplifies the
calculations within Eliashberg theory: the Landau damping of phonons can be
neglected in the calculation of the self-energy, and the self-energy
$\Sigma(\mathbf{k},\omega_{m})$ can be approximated by the local
$\Sigma(\omega_{m})$ and computed perturbatively rather than self-
consistently. This holds for $\omega_{m}<E_{F}$. At larger $\omega_{m}$, “fast
electron/slow boson” criterion is not valid, and the Eliashberg equation (3)
has to be re-analyzed.

Third, vertex corrections to Eliashberg theory become $O(1)$ at
$\lambda_{E}=O(1)$, and become parametrically large at $\lambda_{E}>1$, except
for the smallest frequencies $\omega_{m}<\omega_{D}/\lambda_{E}$, where they
remain small. Because $\lambda_{E}=g^{2}/(\omega_{D}E_{F})$, this result
implies that one cannot extend the Eliashberg theory of electron-phonon
interaction to $\omega_{D}=0$. For a finite $E_{F}$, Eliashberg theory is
valid only when $\omega_{D}$ exceeds a certain value. At smaller $\omega_{D}$,
a new description is required.

We summarize these results in Fig. 7.

Figure 7: Illustration of the applicability of Eliashberg theory for electron-
phonon interactions at different energy scales: (a) $\lambda\gg 1$,
$\lambda_{E}\ll 1$; (b) $\lambda\gg 1$, $\lambda_{E}\sim 1$; (c) $\lambda\gg
1$, $\lambda_{E}\gg 1$.

##  III Quantum critical metal

We now analyze the validity of an Eliashberg-type description of a system of
electrons interacting with soft collective bosons representing order parameter
fluctuations. For definiteness, we focus on the system near a ${\bm{q}}=0$
Ising-ferromagnetic or Ising-nematic instability (see e.g., Ref. 58 and
references therein). The two cases differ by the form-factor for fermion-boson
coupling (it has d-wave form in the Ising-nematic case). We verified that the
form-factor does not play a critical role in our considerations and can be
safely set to one (see also Ref. [80]). We do not consider $SU(2)$
ferromagnetic case as the ordering transition there is affected by non-
analytic corrections to spin susceptibility [81].

The set of Eliashberg-type equations is the same as Eqs. 3 and 4. Like for the
electron-phonon case, it can be either derived diagrammatically [13,
*finger_2001, 82] as self-consistent one-loop equations, or obtained as
stationary solutions of the Luttinger-Ward functional [58]. The bare fermionic
Green’s function is the same as in Eq. 2. For the bare bosonic propagator we
choose the conventional Ornstein-Zernike form

$\chi_{0}(\mathbf{q})=\frac{\chi_{0}}{\xi^{-2}+|\mathbf{q}|^{2}}.$ (30)

We will define a related frequency scale $\omega^{*}_{D}=v_{F}\xi^{-1}$. The
full generic $\chi_{0}(\mathbf{q})$ also contains a dynamical
$(\Omega_{n}/v_{F})^{2}$ term, however, this term only becomes relevant at
frequencies above the upper limit of quantum-critical behavior (see below).
For this reason, we neglect the $\Omega^{2}_{n}$ term in $\chi_{0}$ in our
analysis.

We label the fermion-boson coupling as $\tilde{g}$ and introduce the effective
dimension-full interaction $g^{*}=\tilde{g}^{2}\chi_{0}$. Like in the
electron-phonon case, there are three energy scales in this model: the
effective coupling $g^{*}$, the bosonic energy $\omega^{*}_{D}$, and the Fermi
energy $E_{F}$. We again assume that $E_{F}$ is the largest scale in the
problem, i.e., set $E_{F}\gg\omega^{*}_{D},g^{*}$. Like before, one can
construct two dimensionless ratios out of these couplings. We choose them to
be

$\lambda^{*}=\frac{g^{*}}{4\pi\omega^{*}_{D}},\quad\lambda^{*}_{E}=\frac{g^{*}}{E_{F}}.$
(31)

By construction, $\lambda^{*}_{E}\ll 1$, but $\lambda^{*}$ can be either small
or large. We will see that the self-energy at the lowest $\omega_{m}$ is
$\Sigma=\lambda^{*}\omega_{m}$, i.e., $\lambda^{*}\sim 1$ separates weak
coupling ($\lambda^{*}<1$) and strong coupling ($\lambda^{*}>1$) regimes,
respectively.

###  III.1 Solution of Eliashberg equations

The analysis of the Eliashberg equations for fermions near a $q=0$ QCP has
been done before in various contexts [41, *Monien1993, *Nayak1994,
*Altshuler1994, *Kim_1994, 46, 47, *Metzner2003, 49, 50, 51, 52, 53,
*Klein2019, *Klein2020, *Klein2022, 22, 58]. We list the existing results and
present some new ones that will allow a direct comparison with the electron-
phonon case.

We first discuss the calculation of the fermionic self-energy and bosonic
polarization. Like for the electron-phonon case, we assume and then verify
that the self-energy $\Sigma(k,\omega)$ can be approximated by a local
$\Sigma(\omega_{m})$. For such a self-energy, earlier calculations found that
for $v_{F}|\mathbf{q}|\gg|\Omega_{n}|$, and both much smaller than $E_{F}$,
the polarization can be written as a sum of static and dynamic terms
$\Pi(\mathbf{q},\Omega_{n})=\Pi(\mathbf{q},0)+\delta\Pi(\mathbf{q},\Omega_{n})$,
where $\delta\Pi(\mathbf{q},\Omega_{n})$ has the form of Landau damping:

$\delta\Pi(\mathbf{q},\Omega_{n})=\frac{g^{*}k_{F}}{\chi_{0}\pi
v^{2}_{F}}\frac{|\Omega_{n}|}{|\mathbf{q}|}.$ (32)

This is similar to the electron-phonon polarization, Eq. 8 at $q\ll k_{F}$.
Like there, the prefactor for the Landau damping term does not depend on the
fermionic self-energy, i.e., Eq. 32 has the same form for free fermions and
for interacting fermions. We incorporate the static $\Pi(\mathbf{q},0)$ into
$\chi_{0}(|\mathbf{q}|)$, like we did for the electron-phonon case.
Substituting into $\chi(q)$, we then obtain

$\chi(q)=\frac{\chi_{0}}{\xi^{-2}+|\mathbf{q}|^{2}+\alpha\frac{|\Omega_{n}|}{|\mathbf{q}|}},$
(33)

where $\alpha=\frac{g^{*}k_{F}}{\pi v_{F}^{2}}$. Because relevant $\mathbf{q}$
are much smaller than $k_{F}$, the dynamical Landau damping term in $\chi(q)$
becomes relevant when $\Omega_{n}$ exceeds the scale

$\omega^{*}=\frac{(\omega^{*}_{D})^{3}}{g^{*}E_{F}}\sim\frac{\omega^{*}_{c}}{(\lambda^{*})^{3}},$
(34)

where $\omega^{*}_{c}\sim E_{F}(\lambda^{*}_{E})^{2}$ (the exact definition
with the prefactor is in Eq. 39 below). Because we consider $\lambda_{E}^{*}$
to be small, we have $\omega^{*}_{c}\ll E_{F}$. We will see that at strong
coupling, when $\lambda^{*}>1$, $\omega^{*}$ is the upper edge of Fermi liquid
behavior, while $\omega^{*}_{c}>\omega^{*}$ is the upper edge of quantum-
critical non-Fermi liquid behavior. This identification implies that the
Landau damping term in Eq. 33 is essential outside of the Fermi liquid regime.
This distinguishes this case from the electron-phonon one, where relevant
$q\sim k_{F}$ and the Landau damping term is irrelevant for all frequencies,
as long as $\lambda_{E}$ is small.

We now substitute $\chi(q)$ from Eq. 33 into the formula for the self-energy,
Eq. 3. As before, we assume and verify that
$\Sigma(\mathbf{k},\omega_{m})\approx\Sigma(\mathbf{k}_{F},\omega_{m})=\Sigma(\omega_{m})$.
The self-consistent equation for $\Sigma(\omega_{m})$ is

$\Sigma(\omega_{m})=g^{*}\int\frac{d\Omega_{n}}{2\pi}\int\frac{d^{2}\mathbf{q}}{4\pi^{2}}\frac{1}{\xi^{-2}+|\mathbf{q}|^{2}+\alpha\frac{|\Omega_{n}|}{|\mathbf{q}|}}\frac{1}{i\tilde{\Sigma}(\omega_{m}+\Omega_{n})-\epsilon_{\mathbf{k}_{F}+\mathbf{q}}},$
(35)

where, as before, $\tilde{\Sigma}(\omega_{m})=\omega_{m}+\Sigma(\omega_{m})$.
A straightforward analysis of the relevant scales in this equation shows that
relevant $\Omega_{n}\sim\omega_{m}$, relevant
$\epsilon_{\mathbf{k}_{F}+\mathbf{q}}\sim\tilde{\Sigma}(\omega_{m})$, and
relevant $|\mathbf{q}|\sim\xi^{-1}$ for $\omega_{m}<\omega^{*}$ and
$|\mathbf{q}|\sim(\alpha|\omega_{m}|)^{1/3}$ for $\omega_{m}>\omega^{*}$. For
the fermionic dispersion $\epsilon_{\mathbf{k}_{F}+\mathbf{q}}$ we will be
using

$\epsilon_{\mathbf{k}_{F}+\mathbf{q}}=v_{F}q_{\perp}+\frac{q_{\parallel}^{2}}{2m},$
(36)

where $q_{\perp}$ and $q_{\parallel}$ are momentum components perpendicular
and parallel to the Fermi surface (along $\mathbf{k}_{F}$ and perpendicular to
$\mathbf{k}_{F}$, respectively). Comparing relevant scales, we immediately see
that for $\lambda^{*}_{E}\ll 1$, relevant $q_{\parallel}$ are much larger than
$q_{\perp}$ as long as
$\omega_{m}<\omega_{\text{max}}\equiv(g^{*}E_{F})^{1/2}\sim\omega^{*}_{c}/(\lambda^{*}_{E})^{3/2}$.
For frequencies below $\omega_{\text{max}}$ we then can factorize the momentum
integration, i.e., re-express $d^{2}\mathbf{q}$ as
$(1/v_{F})d\epsilon_{\mathbf{k}_{F}+\mathbf{q}}dq_{\parallel}$ and integrate
the fermionic propagator over $\epsilon_{\mathbf{k}_{F}+\mathbf{q}}$ and
bosonic $\chi(|\mathbf{q}|,\Omega_{n})\approx\chi(|q_{\parallel}|,\Omega_{n})$
over $q_{\parallel}$. Using

$\int\frac{d\epsilon}{i\tilde{\Sigma}(\omega_{m}+\Omega_{n})-\epsilon}=-i\pi{\operatorname{sgn}}(\omega_{m}+\Omega_{n})$
(37)

we find that the right-hand side of Eq. 3 does not depend on the self-energy,
i.e., has the same form as for free fermions. Completing the integration, we
obtain

$\Sigma^{(E)}(\omega_{m})=\lambda^{*}\omega_{m}F\left(\frac{\omega_{m}}{\omega^{*}}\right)$
(38)

in terms of a function $F$, where $F(0)=1$ and $F(x\gg
1)=((4\pi)^{1/3}/\sqrt{3})/x^{1/3}$. One can obtain the full analytical
formula for $F(x)$, but it is rather cumbersome and not particularly
enlightening so we do not present it here.

Figure 8:  (Color online) Numerical results for the self-energy in Eliashberg
theory for $\lambda^{*}_{E}=0.29$ at (a) intermediate coupling
$\lambda^{*}=1.35$ and (b) infinitely strong coupling $\lambda^{*}=\infty$ at
a QCP. The solid line is the perturbative expression $\Sigma^{(E)}$, Eq. (38),
obtained by factorizing the momentum integration, and squares are the
numerical solutions of the full equation (35) for $\Sigma(\omega_{m})$. (c)
The difference between $\Sigma(\omega_{m})$ and $\Sigma^{(E)}(\omega_{m})$ at
the QCP ($\lambda^{*}=\infty$). Red solid line is the analytic result, Eq. 41.
(d) Verification of the scaling relation
$\Sigma(\omega_{m})-\Sigma^{(E)}(\omega_{m})\propto(\lambda_{E}^{*})^{1/2}$
(red line) at a particular $\omega_{m}$.

At small frequencies, $\Sigma^{(E)}(\omega)$ has the Fermi liquid form,
$\Sigma(\omega_{m})=\lambda^{*}\omega_{m}$. For large $\lambda^{*}$,
$\Sigma^{(E)}(\omega_{m})\gg\omega_{m}$. In this regime, relevant
$q_{\parallel}\sim\xi^{-1}$ while relevant
$q_{\perp}\sim(\omega_{m}/\omega^{*})(\omega_{D}^{*}/E_{F})\xi^{-1}$ are far
smaller. This is consistent with the “fast electrons/slow bosons” criterion.
In this regime, we also have typical
$q^{2}_{\parallel}/k_{F}\sim(\omega_{D}^{*}/E_{F})\xi^{-1}$. This will be
relevant to our analysis of vertex corrections below.

At larger $\omega_{m}>\omega^{*}$, the self-energy crosses over to a non-Fermi
liquid, quantum-critical form
$\Sigma^{(E)}(\omega_{m})=\omega^{2/3}_{m}(\omega^{*}_{c})^{1/3}$, where

$\omega^{*}_{c}=\frac{1}{16\pi^{2}\sqrt{27}}\frac{(g^{*})^{2}}{E_{F}}=\frac{1}{16\pi^{2}\sqrt{27}}E_{F}(\lambda^{*}_{E})^{2}.$
(39)

In terms of $\omega^{*}_{c}$,

$\omega^{*}=\frac{\sqrt{27}}{8}\frac{\omega^{*}_{c}}{(\lambda^{*})^{3}},\omega_{max}=16\pi^{2}\sqrt{27}\frac{\omega^{*}_{c}}{(\lambda^{*}_{E})^{3/2}}$
(40)

In the quantum-critical regime, relevant
$q_{\parallel}\sim(\alpha\omega_{m})^{1/3}$ and relevant
$q_{\perp}\sim\omega^{2/3}_{m}\omega^{1/3}_{0}/v_{F}$. These can be re-
expressed as $q_{\parallel}\sim
k_{F}(\omega_{m}/\omega^{*}_{c})^{1/3}\lambda^{*}_{E}$ and $q_{\perp}\sim
k_{F}(\omega_{m}/\omega^{*}_{c})^{2/3}(\lambda^{*}_{E})^{2}$. We see that
relevant $q_{\parallel}$ are again larger than $q_{\perp}$, i.e., the “fast
electrons/slow bosons” criterion is well satisfied when $\lambda^{*}_{E}$ is
small. The conditions look similar to the Fermi liquid regime, but there is
one important difference: relevant values of $q^{2}_{\parallel}/k_{F}$ are now
comparable to relevant $q_{\perp}$. We will discuss below how this affects
vertex corrections.

At even larger $\omega_{m}>\omega^{*}_{c}$, the self-energy becomes smaller
than $\omega_{m}$, although it still preserves its $\omega^{2/3}_{m}$ form. In
this last regime relevant $q_{\parallel}\sim(\alpha\omega_{m})^{1/3}$, like
before, but relevant $q_{\perp}\sim\omega_{m}/v_{F}$. These can be re-
expressed as $q_{\parallel}\sim
k_{F}(\lambda^{*}_{E})^{1/2}(\omega_{m}/\omega_{\text{max}})^{1/3}$ and
$q_{\perp}\sim k_{F}(\lambda^{*}_{E})^{1/2}(\omega_{m}/\omega_{\text{max}})$.
The “fast electrons/slow bosons” criterion is satisfied as long as
$\omega_{m}<\omega_{\text{max}}$.

At $\omega_{m}>\omega_{\text{max}}$ the criterion of fast electrons and slow
bosons is no longer valid and, as a result, one cannot factorize the momentum
integration. In this regime, however, the $\Omega^{2}_{m}$ term in the bare
$\chi_{0}(q)$ cannot be neglected. We don’t analyze this high-frequency regime
here.

###  III.2 Subleading terms in the self-energy

As for the electron-phonon case, there are three corrections to the
perturbative one-loop self-energy. They come from (i) the momentum dependence
of $\Sigma(\mathbf{k},0)$, (ii) from non-factorization of momentum integration
and (iii) from keeping $\tilde{\Sigma}(\omega_{m}+\Omega_{n})$ in the right-
hand side of Eq. 35. The momentum dependent part of the self-energy is
$\Sigma(\mathbf{k},0)\sim\epsilon_{k}(\lambda^{*}_{E})^{1/2}$ [46]. It gives a
small renormalization of the dispersion at $\lambda^{*}_{E}\ll 1$. The
correction from non- factorization of momentum integration is of order of the
ratio of relevant scales $(q_{\perp}/q_{\parallel})^{2}$. In the quantum-
critical regime this yields a relative correction to the self-energy of order
$(\omega/\omega^{*}_{c})^{2/3}(\lambda^{*}_{E})^{2}$, which is at most
$O((\lambda^{*}_{E})^{2})$. In the Fermi liquid regime the correction is even
smaller. In the Fermi gas regime the correction increases and becomes of order
one at $\omega\sim\omega_{\text{max}}$, which, we remind, is the largest
frequency up to which the concept the “fast electron/slow boson” is
applicable.

The correction $\delta\Sigma(\omega_{m})$ from keeping
$\tilde{\Sigma}(\omega_{m}+\Omega_{n})$ is of order
$\omega_{m}(\lambda_{E})^{1/2}$ for all $\omega_{m}<\omega_{{\text{max}}}$. We
verified that it comes from internal $\Omega_{n}\sim\omega_{\text{max}}$ and
momenta $q_{\perp}\sim q_{\parallel}\sim k_{F}(\lambda_{E}^{*})^{1/2}$.For
such $\Omega_{n}$,
$\tilde{\Sigma}(\omega_{m}+\Omega_{n})\approx\omega_{m}+\Omega_{n}$, i.e., the
actual self-energy of an internal fermion is irrelevant. In explicit form we
find for $\delta\Sigma(\omega_{m})$

$\delta\Sigma(\omega_{m})=-\omega_{m}(\lambda_{E})^{1/2}J$ (41)

where

$J=\frac{1}{2\pi^{2}}\int_{0}^{\infty}dx\int_{0}^{\infty}dy\frac{x^{2}y}{(4x^{2}+y^{2})^{3/2}\left(x^{3}+\frac{y}{4\pi}\right)}\approx
0.084.$ (42)

and the variables are $y=\Omega_{n}/\omega_{\text{max}}$ and
$x=q/(k_{F}(\lambda_{E}^{*})^{1/2})$.We caution, however, that at $y=O(1)$,
i.e., at $\Omega_{n}\sim\omega_{\text{max}}$, the $(\Omega_{m}/v_{F}^{2})$
term in $\chi(q)$, which we neglected, becomes comparable with the Landau
damping, and this may change the value of $J$. Still, it remains a number of
order one, and $\delta\Sigma(\omega_{m})$ is much smaller than
$\Sigma^{(E)}(\omega_{m})$.

In Fig. 8, we show the numerical result of the self-energy for a small but
finite $\lambda_{E}^{*}$. The solid line is $\Sigma^{(E)}(\omega_{m})$ at
$k=k_{F}$ and the dotted line is the full self-consistent numerical result
$\Sigma(\omega_{m})=\Sigma^{(E)}(\omega_{m})+\delta\Sigma(\omega_{m})$. We see
that $\delta\Sigma(\omega_{m})$ is linear in $\omega_{m}$, and the prefactor
scales as $(\lambda_{E}^{*})^{1/2}$, as in (41). The numerical data matches
Eq. 41 very well.

###  III.3 Vertex correction

Vertex corrections to Eliashberg theory have been discussed in detail in Ref.
46. We present several new results below.

The lowest order vertex correction is given by the diagram in Fig. 4 (a). In
explicit form

$\frac{\Delta
g(k,q)}{g}=g^{*}\int\frac{d\Omega^{\prime}_{m}d^{2}\mathbf{q}^{\prime}}{(2\pi)^{3}}\frac{1}{i\tilde{\Sigma}(\omega_{m}+\Omega^{\prime}_{n})-\epsilon_{\mathbf{k}+\mathbf{q}^{\prime}}}\frac{1}{i\tilde{\Sigma}(\omega_{m}+\Omega^{\prime}_{n}+\Omega_{n})-\epsilon_{\mathbf{k}+\mathbf{q}^{\prime}+\mathbf{q}}}\frac{1}{\left|\mathbf{q}^{\prime}\right|^{2}+\alpha\frac{\left|\Omega^{\prime}_{n}\right|}{\left|\mathbf{q}^{\prime}\right|}},$
(43)

where we recall, $q=(\mathbf{q},\Omega_{n})$ is the external bosonic momentum
and $k=(\mathbf{k},\omega_{m})$ is the external fermionic momentum.

Like for the electron-phonon case, vertex correction at $\mathbf{q}=0$ has to
obey the Ward identity, Eq. 19, associated with the conservation of the
particle number. We show that this indeed holds.

In the Fermi liquid regime at $\omega_{m}<\omega^{*}$, an explicit calculation
of $\Delta g(k,\Omega_{n},\mathbf{q}=0)/g$ yields,

$\frac{\Delta g(k,\Omega_{n},\mathbf{q}=0)}{g}=\frac{\lambda}{1+\lambda}$ (44)

At large $\lambda$, the one-loop vertex correction is close to one. To get the
full vertex $g_{\text{full}}(k,\Omega_{m},\mathbf{q}=0)$ one has to sum the
series of ladder vertex correction diagrams, just as we did for the electron-
phonon case. The summation yields the expected result

$\frac{g_{\text{full}}(k,\Omega_{n},\mathbf{q}=0)}{g}=\frac{1}{1-\frac{\lambda}{1+\lambda}}=1+\lambda,$
(45)

which is equal to
$(\tilde{\Sigma}(\omega_{m}+\Omega_{n})-\tilde{\Sigma}(\omega_{m}))/\Omega_{n}$
in the FL frequency regime, and thus satisfies the Ward identity, Eq. 19.

In the quantum-critical regime, the one-loop vertex correction is again of
$O(1)$, but has a more complicated structure. Introducing dimensionless
variables $x=\omega_{m}/\Omega_{n}$ and
$x^{\prime}=-(\omega_{m}+\Omega^{\prime}_{n})/\Omega_{n}$ and setting both
$\omega_{m}$ and $\Omega_{n}$ to be positive, we find one-loop vertex
correction in the form

$\frac{\Delta
g(x,\Omega_{n})}{g}=\frac{2}{3}\int_{0}^{1}\frac{dx^{\prime}}{\left|x^{\prime}-x\right|^{1/3}}\\\
\times\frac{1}{(1-x^{\prime})^{2/3}+(x^{\prime})^{2/3}+\left(\frac{\Omega_{n}}{\omega^{*}_{c}}\right)^{1/3}}.$
(46)

The ladder series of vertex corrections in this situation can be re-expressed
as an integral equation for $g_{\text{full}}(x,\Omega_{n})$:

$g_{\text{full}}(x,\Omega_{n})=1+\frac{2}{3}\int_{0}^{1}\frac{dx^{\prime}}{\left|x^{\prime}-x\right|^{1/3}}\frac{g_{\text{full}}(x^{\prime},\Omega_{n})}{(1-x^{\prime})^{2/3}+(x^{\prime})^{2/3}+\left(\frac{\Omega_{n}}{\omega^{*}_{c}}\right)^{1/3}}$
(47)

This is an integral equation in the variable $x$, while $\Omega_{n}$ just acts
as a parameter. One can easily verify that the solution of Eq. 47 is

$g_{\text{full}}(x,\Omega_{n})=\left(\frac{\omega^{*}_{c}}{\Omega_{n}}\right)^{1/3}\left((1-x)^{2/3}+x^{2/3}+\left(\frac{\Omega_{n}}{\omega^{*}_{c}}\right)^{1/3}\right).$
(48)

In the original variables Eq. 48 becomes

$g_{\text{full}}(\omega_{m},\Omega_{n})=1+\frac{\left((\omega_{m}+\Omega_{n})^{2/3}-(\omega_{m})^{2/3}\right)\omega^{2/3}_{0}}{\Omega_{n}}=\frac{\tilde{\Sigma}(\omega_{m}+\Omega_{n})-\tilde{\Sigma}(\omega_{m})}{\Omega_{n}}$
(49)

as in Eq. 19.

For finite $\mathbf{q}$, the result is more involved. The one-loop vertex
correction for $\mathbf{k}$ on the Fermi surface is a scaling function of five
variables

$\frac{\Delta
g(k,q)}{g}=\Phi\left(\frac{\omega_{m}}{\Omega_{n}},\frac{v_{F}q_{\perp}}{\tilde{\Sigma}(\omega_{m})},\frac{q_{\parallel}^{2}/2m}{\tilde{\Sigma}(\omega_{m})},\frac{\omega_{m}}{{\tilde{\Sigma}}(\omega_{m})},\frac{(\omega^{*}_{D})^{2}}{E_{F}{\tilde{\Sigma}}(\omega_{m})}\right)$
(50)

In the Fermi liquid regime at $\omega_{m}<\omega^{*}$, typical
$v_{F}q_{\perp}\sim\Sigma(\omega_{m})=\lambda^{*}\omega_{m}$ and typical
$(q^{2}_{\parallel}/2m)/\Sigma(\omega_{m})\sim\omega^{*}/\omega_{m}$. We
verified numerically that in this case $\Phi$ is of order
$\omega_{m}/\omega^{*}\ll 1$, i.e., $\Delta
g(k,q)/g\sim\omega_{m}/\omega^{*}\ll 1$.

In the quantum-critical regime the last two variables in Eq. 50 can be set to
zero. The scaling function of the other three variables, $\Phi(x,y,z,0,0)$, is

$\Phi(x,y,z,0,0)=\frac{3^{3/4}}{4\pi}\int_{0}^{1}dr\int_{0}^{\infty}\frac{ds}{s^{3/2}+3^{3/4}\left|r+x\right|}\\\
\times\left[\frac{1}{(1-r)^{2/3}+r^{2/3}+i(y+z+2\sqrt{\left|z\right|s})}\right.\\\
\left.+\frac{1}{(1-r)^{2/3}+r^{2/3}+i(y+z-2\sqrt{\left|z\right|s})}\right].$
(51)

The scaling function $\Phi$ in Eq. 50 is generally complex and can be
parametrized as $\Phi=|\Phi|e^{i\psi_{\Phi}}$. We plot $|\Phi|$ and
$\psi_{\Phi}$ in Fig. 9 for several values of parameters. We see that in
general $\Phi$ is of order one, but not particularly close to one, in
distinction to the case $q=0$. In this situation, we expect that higher-order
vertex corrections remain of order one, but do not change substantially the
value of $\Delta g/g$ compared to one-loop result.

Figure 9: Scaling function $\Phi(x,y,z,0,0)=|\Phi|e^{i\psi_{\Phi}}$,
describing the one-loop vertex correction in the quantum-critical regime (cf.
Eq. 50). The functions $|\Phi|$ and $\psi_{\Phi}$ are plotted in $y,z$ plane
for $x=-0.5$ (a,b) and in (x,y) plane for $z=0$ (c,d).

Finally, in the Fermi gas regime
$\omega^{*}_{c}<\omega_{m}<\omega_{\text{max}}$, we have for typical
$\mathbf{q}$, and $\Omega_{m}\sim\omega_{D}^{*}$,
$q_{\parallel}^{2}/(2m)/\tilde{\Sigma}(\omega_{m})\ll 1$,
$(\omega^{*}_{D})^{2}/(E_{F}{\tilde{\Sigma}}(\omega_{m}))\ll 1$,
$v_{F}q_{\perp}/{\tilde{\Sigma}}(\omega_{m})\sim 1$ and
$\omega_{m}/{\tilde{\Sigma}}(\omega_{m})\approx 1$. In this situation, we find
$\Phi\sim\Sigma(\omega_{m})/\omega_{m}\sim(\omega^{*}_{c}/\omega)^{1/3}\ll 1$,
and hence $\Delta g(k,q)/g\ll 1$.

For qualitative analysis we do the same as for the electron-phonon case and
estimate $\Delta g(k,q)/g$ by using typical values of internal momenta and
frequency. A simple experimentation shows that by order of magnitude

$\frac{\Delta
g(k,q)}{g}\sim\frac{\Sigma(\omega_{m})}{\tilde{\Sigma}(\omega_{n})+v_{F}q^{\text{typ}}_{\perp}+\frac{(q^{\text{typ}}_{\parallel})^{2}}{2m}}$
(52)

In the Fermi liquid and Fermi gas regimes the vertex correction is small
because either $(q^{\text{typ}}_{\parallel})^{2}/(2m)\gg\Sigma(\omega_{m})$ or
$\tilde{\Sigma}(\omega_{n})\gg\Sigma(\omega_{n})$. In the quantum-critical,
non-Fermi liquid regime all parameters in Eq. 52 are of the same order, and
$\Delta g(k,q)/g$ is generally of order one.

The quantitative measure of the strength of the vertex correction is the
magnitude of the two-loop contribution to the self-energy with the vertex
correction included, $\Sigma^{(2)}(\omega_{m})$ compared to the one-loop
$\Sigma^{(E)}(\omega_{m})$. We show the corresponding diagram in Fig. 4 (b).
In the Fermi liquid and Fermi gas regimes, we find that
$\Sigma^{(2)}(\omega_{m})\ll\Sigma^{(E)}(\omega_{m})$, consistent with the
smallness of $\Delta g/g$. In the quantum-critical regime, earlier order of
magnitude studies [44, 46, 83, 66, 65] have found that
$\Sigma^{(2)}(\omega_{m})$ is of the same order as $\Sigma^{(E)}(\omega_{m})$,
unless one extends the theory to large $N$ or assumes that the prefactor for
$q^{2}_{\parallel}$ in $\epsilon_{\mathbf{k}_{F}+\mathbf{q}}$ is much larger
than $v_{F}/k_{F}$.

We confirmed this results, but went further and computed
$\Sigma^{(2)}(\omega_{m})$ numerically with the prefactor. We found

$\Sigma^{(2)}(\omega_{m})\simeq 0.038\Sigma^{(E)}(\omega_{m}).$ (53)

We see that while by order of magnitude $\Sigma^{(2)}(\omega_{m})$ is of order
$\Sigma^{(E)}(\omega_{m})$, it is far smaller numerically. It has been argued
[66, *Lee2018, 65, 68, 84, 85] that self-energies with higher-loop order
vertex corrections included contain logarithmic singularities We didn’t
compute these terms explicitly, but based on Eq. 53 we expect them to contain
small prefactors and remain small down to very small frequencies, before
logarithmical singularities become relevant. We recall in this regard that the
ground state near an Ising-nematic/Ising-ferromagnetic transition is a
superconductor, hence in practice the behavior at the lowest frequencies is
relevant only if for some reason superconductivity does not develop.

For completeness, we also present results at weak coupling, when both
$\lambda^{*}$ and $\lambda^{*}_{E}$ are small. There are two weak coupling
regimes: $\lambda^{*}\ll(\lambda^{*}_{E})^{1/2}\ll 1$ and
$(\lambda^{*}_{E})^{1/2}\ll\lambda^{*}\ll 1$. We focus on the second regime as
it borders strong-coupling regime at $\lambda^{*}=O(1)$. In this regime,
$\omega^{*}_{c}\ll\omega^{*}$, i.e., there is no range of non-Fermi liquid
behavior, although the self-energy still interpolates between
$\lambda^{*}\omega_{m}$ at small frequencies and
$(\omega_{m})^{2/3}(\omega_{c}^{*})^{1/3}$ at higher frequencies (the relation
between all characteristic frequencies is
$\omega^{*}_{c}<\omega^{*}<\omega_{D}<\omega_{max}$). A simple calculation
shows that in this case vertex correction is small in $\lambda^{*}$ and
typical $q^{\text{typ}}_{\perp}$ are parametrically larger than
$q^{\text{typ}}_{\parallel}$ for all $\omega_{m}<\omega_{max}$ (see Fig. 11a
for their values). Hence, Eliashberg theory is applicable and self-energy can
be computed within perturbative one-loop approximation.

Figure 10: Two-loop fermionic self energy with vertex correction included,
$\Sigma^{(2)}(\omega_{m})$ (see Fig. 4 (b)). Left axis:
$\Sigma^{(2)}(\omega_{m})$ in units of the characteristic frequency
$\omega^{*}_{c}$ (the upper edge of non-Fermi liquid behavior). Right axis:
the ratio of $\Sigma^{(2)}(\omega_{m})$ and the full one-loop Eliashberg self-
energy $\Sigma(\omega_{m})=\Sigma^{(E)}(\omega_{m}+\delta\Sigma(\omega_{m})$
(see Section III.2).

###  III.4 Summary of Section III

Like for the electron-phonon system, there are three energy scales in the
problem: the bosonic energy $\omega^{*}_{D}$, the coupling $g^{*}$ and the
Fermi energy $E_{F}$. This allows one to introduce two dimensionless ratios
$\lambda^{*}=g^{*}/(4\pi\omega^{*}_{D})$ and $\lambda^{*}_{E}=g^{*}/E_{F}$.
The latter is a small parameter for the low-energy theory. The strong coupling
regime occurs at $\lambda^{*}>1$, $\lambda^{*}_{E}<1$. In this regime, the
system displays Fermi liquid behavior at
$\omega<\omega^{*}\sim\omega^{*}_{D}\lambda^{*}_{E}/(\lambda^{*})^{2}$,
quantum-critical, non-Fermi liquid behavior with
$\Sigma(\omega_{m})\propto\omega_{m}^{2/3}$ at
$\omega^{*}<\omega_{m}<\omega^{*}_{c}$, where $\omega^{*}_{c}\sim
g^{*}\lambda^{*}_{E}$, and Fermi-gas behavior at larger frequencies.

The three key results for the system near an Ising-nematic/Ising-ferromagnetic
QCP are the following.

First, Eliashberg theory is rigorously justified at strong coupling
$\lambda^{*}\gg 1$ in the Fermi liquid and Fermi gas regimes, but not in the
quantum-critical regime. In the latter, the leading vertex correction is
$O(1)$ even when $\lambda^{*}_{E}$ is small. It is nevertheless small
numerically, as evidenced by numerical smallness of the two-loop self-energy
with vertex correction included.

Second, the small parameter $\lambda_{E}^{*}$ simplifies the calculations
within Eliashberg theory: the self-energy
$\Sigma^{(E)}(\mathbf{k},\omega_{m})$ is well approximated by the local
$\Sigma^{(E)}(\omega_{m})$, and the momentum integration in the computation of
$\Sigma^{(E)}(\omega_{m})$ can be factorized by invoking the Migdal “fast
electron/slow boson” criterion. This factorization is rigorously justified at
all $\omega_{m}$ up to
$\omega_{\text{max}}\sim\omega^{*}_{c}/(\lambda^{*}_{E})^{3/2}\gg\omega^{*}_{c}$.
In this respect, $\lambda^{*}_{E}$ plays the same role as $\lambda_{E}$ for
the electron-phonon case.

Third, and most important, the corrections to Eliashberg theory remain $O(1)$
(and numerically small) even at a QCP, where $\omega^{*}_{D}=0$ and
$\lambda^{*}=\infty$. Furthermore, the “fast electron/slow boson” criterion
also remains valid at a QCP as long as $\lambda^{*}_{E}$ is small. In other
words, Eliashberg theory near a nematic QCP, while not rigorously justified,
is quite accurate numerically and to the same numerical accuracy can be
extended right to the QCP.

We summarize the results for the Ising-nematic case in Fig. 11.

Figure 11:  Illustration of the applicability of Eliashberg theory near Ising-
nematic/Ising-ferromagnetic critical point in a metal in (a) weak coupling
regime at some distance from a critical point and (b) strong coupling regime
near and at a critical point.

##  IV Conclusions

In this paper we compared the validity of Eliashberg theory for electrons
interacting with an Einstein phonon and with soft nematic fluctuations near an
Ising-nematic QCP and soft magnetic fluctuations near Ising-ferromagnetic QCP.
Eliashberg theory is the set of coupled one-loop self-consistent equations for
the fermionic self-energy and polarization operator, with vertex corrections
neglected.

For electron-phonon interaction, Eliashberg theory has been justified by the
argument that an Einstein phonon is a slow mode compared to a fermion (the
effective boson velocity is smaller than Fermi velocity). In the Ising-
nematic/Ising-ferromagnetic case this argument is not valid as soft
fluctuations are collective modes of electrons and their velocity is of the
same order as $v_{F}$.

We examined self-consistent Eliashberg theory and two-loop corrections to it
for both cases in the strong coupling regime, where the system displays Fermi
liquid behavior at the lowest energies, non-Fermi liquid behavior in a wide
range of intermediate energies, and Fermi gas behavior at the largest
energies.

For the electron-phonon case, Eliashberg theory is rigorously justified when
the Eliashberg parameter $\lambda_{E}$ is small. Namely, the two-loop self-
energy with vertex correction is small in $\lambda_{E}$. Simultaneously,
Eliashberg equations can be simplified using the same smallness of
$\lambda_{E}$ to un-coupled perturbative one-loop equations, which can be
easily solved. In physical terms, the smallness of $\lambda_{E}$ follows
directly from the condition that an Einstein phonon is a slow mode compared to
electrons.

For the Ising-nematic/Ising-ferromagnetic case, soft bosons are collective
modes of the electrons, and in the non-Fermi liquid quantum-critical regime
there is no parametric smallness of the two-loop self-energy with vertex
correction compared to the one-loop self-energy in the Eliashberg theory.
Nevertheless, we found that the two-loop self-energy is numerically much
smaller than the one-loop one. Additionally, the low-energy theory contains a
small parameter $\lambda^{*}_{E}$, which plays the role of $\lambda_{E}$ in
the sense that it again allows one to reduce the coupled self-consistent one-
loop Eliashberg equations to un-coupled perturbative one-loop equations, which
one can easily solve. The implication of these results is that Eliashberg
theory for the Ising-nematic/Ising-ferromagnetic case is on rather solid
grounds. It is very likely that this holds also for other cases when fermions
interact with their soft collective excitations in the charge or spin channel.

There is one aspect in which the Eliashberg description of fermions coupled to
soft collective bosons works even better than for fermions interacting with an
Einstein phonon. Namely, for the collective boson case, the parameter
$\lambda^{*}_{E}$ is independent of the distance to a QCP, and two-loop self-
energy remains numerically smaller than the one-loop one even at a QCP. As a
result, Eliashberg theory can be extended right to the QCP. For the electron-
phonon case, the Eliashberg parameter $\lambda^{*}_{E}$ is inversely
proportional to the dressed Debye frequency, and Eliashberg theory inevitably
breaks down at some distance from the point where $\omega_{D}$ would vanish.
Beyond this point, a completely new description in terms of polarons is needed
[70, 86, 72, 73, 87].

###### Acknowledgements.

We thank Ar. Abanov, E. Berg, H. Goldman, R. Fernandes, M. Foster, I.
Esterlis, P.A. Lee, D. Maslov, A. Millis, C. Murthy, S. Kivelson, A. Klein, P.
Nosov, N. Prokofiev, S. Raghu, B. Svistunov, S. Sachdev, J. Schmalian, Y. Wang
and Y-M. Wu for helpful discussions. The work of A.Ch. was supported by U.S.
Department of Energy, Office of Science, Basic Energy Sciences, under Award
No. DE-SC0014402. Part of the work was done at the Kavli Institute for
Theoretical Physics (KITP) in Santa Barbara, CA. KITP is supported by the
National Science Foundation under Grants No. NSF PHY-1748958 and PHY-2309135.

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