# Obstructions to odd-frequency superconductivity in Eliashberg theory

Edwin Langmann  Department of Physics, KTH Royal Institute of Technology,
SE-106 91 Stockholm, Sweden  Nordita, KTH Royal Institute of Technology and
Stockholm University, SE-106 91 Stockholm, Sweden  Christian Hainzl
Mathematisches Institut, Ludwig-Maximilians-Universität München, 80333 Munich,
Germany  Robert Seiringer  IST Austria, Am Campus 1, 3400 Klosterneuburg,
Austria  Alexander V. Balatsky  Nordita, KTH Royal Institute of Technology and
Stockholm University, SE-106 91 Stockholm, Sweden  Department of Physics,
University of Connecticut, Storrs, CT 06268, USA

(July 4, 2022)

###### Abstract

We present a necessary condition for odd-frequency (odd-f) superconductivity
(SC) to occur in a large class of materials described by Eliashberg theory. We
use this condition to prove a no-go theorem ruling out the occurrence of odd-f
SC in standard one-band superconductors with pairing interactions mediated by
phonon exchange. We also present a corresponding no-go theorem for
superconductors with interactions mediated by spin-fluctuations. Our results
explain why odd-f SC is rare in conventional materials, and they open up the
possibility for a search for materials with interactions designed so as to
allow for odd-f SC.

Introduction. Superconductivity (SC) is a fascinating phenomenon where quantum
coherence is developed on a macroscopic scale. The usual narrative is that the
condensation of Cooper pairs leads to coherence that is responsible for the
striking phenomena exhibited by superconductors, including resistance-less
flow of an electron fluid, Meissner effect, superconducting vortices, and the
Josephson effect. Ginzburg-Landau theory for SC uses the pair condensate wave
function, $\Psi({\bf r},t)=\langle c_{\uparrow}({\bf r},t)c_{\downarrow}({\bf
r},t)\rangle$, as an order parameter [1, 2] ($c_{\sigma}({\bf r},t)$ are
electron operators at position ${\bf r}$, time $t$, and with spin index
$\sigma=\uparrow,\downarrow$). This BCS-Ginzburg-Landau paradigm has been
tremendously successful; ultimately, it enabled the viable platform for
quantum computing based on Josephson junctions [3].

With all the success of the current paradigm, there is a major omission in the
discussion of SC: it omits the challenge posed by the existence of a whole
class called odd-frequency (odd-f) superconductors, see e.g. [4, 5]. The name
odd-f is a consequence of highly nontrivial pairing correlations where
amplitudes $F_{\sigma,\sigma^{\prime}}({\bf r},t|{\bf
r^{\prime}},t^{\prime})=\langle c_{\sigma}({\bf r},t)c_{\sigma^{\prime}}({\bf
r^{\prime}},t^{\prime})\rangle$ are odd functions under permutation of
(Matsubara imaginary) times $t$ and $t^{\prime}$; hence, the equal time order
parameter $\Psi({\bf r},t)$, which is proportional to the equal time pairing
amplitude, vanishes. This type of pairing correlations, along with the title
odd-f (sometimes called Berezinskii pairing), was initially proposed for spin
triplet superfluid ${}^{3}\mathrm{He}$ by Berezinskii [6]. One can extend
these pairing states to the case of superconductors with electronic condensate
[7]. By now, we know that odd-f pairing correlations can be induced in SCFM
(ferromagnetic heterostructures), in the vortex cores, in superfluid 3He near
boundaries, in Josephson junctions, in interacting Majorana fermions, and in
Dirac semimetals, provided that the pairing interactions are strong enough
[5].

With the literature on emergent odd-f pairing states growing, it is still
viewed as an exotic and remote possibility. Indeed, in a majority of
superconducting states, the conventional even-frequency (even-f) BCS pairing
channel is dominant. Moreover, after unsuccessful attempts to find odd-f SC by
solving the Eliashberg equations numerically, it was argued already early on
that pairing by phonons is unlikely to give odd-f SC [8]; see also [9] for a
recent such argument. Hence the question is: why are odd-f pairing states so
rare and hard to stabilize? Or more precisely: if the odd-f pairing is a
fundamentally allowed pairing state, why is it so hard to observe in nature?
One possible answer is that we simply do not understand all the physical
properties of odd-f superconductors and how they are different from even-f
BCS-like superconductors. Here, we present an alternative explanation for the
relative scarcity of odd-f pairing states found to date: conventional
materials in nature typically have interaction potentials that do not allow
for odd-f superconducting solutions of the pertinent Eliashberg equations.
Since odd-f pairing requires strong interactions, one needs to keep track of
both normal selfenergy renormalizations as well as pairing self energies on
equal footing. We find an obstruction of the gap equation and selfenergy
correction to have a non-trivial solution in the odd-f channel for the
specific cases of spin independent electron-phonon interaction: The strong
coupling required to produce odd-f pairing states at the same time leads to a
strong selfenergy renormalization that ultimately prohibits odd-f states. We
formulate this obstruction as a no-go theorem for odd-f pairing states within
the Eliashberg formalism for specific cases. This theorem explains why
numerous attempts to produce odd-f states in the Eliashberg framework have
failed. It also clarifies how one would have to change the interactions to
allow for odd-f SC. We suggest to turn this theorem into a design principle
for pairing interactions that lead to odd-f SC states in the Eliashberg
approach. Thus, the challenge is shifted to finding materials where such an
interaction is realized.

It is important to note that our results hold true for the general Eliashberg
equations, taking into account the (Matsubara) frequency dependence and
momentum dependence of the interactions, without any further approximation.
Commonly, these general Eliashberg equations are simplified by the local
approximation, eliminating the momentum dependence [10]; in particular, the
original works on odd-f SC were based on these simplified Eliashberg equations
[7, 8]. We derive our no-go theorem for the simplified Eliashberg equations in
the main text; the extension to the general Eliashberg equations is explained
in the main text, with the detailed proof deferred to Appendix A. For
simplicity, we only consider the linearized Eliashberg equations determining
the superconducting critical temperature, $T_{c}$, in the main text; as shown
in Appendix A, it is easy to drop this restriction. In the main text, we also
present another no-go theorem derived in Appendix A, which applies to systems
that, in addition to Coulomb- and phonon interactions, also have interactions
mediated by spin fluctuations.

SP∗T∗-rule. We start by discussing symmetries that allow to classify different
possibilities for superconducting states in one-band models, including odd-f
SC [5]. We consider models with fermion operators,
$c_{\sigma}(\mathrm{i}\omega_{n},\mathbf{k})$ [11], labeled by a spin index,
$\sigma=\uparrow,\downarrow$, Masubara frequencies, $\omega_{n}=(2n+1)\pi T$
with integers $n$ and $T>0$ the temperature, and pseudo-momenta in the
Brillouin zone, $\mathbf{k}$. A possible superconducting state is described by
the fermion correlation functions

$F_{\sigma,\sigma^{\prime}}(\mathrm{i}\omega_{n},\mathbf{k})\equiv\langle
c_{\sigma}(\mathrm{i}\omega_{n},\mathbf{k})c_{\sigma^{\prime}}(-\mathrm{i}\omega_{n},-\mathbf{k})\rangle,$
(1)

and $G_{\sigma,\sigma^{\prime}}(\mathrm{i}\omega_{n},\mathbf{k})\equiv\langle
c^{\dagger}_{\sigma}(\mathrm{i}\omega_{n},\mathbf{k})c_{\sigma^{\prime}}(\mathrm{i}\omega_{n},\mathbf{k})\rangle$,
assuming thermal equilibrium and translational invariance. We consider
transformations $S$, $P^{*}$, and $T^{*}$ that swap the spin index, the sign
of the momentum, and the sign of the Matsubara frequency, respectively:

$\begin{split}S:\;\sigma\leftrightarrow\sigma^{\prime},\quad
P^{*}:\;\mathbf{k}\to-\mathbf{k},\quad
T^{*}:\;\omega_{n}\to-\omega_{n}.\end{split}$ (2)

Note that all these transformations square to the identity $1$, i.e., the
possible eigenvalues of $S$, $P^{*}$ and $T^{*}$ are $+1$ and $-1$. Obviously
the combined transformation $SP^{*}T^{*}$ interchanges the two fermion
operators in (1) and, by the anti-commutativity of fermion operators,
$SP^{*}T^{*}=-1$. We are mainly interested in odd-f SC where $T^{*}=-1$, i.e.,
$F$ is a odd function of Matsubara frequencies. The one-band models we
consider are invariant under the transformation $S$, $P^{*}$ and $T^{*}$
separately, which implies that the superconducting gap, $\Delta$, has the same
transformation properties under these transformations as $F$. In the odd-f SC
case there are two possibilities: $(S,P^{*},T^{*})=(-,-,-)$ and $(+,+,-)$,
which correspond to spin-singlet odd-parity odd-f SC and spin-triplet even-
parity odd-f SC, respectively. We recall for later reference that, in a
rotation invariant system, $P^{*}=+1$ corresponds to $s$\- or $d$-wave SC,
whereas $P^{*}=-1$ corresponds to $p$-wave SC.

No-go theorem in special case. We consider the simplified Eliashberg equations
obtained by projection to the $s$-wave channel and ignoring the
$|\mathbf{k}|-$dependence of the interactions (local approximation); see [10,
12, 13, 14].

The simplified Eliashberg equations for a one-band superconductor in the local
approximation determine the superconducting order parameter (also known as
gap), $\Delta(\mathrm{i}\omega_{n})$, and the normal-state renormalization,
$Z(\mathrm{i}\omega_{n})$, as follows (we use units such that $k_{B}=\hbar=1$)

$\begin{split}Z(\mathrm{i}\omega_{n})\Delta(\mathrm{i}\omega_{n})&=-T\sum_{m}V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\frac{\pi}{|\omega_{m}|}\Delta(\mathrm{i}\omega_{m}),\\\
Z(\mathrm{i}\omega_{n})&=1-\frac{\pi}{\omega_{n}}T\sum_{m}V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\frac{\omega_{m}}{|\omega_{m}|}.\end{split}$
(3)

The standard potential describing interactions mediated by phonons is given by

$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\mu^{*}\theta(1-|\omega_{n}|/\omega_{c})\theta(1-|\omega_{m}|/\omega_{c})\\\
-\int_{0}^{\infty}\alpha^{2}F(\Omega)\frac{2\Omega}{\Omega^{2}+(\omega_{n}-\omega_{m})^{2}}d\Omega$
(4)

where $\mu^{*}>0$ is the Coulomb pseudo-potential, $\theta(x)$ the Heaviside
function, $\omega_{c}>0$ an energy cutoff, and $\alpha^{2}F(\Omega)\geq 0$ the
Eliashberg function [10].

By writing the gap as a sum of even-f and odd-f parts, $\Delta^{+}$ and
$\Delta^{-}$, respectively: $\Delta=\Delta^{+}+\Delta^{-}$ with

$\Delta^{\pm}(\mathrm{i}\omega_{n})\equiv\frac{1}{2}[\Delta(\mathrm{i}\omega_{n})\pm\Delta(-\mathrm{i}\omega_{n})]=\pm\Delta^{\pm}(-\mathrm{i}\omega_{n}),$
(5)

the first equation in (3) decouples:
$Z(\mathrm{i}\omega_{n})\Delta^{\pm}(\mathrm{i}\omega_{n})=-T\sum_{m\geq
0}V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\pi\Delta^{\pm}(\mathrm{i}\omega_{m})/\omega_{m}$
for $n\geq 0$ with

$V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\equiv
V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\pm
V(\mathrm{i}\omega_{n},-\mathrm{i}\omega_{m})$ (6)

(we used that
$V(\mathrm{i}\omega_{n},-\mathrm{i}\omega_{m})=V(-\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})$,
which implies that $Z$ is even:
$Z(-\mathrm{i}\omega_{n})=Z(\mathrm{i}\omega_{n})$). Moreover, the second
equation in (3) can be written as $Z(\mathrm{i}\omega_{n})=1-(\pi
T/\omega_{n})\sum_{m\geq 0}V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})$,
and by inserting this into the equation for $\Delta^{-}$, we obtain

$\Delta^{-}(\mathrm{i}\omega_{n})=\pi T\sum_{m\geq
0}V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\left(\frac{\Delta^{-}(\mathrm{i}\omega_{n})}{\omega_{n}}-\frac{\Delta^{-}(\mathrm{i}\omega_{m})}{\omega_{m}}\right),$
(7)

which is an eigenvalue equation of a temperature dependent matrix determining
$T_{c}$. Multiplying both sides of this equation by the complex conjugate of
$\Delta^{-}(\mathrm{i}\omega_{n})/\omega_{n}$, summing over $n\geq 0$, and
symmetrizing the resulting double sum on the right-hand side using
$V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=V^{-}(\mathrm{i}\omega_{m},\mathrm{i}\omega_{n})$
we obtain (further details can be found in Appendix A.1)

$T\sum_{n\geq
0}\frac{|\Delta^{-}(\mathrm{i}\omega_{n})|^{2}}{\omega_{n}}=T^{2}\frac{\pi}{2}\sum_{m,n\geq
0}V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\left|\frac{\Delta^{-}(\mathrm{i}\omega_{n})}{\omega_{n}}-\frac{\Delta^{-}(\mathrm{i}\omega_{m})}{\omega_{m}}\right|^{2}.$
(8)

Note that this is an exact consequence of the Eliashberg equations (3). Our
odd-f-condition is as follows: To obtain odd-f SC, the interaction potential
$V$ must be such that the identity in Eq. (8) does not lead to a
contradiction. As we now show, this condition is powerful: We compute $V^{-}$
for the interaction (4) and obtain

$V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=-\int_{0}^{\infty}\alpha^{2}F(\Omega)\frac{8\Omega\omega_{n}\omega_{m}}{[\Omega^{2}+(\omega_{n}-\omega_{m})^{2}][\Omega^{2}+(\omega_{n}+\omega_{m})^{2}]}d\Omega\quad(n,m>0),$
(9)

which is manifestly non-positive (note that minus sign and recall that
$\alpha^{2}F(\Omega)\geq 0$). Clearly, this non-positiveness leads to a
contradiction in (8) unless $\Delta^{-}=0$. This rules out spin triplet
$s$-wave odd-f SC for the simplified Eliashberg equations (3)–(4). As
explained below, one can adapt this argument to general Eliashberg theory and,
by that, rule out odd-f SC in any channel: our general result is as follows.

No-go Theorem I: In $P^{*}T^{*}$-invariant Eliashberg theory, odd-f SC is
impossible in a standard one-band superconductor with non-dynamically screened
Coulomb interactions and attractive interactions mediated by phonons.

No-go theorem in general case. We proceed by explaining the precise meaning of
No-go Theorem I for Eliashberg theory without local approximation.

The general (linearized) Eliashberg equations which determine the
superconducting order parameter, $\phi$, and the normal-state self-energy,
$\Sigma$, depending on (Matsubara) frequency, $\omega_{n}$, and momentum,
$\mathbf{k}$, are given by

$\begin{split}\phi(\mathrm{i}\omega_{n},\mathbf{k})&=T\sum_{m}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})F(\mathrm{i}\omega_{m},\mathbf{k}^{\prime}),\\\
\Sigma(\mathrm{i}\omega_{n},\mathbf{k})&=-T\sum_{m}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})G(\mathrm{i}\omega_{m},\mathbf{k}^{\prime}),\end{split}$
(10)

with $\int\\!\\!{\rm d}\mathbf{k}^{\prime}\,$ short for
$\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}$, where the superconducting and normal
state parts of the fermion correlations functions, $F$ and $G$, are determined
as follows,
$F(\mathrm{i}\omega_{n},\mathbf{k})=-\phi(\mathrm{i}\omega_{n},\mathbf{k})/[\tilde{\omega}_{n}(\mathbf{k})^{2}+\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})^{2}]$
and
$G(\mathrm{i}\omega_{n},\mathbf{k})=1/[\mathrm{i}\tilde{\omega}_{n}(\mathbf{k})-\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})]$;
see [10, 12, 13] and references therein. We use standard notation:
$\tilde{\omega}_{n}(\mathbf{k})\equiv\omega_{n}Z(\mathrm{i}\omega_{n},\mathbf{k})$
and
$\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})\equiv\epsilon(\mathbf{k})+\chi(\mathrm{i}\omega_{n},\mathbf{k})$
with the mass and band renormalizations, $Z$ and $\chi$, determined by the
self-energy as follows [10],
$1-Z(\mathrm{i}\omega_{n},\mathbf{k})\equiv[\Sigma(\mathrm{i}\omega_{n},\mathbf{k})-\Sigma(-\mathrm{i}\omega_{n},\mathbf{k})]/2\mathrm{i}\omega_{n}$
and
$\chi(\mathrm{i}\omega_{n},\mathbf{k})\equiv[\Sigma(\mathrm{i}\omega_{n},\mathbf{k})+\Sigma(-\mathrm{i}\omega_{n},\mathbf{k})]/2$.
Note that the model is defined by the band relation, $\epsilon(\mathbf{k})$,
and the two-body potential,
$V=V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})$;
see Appendix A.2 for examples. Moreover, the equations hold true for spin-
singlet and spin-triplet SC, and for $s$-, $p$-, $d$\- or $f$-wave SC [10].
The only assumption we make is invariance under the transformations $T^{*}$
and $P^{*}$, i.e.,

$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=V(-\mathrm{i}\omega_{n},-\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};-\mathbf{k},-\mathbf{k}^{\prime})$
(11)

and $\epsilon(\mathbf{k})=\epsilon(-\mathbf{k})$. We also recall the
definition of the superconducting gap [10],
$\Delta(\mathrm{i}\omega_{n},\mathbf{k})\equiv\phi(\mathrm{i}\omega_{n},\mathbf{k})/Z(\mathrm{i}\omega_{n},\mathbf{k})$.

Similarly as in the simplified case with the local approximation above, one
can show that these Eliashberg equations lead to decoupled equations for the
even-f and odd-f gap solutions $\Delta^{+}$ and $\Delta^{-}$, respectively,
with $\Delta^{-}$ satisfying the following exact identity

$T\sum_{n\geq 0}\int\\!\\!{\rm
d}\mathbf{k}\,\frac{|\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k})|^{2}}{W(\mathrm{i}\omega_{n},\mathbf{k})\omega_{n}}=T^{2}\sum_{n,m\geq
0}\int\\!\\!{\rm d}\mathbf{k}\,\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{2W(\mathrm{i}\omega_{n},\mathbf{k})W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}\Bigl{|}\frac{\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k})}{\omega_{n}}-\frac{\Delta^{-}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}{\omega_{m}}\Bigr{|}^{2},$
(12)

$V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\equiv
V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})-V(-\mathrm{i}\omega_{n},\mathrm{i}\omega_{m},\mathbf{k},\mathbf{k}^{\prime})$,
$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\equiv\left[\tilde{\omega}(\mathbf{k}^{\prime})^{2}+\tilde{\epsilon}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})^{2}\right]/\omega_{m}Z(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})$;
the interested reader can find a detailed derivation of this fact in Appendix
A.1; see Proposition 1. Thus, we get, again, the odd-f condition as formulated
above with Eq. (8) replaced by Eq. (12). Moreover, similarly as for the
special case discussed in more detail above, for standard potentials, $V$,
which describe screened Coulomb interactions and attractive interactions
mediated by phonons, one finds that $V^{-}$ is manifestly non-positive, and
this proves No-go theorem I as it stands for the general Eliashberg equations;
the interested reader can find details of our proof in Appendix A.2. We stress
that this no-go theorem is very general: it is true for any kind of odd-f SC,
i.e., the spin-structure and angular dependence of the gap make no difference.

Spin dependent interactions. Our results above generalize to models with spin
dependent interaction potentials. In Appendix A.1, we discuss more general
model potentials which not only include screened Coulomb interactions,
$V_{\mathrm{C}}$, and phonon interactions, $V_{\mathrm{ph}}$, but also
interactions mediated by spin fluctuations, $V_{\mathrm{sf}}$. In this case,
we get another no-go theorem (No-go Theorem II in Appendix A.3) ruling out
spin-triplet odd-f SC in a one-band superconductor with phonon- and spin-
fluctuation mediated interactions.

Recall that, for odd-f SC, spin-singlet corresponds to $S=-1$ and $P^{*}=-1$,
and spin-triplet corresponds to $S=+1$ and $P^{*}=+1$. Thus, the no-go theorem
allows for spin-singlet $p$-wave odd-f SC, in agreement with results in [15].
However, by our result, spin-triplet $s$\- and $d$\- wave odd-f SC are
impossible.

Interactions leading to odd-f SC. Our results above rule out odd-f SC in a
large class of models describing real materials. However, they can also be
used to design interactions leading to odd-f SC. To give examples, we mainly
restrict ourselves to the Eliashberg equations (3) for simplicity (local
approximation and spin-independent interactions).

A large class of examples avoiding our no-go theorems are repulsive
interactions, $V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\geq 0$, which are
monotonically decreasing functions of $|\omega_{n}-\omega_{m}|$. We study
three such examples. Our first example is
$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\lambda\Omega^{2}/[\Omega^{2}+(\omega_{n}-\omega_{m})^{2}]$
for $\lambda>0$ ($\Omega>0$ is an energy scale, here and in the following);
this is like a phonon interaction, except that the coupling constant has the
opposite sign. To obtain a more physical such example, we start with the
momentum-dependent potential given in [16, Eq. (2)] describing screened
Coulomb interactions close to a quantum critical point,

$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=\frac{g_{\text{eff}}}{|\mathbf{k}-\mathbf{k}^{\prime}|^{2}+\Gamma\frac{|\omega_{n}-\omega_{m}|}{|\mathbf{k}-\mathbf{k}^{\prime}|}}$
(13)

for $g_{\text{eff}}>0$ and $\Gamma>0$; see [17] for other systems described by
this and similar such potentials. We note that this potential satisfies our
odd-f condition after (12). In three dimensions, by restricting the momenta to
the Fermi surface, $|\mathbf{k}|=|\mathbf{k}^{\prime}|=k_{F}$ (Fermi
momentum), and angular averaging, one can reduce this potential to
$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\lambda\log(1+\Omega/|\omega_{n}-\omega_{m}|)$
with $\lambda=g_{\text{eff}}N(0)/6k_{F}^{2}>0$ and $\Omega=8k_{F}^{3}/\Gamma$,
where $N(0)$ is the electronic density of states at the Fermi surface (this
standard approximation reducing the general Eliashberg equation to (3) is
discussed in [10]); the latter potential is our second example. Our third
example is the potential
$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\lambda(\Omega/|\omega_{n}-\omega_{m}|)^{\gamma}$
for $\gamma>0$, which is similar to the potential of the $\gamma$-model
studied recently in [18, 16] but with the opposite sign of the coupling
constant (we rename $g_{\text{eff}}$ to $\Omega$ and introduce a coupling
constant $\lambda>0$). We solved the $T_{c}$-equation (7) for these three
examples numerically and found robust odd-f solutions, with $T_{c}$ increasing
monotonically with the coupling constant in all three examples. We also
obtained a simple exact $T_{c}$-equation for the third example:
$T_{c}=K\Omega\lambda^{1/\gamma}$, where $K$ is a constant that can be
computed numerically. The interested reader can find details and numerical
results in Appendix B.

Conclusions. We present a new constraint on pairing interactions to allow for
odd-f solutions within the Eliashberg framework (odd-f condition). Odd-f SC
requires strong coupling to stabilize the odd-f channel and favor it over
even-f channel. However, both anomalous and normal selfenergies are
renormalized by interactions. By combining the corresponding two Eliashberg
equations, we find a strong constraint on pairing interactions: for spin
independent interactions mediated by phonons, we find odd-f solutions are
impossible within the Eliashberg formalism (No-go theorem I). We also consider
spin dependent interactions mediated by spin-fluctuations, and we find that
single band models cannot support $s$-wave odd-f solutions. Since our no-go
theorems apply to a large class of models used to describe real materials,
they explain why it has been so difficult to find odd-f SC.

We note that multiband systems do not suffer from this constraint and hence
represent a fertile ground to pursue odd-f solutions; for experimentally
observable signatures of odd-frequency pairing in multiband superconductors,
see e.g. [19, 20, 21, 22, 23] and references therein.

Our no-go theorems are about models with standard interactions occurring in
nature, but there is no general mathematical obstruction to odd-f SC if one
drops this latter restriction. As we show in examples, using our results, one
can construct models with pairing interactions that allow for odd-f SC. Thus,
our results can be used as a design principle to construct models for odd-f SC
which, as we hope, can guide the search for real odd-f materials.

Our results are an example of how mathematics can contribute to the theory of
superconductivity; see, e.g., [24, 25, 26] for recent other such examples.

Acknowledgements. We are grateful to A. Black-Schaffer, M. Geilhufe, P.
Coleman, P. Sukhachov, and C. Triola, for useful discussions. We thank A.
Chubukov for pointing out an important detail in Ref. [16] and valuable
comments. The work of AVB was supported by VILLUM FONDEN via the Centre of
Excellence for Dirac Materials (Grant No. 11744) and the European Research
Council ERC-2018-SyG HERO.

##  Appendix A Appendix A. Proof of no-go theorem for general Eliashberg
equations

We give detailed derivations of our results for the general case: a
$P^{*}T^{*}$-invariant one-band model given by a dispersion relation,
$\epsilon$, and two-body interaction potentials, $V$ and $\tilde{V}$. As
explained in Appendix A.2, for conventional superconductors with Coulomb
interactions and attractive interactions due to phonons, $V$ is the same for
spin singlet and spin triplet SC and $V=\tilde{V}$; see (40). However, our
derivation also applies to non-conventional superconductors discussed in
Appendix A.3 where, in addition to Coulomb- and phonon interactions, also
attractive interactions due to spin fluctuations are present. In the latter
case, there are different potentials $V$ for the spin singlet and spin triplet
cases, and $\tilde{V}$ differs from $V$; see (42)–(43). One specific example
for $\epsilon$ would be the Sommerfeld dispersion relation,
$\epsilon(\mathbf{k})=\mathbf{k}^{2}/2m^{*}-\mu$ with the effective mass
$m^{*}$ and the chemical potential $\mu$. However, we stress that our results
hold true also also for non-rotation invariant systems and systems with a
finite Brillouin zone.

###  A.1 A.1 Derivation of odd-f condition

In the general case, the Eliashberg equations given in the main text have to
be generalized by replacing (10) with

$\begin{split}\phi(\mathrm{i}\omega_{n},\mathbf{k})&=T\sum_{m}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})F(\mathrm{i}\omega_{m},\mathbf{k}^{\prime}),\\\
\Sigma(\mathrm{i}\omega_{n},\mathbf{k})&=-T\sum_{m}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\tilde{V}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})G(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\end{split}$
(14)

(our notation is explained in the main text), while we use generalizations of
formulas for $F$ and $G$ in the main text to $0\leq T\leq T_{c}$,

$\begin{split}F(\mathrm{i}\omega_{n},\mathbf{k})=-\frac{\phi(\mathrm{i}\omega_{n},\mathbf{k})}{\tilde{\omega}_{n}(\mathbf{k})^{2}+\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})^{2}+|\phi(\mathrm{i}\omega_{n},\mathbf{k})|^{2}},\\\
G(\mathrm{i}\omega_{n},\mathbf{k})=-\frac{\mathrm{i}\tilde{\omega}_{n}(\mathbf{k})+\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})}{\tilde{\omega}_{n}(\mathbf{k})^{2}+\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})^{2}+|\phi(\mathrm{i}\omega_{n},\mathbf{k})|^{2}}\end{split}$
(15)

(note that, for $|\phi(\mathrm{i}\omega_{n},\mathbf{k})|^{2}=0$, this
simplifies to the formulas for $F$ and $G$ given in the main text); the
definitions

$\tilde{\omega}_{n}(\mathbf{k})\equiv\omega_{n}Z(\mathrm{i}\omega_{n},\mathbf{k}),\quad\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})\equiv\epsilon(\mathbf{k})+\chi(\mathrm{i}\omega_{n},\mathbf{k}),\quad\Delta(\mathrm{i}\omega_{n},\mathbf{k})=\frac{\phi(\mathrm{i}\omega_{n},\mathbf{k})}{Z(\mathrm{i}\omega_{n},\mathbf{k})},$
(16)

with

$\begin{split}1-Z(\mathrm{i}\omega_{n},\mathbf{k})&\equiv\frac{1}{2\mathrm{i}\omega_{n}}[\Sigma(\mathrm{i}\omega_{n},\mathbf{k})-\Sigma(-\mathrm{i}\omega_{n},\mathbf{k})],\\\
\chi(\mathrm{i}\omega_{n},\mathbf{k})&\equiv\frac{1}{2}[\Sigma(\mathrm{i}\omega_{n},\mathbf{k})+\Sigma(-\mathrm{i}\omega_{n},\mathbf{k})]\end{split}$
(17)

are as in the main text; see [10, 12, 13, 14] and references therein for
background to these equations. Note that the spin dependence is eliminated
from these equations, and they hold true for spin-singlet and spin-triplet SC
with interaction potentials given below. We assume invariance under the
transformations $P^{*}$ and $T^{*}$, i.e., we assume (11) and the same
relations with $\tilde{V}$ instead of $V$ hold true, and, in addition,
$\epsilon(-\mathbf{k})=\epsilon(\mathbf{k})$.

We use the definition of the gap $\Delta$ in (16) and, similarly as before, we
divide the gap in even-f and odd-f parts,

$\Delta(\mathrm{i}\omega_{n},\mathbf{k})=\Delta^{+}(\mathrm{i}\omega_{n},\mathbf{k})+\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k}),\quad\Delta^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})=\frac{1}{2}\left(\Delta(\mathrm{i}\omega_{n},\mathbf{k})\pm\Delta(-\mathrm{i}\omega_{n},\mathbf{k})\right)=\pm\Delta^{\pm}(-\mathrm{i}\omega_{n},\mathbf{k}).$
(18)

In the following, we use the following definitions of the even- and odd parts
of the interaction potential $V$,

$V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\equiv
V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\pm
V(-\mathrm{i}\omega_{n},\mathrm{i}\omega_{m},\mathbf{k},\mathbf{k}^{\prime}),$
(19)

and similarly for $\tilde{V}$.

Lemma 1: The general Eliashberg equations above imply

$Z(-\mathrm{i}\omega_{n},\mathbf{k})=Z(\mathrm{i}\omega_{n},\mathbf{k}),\quad\chi(-\mathrm{i}\omega_{n},\mathbf{k})=\chi(\mathrm{i}\omega_{n},\mathbf{k}),$
(20)

and

$\begin{split}Z(\mathrm{i}\omega_{n},\mathbf{k})\Delta^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})&=-T\sum_{m\geq
0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}\frac{\Delta^{\pm}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}{\omega_{m}},\\\
\chi(\mathrm{i}\omega_{n},\mathbf{k})&=T\sum_{m\geq 0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{\tilde{V}^{+}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}\frac{\tilde{\epsilon}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}{\omega_{m}Z(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})},\\\
Z(\mathrm{i}\omega_{n},\mathbf{k})&=1-\frac{T}{\omega_{n}}\sum_{m\geq
0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{\tilde{V}^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})},\end{split}$
(21)

with

$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\equiv\frac{\tilde{\omega}(\mathbf{k}^{\prime})^{2}+\tilde{\epsilon}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})^{2}+|\phi(\mathrm{i}\omega_{n},\mathbf{k})|^{2}}{\omega_{m}Z(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})};$
(22)

for $T=T_{c}$, these results holds true for any solution of the Eliashberg
equations, and for $0<T<T_{c}$ they holds true for any solution such that
either $\Delta=\Delta^{+}$ or $\Delta=\Delta^{-}$.

Note that, by the transformation properties of $\Delta^{\pm}$, $Z$ and $\chi$
under $T^{*}:\;\omega_{n}\to-\omega_{n}$, one can restrict (21) to positive
Matsubara frequences, $n\geq 0$, and
$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\geq 0$ for all $m\geq 0$.

Remark 1: To explain the distinctions between $T=T_{c}$ and $T<T_{c}$ in Lemma
1, we note that the linearized Eliashberg equations determining $T_{c}$ are
obtained from this by simplifying
$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})$ in (22) to

$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\equiv\frac{\tilde{\omega}(\mathbf{k}^{\prime})^{2}+\tilde{\epsilon}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})^{2}}{\omega_{m}Z(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})},$
(23)

and this satisfies
$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})=-W(-\mathrm{i}\omega_{m},\mathbf{k}^{\prime})$
without further assumption; however, for $T<T_{c}$, this only holds true if
$|\phi(\mathrm{i}\omega_{n},\mathbf{k})|^{2}=|Z(\mathrm{i}\omega_{n},\mathbf{k})\Delta(\mathrm{i}\omega_{n},\mathbf{k})|^{2}=|\phi(-\mathrm{i}\omega_{n},\mathbf{k})|^{2}$
and, for this reason, our results hold true only if $\Delta=\Delta^{\pm}$.
Thus, in principle, our no-go theorems do not rule out the possibility of a
mixed gap at temperatures below $T_{c}$, i.e., they leave open the possibility
that, at $T<T_{c}$, $\Delta=\Delta^{+}+\Delta^{-}$ with $\Delta^{+}$ and
$\Delta^{-}$ both non-zero.

###### Proof of Lemma 1.

In the following, we use the notation
$\phi^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})\equiv[\phi(\mathrm{i}\omega_{n},\mathbf{k})\pm\phi(-\mathrm{i}\omega_{n},\mathbf{k})]/2$
and similarly for $\Sigma$, $F$ and $G$.

The symmetry properties stated in (20) are consequences of the definitions of
$Z$ and $\chi$ in (17). Moreover, (15), (16) and (20) imply
$\phi^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})=Z(\mathrm{i}\omega_{n},\mathbf{k})\Delta^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})$
and

$F^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})=-\frac{\Delta^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})}{\omega_{n}W(\mathrm{i}\omega_{n},\mathbf{k})},\quad
G^{+}(\mathrm{i}\omega_{n},\mathbf{k})=-\frac{\tilde{\epsilon}(\mathrm{i}\omega_{n},\mathbf{k})}{\omega_{n}Z(\mathrm{i}\omega_{n},\mathbf{k})W(\mathrm{i}\omega_{n},\mathbf{k})},\quad
G^{-}(\mathrm{i}\omega_{n},\mathbf{k})=-\frac{\mathrm{i}}{W(\mathrm{i}\omega_{n},\mathbf{k})}.$
(24)

The first equation in (14) and the definitions of $\phi^{\pm}$ and $V^{\pm}$
imply

$\begin{split}\phi^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})=&T\sum_{m}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{1}{2}V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})F(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\\\
&=T\sum_{m\geq 0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{1}{2}\left[V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})F(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})+V^{\pm}(\mathrm{i}\omega_{n},-\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})F(-\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\right].\end{split}$
(25)

Inserting
$V^{\pm}(\mathrm{i}\omega_{n},-\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=\pm
V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})$
implied by (11) and (19), we compute this as follows,

$\begin{split}\phi^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})&=T\sum_{m\geq
0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\frac{1}{2}[F(\mathrm{i}\omega;m,\mathbf{k}^{\prime})\pm
F(-\mathrm{i}\omega,\mathbf{k}^{\prime})]\\\ &=T\sum_{m\geq 0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})F^{\pm}(\mathrm{i}\omega_{m},\mathbf{k})=-T\sum_{m\geq
0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\frac{\Delta^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})}{\omega_{n}W(\mathrm{i}\omega_{n},\mathbf{k})},\end{split}$
(26)

recalling the definition of $F^{\pm}$ in the second identity and inserting
$F^{\pm}$ in (24) in the third. Since
$\phi^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})=Z(\mathrm{i}\omega_{n},\mathbf{k})\Delta^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})$,
this proves the first equation in (21).

In the same way, the second equation in (14) implies
$\Sigma^{\pm}(\mathrm{i}\omega_{n},\mathbf{k})=-T\sum_{m\geq 0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\tilde{V}^{\pm}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})G^{\pm}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})$,
and thus with (17) and (24),

$\begin{split}\chi(\mathrm{i}\omega_{n},\mathbf{k})&=\Sigma^{+}(\mathrm{i}\omega_{n},\mathbf{k})=-T\sum_{m\geq
0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\tilde{V}^{+}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})G^{+}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\\\
&=-T\sum_{m\geq 0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\tilde{V}^{+}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\frac{\tilde{\epsilon}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}{\omega_{n}Z(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}\end{split}$
(27)

and

$\begin{split}Z(\mathrm{i}\omega_{n},\mathbf{k})&=1-\frac{1}{\mathrm{i}\omega_{n}}\Sigma^{-}(\mathrm{i}\omega_{n},\mathbf{k})=1+\frac{T}{\mathrm{i}\omega_{n}}\sum_{m\geq
0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\tilde{V}^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})G^{-}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\\\
&=1-\frac{T}{\omega_{n}}\sum_{m\geq 0}\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\frac{\tilde{V}^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}.\end{split}$
(28)

This proves the second and the third equations in (21). ∎

We now are ready to state and prove the odd-f condition for the general
Eliashberg equations.

Proposition 1 (Odd-f Condition): The equations in (21) imply

$\begin{split}T\sum_{n\geq 0}\int\\!\\!{\rm
d}\mathbf{k}\,\frac{|\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k})|^{2}}{W(\mathrm{i}\omega_{n},\mathbf{k})\omega_{n}}=T^{2}\sum_{n,m\geq
0}\int\\!\\!{\rm d}\mathbf{k}\,\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,\Biggl{[}\frac{V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{2W(\mathrm{i}\omega_{n},\mathbf{k})W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}\Bigl{|}\frac{\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k})}{\omega_{n}}-\frac{\Delta^{-}(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}{\omega_{m}}\Bigr{|}^{2}\\\
+\frac{(\tilde{V}-V)^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})}{W(\mathrm{i}\omega_{n},\mathbf{k})W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})}\Bigl{|}\frac{\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k})}{\omega_{n}}\Bigr{|}^{2}\Biggr{]}\end{split}$
(29)

where

$(\tilde{V}-V)^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\equiv\tilde{V}^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})-V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime}).$
(30)

Thus, to to obtain odd-f SC, the interactions $V$ and $\tilde{V}$ must be such
that (29) does not lead to a contradiction.

Note that, for $\tilde{V}=V$ and $W$ in (23) (which corresponds to spin-
independent interactions at $T=T_{c}$), (29) reduces to (12) in the main text.

###### Proof of Proposition 1:.

In our argument below, we use the following property of interaction potentials
$V$:

$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=V(\mathrm{i}\omega_{m},\mathrm{i}\omega_{n};\mathbf{k}^{\prime},\mathbf{k}).$
(31)

It is important to note that this always holds true (for the convenience of
the reader, we recall the argument for spin independent interactions: in the
functional integral formalism (see e.g. Ref. 27), the interaction part of the
action defining the interacting fermion model is
$\frac{1}{2}T^{2}\sum_{n,m}\int\\!\\!{\rm d}\mathbf{k}\,\int\\!\\!{\rm
d}\mathbf{k}^{\prime}\,V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})\hat{n}(\mathrm{i}\omega_{n},\mathbf{k})\hat{n}(\mathrm{i}\omega_{m};\mathbf{k}^{\prime})$
where $\hat{n}(\mathrm{i}\omega_{n},\mathbf{k})$ are Fourier transformed
fermion densities which commute and thus, clearly, one can assume (31) without
loss of generality).

We find it convenient to abbreviate the variables $(\omega_{n},\mathbf{k})$
and $(\omega_{m},\mathbf{k}^{\prime})$ by $N$ and $M$, respectively, and
$T\sum_{m\geq 0}\int\\!\\!{\rm d}\mathbf{k}^{\prime}\,$ by $\sum_{M}$ etc.
This allows us to write the first and third equations in (21) in the odd-f
case short as

$Z_{N}\Delta^{-}_{N}=-\sum_{M}\frac{V^{-}_{NM}}{W_{M}}\frac{\Delta^{-}_{M}}{\omega_{M}},\quad
Z_{N}=1-\frac{1}{\omega_{N}}\sum_{M}\frac{\tilde{V}^{-}_{NM}}{W_{M}}$ (32)

with $\omega_{N}\equiv\omega_{n}$, $Z_{N}\equiv
Z(\mathrm{i}\omega_{n},\mathbf{k})$, $V^{-}_{NM}\equiv
V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})$
etc. Inserting the second in the first equation we obtain

$\Delta_{N}^{-}-\sum_{M}\frac{\tilde{V}^{-}_{NM}}{W_{M}}\frac{\Delta_{N}^{-}}{\omega_{N}}=-\sum_{M}\frac{V^{-}_{NM}}{W_{M}}\frac{\Delta^{-}_{M}}{\omega_{M}},$
(33)

which is equivalent to

$\Delta_{N}^{-}=\sum_{M}\left[\frac{V^{-}_{NM}}{W_{M}}\left(\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right)+\frac{(\tilde{V}-V)^{-}_{NM}}{W_{M}}\frac{\Delta^{-}_{N}}{\omega_{N}}\right]$
(34)

where $(\tilde{V}-V)^{-}_{NM}\equiv\tilde{V}^{-}_{NM}-V^{-}_{NM}$. Multiplying
both sides of this identity with $\overline{\Delta^{-}_{N}}/W_{N}\omega_{N}$
and summing over $N$ we find (the bar indicates complex conjugation),

$\sum_{N}\frac{|\Delta^{-}_{N}|^{2}}{W_{N}\omega_{N}}=\sum_{M,N}\left[\frac{V^{-}_{NM}}{W_{N}W_{M}}\left(\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right)\frac{\overline{\Delta^{-}_{N}}}{\omega_{N}}+\frac{(\tilde{V}-V)^{-}_{NM}}{W_{N}W_{M}}\left|\frac{\Delta^{-}_{N}}{\omega_{N}}\right|^{2}\right].$
(35)

We observe that (11), the definition of $V^{-}$ in (19), and (31) imply that
$V^{-}_{NM}$ is even under the exchange $N\leftrightarrow M$:
$V^{-}_{MN}=V^{-}_{NM}$. We use this to anti-symmetrize the first sum on the
right-hand side in (35):

$\sum_{M,N}\frac{V^{-}_{NM}}{W_{N}W_{M}}\left(\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right)\frac{\overline{\Delta^{-}_{N}}}{\omega_{N}}=\sum_{N,M}\frac{V^{-}_{MN}}{W_{M}W_{N}}\left(\frac{\Delta^{-}_{M}}{\omega_{M}}-\frac{\Delta^{-}_{N}}{\omega_{N}}\right)\frac{\overline{\Delta^{-}_{M}}}{\omega_{M}}=-\sum_{M,N}\frac{V^{-}_{NM}}{W_{N}W_{M}}\left(\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right)\frac{\overline{\Delta^{-}_{M}}}{\omega_{M}}$
(36)

(in the first identity, we only renamed summation variables $(N,M)\to(M,N)$),
and by equating the left-hand side with the average of the left- and right-
hand sides of this identity we obtain

$\begin{split}\sum_{M,N}\frac{V^{-}_{NM}}{W_{N}W_{M}}\left(\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right)\frac{\overline{\Delta^{-}_{N}}}{\omega_{N}}=&\frac{1}{2}\sum_{M,N}\frac{V^{-}_{NM}}{W_{N}W_{M}}\left(\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right)\left(\frac{\overline{\Delta^{-}_{N}}}{\omega_{N}}-\frac{\overline{\Delta^{-}_{M}}}{\omega_{M}}\right)\\\
=&\sum_{M,N}\frac{V^{-}_{NM}}{2W_{N}W_{M}}\left|\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right|^{2}.\end{split}$
(37)

Inserting this into (35), we obtain

$\sum_{N}\frac{|\Delta^{-}_{N}|^{2}}{W_{N}\omega_{N}}=\sum_{M,N}\Biggl{[}\frac{V^{-}_{NM}}{2W_{N}W_{M}}\left|\frac{\Delta^{-}_{N}}{\omega_{N}}-\frac{\Delta^{-}_{M}}{\omega_{M}}\right|^{2}+\frac{(\tilde{V}-V)^{-}_{NM}}{W_{N}W_{M}}\left|\frac{\Delta^{-}_{N}|}{\omega_{N}}\right|^{2}\Biggr{]},$
(38)

which is the identity in (29).

We showed that (29) is an exact consequence of the Eliashberg equations. Thus,
clearly, any odd-f solution $\Delta=\Delta^{-}$ of the Eliashberg equations
must satisfy (29). ∎

###  A.2 A.1 Spin independent interactions

We first consider a standard superconductor where the electrons interact with
screened Coulomb interactions,
$V_{\mathrm{C}}(\mathbf{k},\mathbf{k}^{\prime})$, independent of the
frequencies $\omega_{n}$ and $\omega_{m}$, and with attractive interactions
mediated by phonons,

$V_{\mathrm{ph}}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=-\frac{1}{N(0)}\int_{0}^{\infty}\alpha^{2}F(\Omega;\mathbf{k},\mathbf{k}^{\prime})\frac{2\Omega}{\Omega^{2}+(\omega_{n}-\omega_{m})^{2}}d\Omega,$
(39)

with $N(0)$ the electron density at the Fermi surface and
$\alpha^{2}F(\Omega;\mathbf{k},\mathbf{k}^{\prime})$ the Eliashberg function;
see [10]. Ignoring impurities one has, in this case,

$V=\tilde{V}=V_{\mathrm{C}}+V_{\mathrm{ph}}$ (40)

(note that, since $(\tilde{V}-V)^{-}=0$, (29) simplifies to (12)). By simple
computations we find $(V_{\mathrm{C}})^{-}=0$ (since $V_{\mathrm{C}}$ is
frequency independent) and

$(V_{\mathrm{ph}})^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})=-\frac{1}{N(0)}\int_{0}^{\infty}\alpha^{2}F(\Omega;\mathbf{k},\mathbf{k}^{\prime})\frac{8\Omega\omega_{n}\omega_{m}}{[\Omega^{2}+(\omega_{n}-\omega_{m})^{2}][\Omega^{2}+(\omega_{n}+\omega_{m})^{2}]}d\Omega\quad(n,m\geq
0).$ (41)

We recall that the Eliashberg function is determined by the phonon
frequencies, $\omega_{\lambda}(\mathbf{k})$, and electron-phonon coupling
constants, $g_{\lambda}(\mathbf{k},\mathbf{k}^{\prime})$, as follows,
$\alpha^{2}F(\Omega;\mathbf{k},\mathbf{k}^{\prime})=\sum_{\lambda}|g_{\lambda}(\mathbf{k},\mathbf{k}^{\prime})|^{2}\delta(\Omega-\omega_{\lambda}(\mathbf{k}-\mathbf{k}^{\prime}))$,
and thus, $\alpha^{2}F(\Omega;\mathbf{k},\mathbf{k}^{\prime})\geq 0$. This
implies that
$(V_{\mathrm{ph}})^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})$
in (41) is non-positive. Since $(\tilde{V}-V)^{-}=0$,
$V^{-}=(V_{\mathrm{ph}})^{-}\leq 0$, and
$W(\mathrm{i}\omega_{m},\mathbf{k}^{\prime})\geq 0$ for $n,m\geq 0$, the RHS
in (29) is non-positive, which is in contradiction with the LHS in (29) which
is manifestly non-negative. The only way to avoid this contradiction is
$\Delta^{-}(\mathrm{i}\omega_{n},\mathbf{k})=0$ for all $n\geq 0$ and
$\mathbf{k}$, i.e., the Eliashberg equations do not have a non-trivial odd-f
gap solution. This proves No-go theorem I stated in the main text (note that,
there, it only was proved for the Eliashberg equations in the local
approximation). Note that this no-go theorem is very general: it is true for
any kind of odd-f SC, i.e., the spin-structure and angular dependence of the
gap do not matter.

###  A.3 A.2 Spin dependent interactions

We now consider a model for a non-conventional superconductor where also
interactions mediated by spin-fluctuations are present. In this case, the
pairing interaction $V$ depends on whether the gap is spin-singlet or spin-
triplet,

$V=\begin{cases}V^{\mathrm{s}}=V_{\mathrm{C}}+V_{\mathrm{ph}}-V_{\mathrm{sf}}&\text{(spin
singlet)}\\\
V^{\mathrm{t}}=V_{\mathrm{C}}+V_{\mathrm{ph}}+\frac{1}{3}V_{\mathrm{sf}}&\text{(spin
triplett)}\end{cases},$ (42)

with $V_{\mathrm{sf}}$ the spin-fluctuation interaction, whereas the
interaction affecting the normal state is the same in both cases,

$\tilde{V}=V_{\mathrm{C}}+V_{\mathrm{ph}}+V_{\mathrm{sf}};$ (43)

see [28, Appendix C]. The spin-fluctuation interaction is given by an
expression as in Eq. (39) but with
$\alpha^{2}F(\Omega;\mathbf{k},\mathbf{k}^{\prime})$ replaced by another non-
negative function proportional to the imaginary part of the spin
susceptibility; the only property we need is that
$(V_{\mathrm{sf}})^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m};\mathbf{k},\mathbf{k}^{\prime})$
is non-positive, similarly to (41). In the spin-triplet case we have
$V^{-}=(V_{\mathrm{ph}})^{-}+\frac{1}{3}(V_{\mathrm{sf}})^{-}$ and
$(\tilde{V}-V)^{-}=\frac{2}{3}(V_{\mathrm{sf}})^{-}$, which both are non-
positive, and thus (29) leads to a contradiction, as above. However, in the
spin-singlet case, $V^{-}=(V_{\mathrm{ph}})^{-}-(V_{\mathrm{sf}})^{-}$ and
$(\tilde{V}-V)^{-}=2(V_{\mathrm{sf}})^{-}$: since $-(V_{\mathrm{sf}})^{-}$ is
non-negative, $V^{-}$ can be positive and thus, in this case, (29) cannot rule
out possible odd-f SC.

We summarize this result as follows.

No-go theorem II. In $P^{*}T^{*}$-invariant Eliashberg theory, spin-triplet
odd-f SC is impossible in a one-band superconductor with non-dynamically
screened Coulomb interactions and attractive interactions mediated by phonons
and spin fluctuations.

##  Appendix B Appendix B: Solutions of odd-f SC models

We give details and numeric results demonstrating odd-f SC solutions of the
Eliashberg equations (3) with interaction potentials

$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\lambda
f(|\omega_{n}-\omega_{m}|/\Omega)$ (44)

depending some function $f(x)$ of $x\geq 0$, together with two parameters
$\lambda>0$ and $\Omega>0$.

Figure 1: Superconducting critical temperature for odd-f SC, $T_{c}$, versus
coupling, $\lambda$, for the Eliashberg equations (3) with the interaction
potential in (44) for two different examples: (a) $f(x)=1/(1+x^{2})$ (inverted
phonon interaction), (b) $f(x)=\log(1+1/x)$ (screened Coulomb interaction). We
checked that, in both cases, $T_{c}$ grows towards $+\infty$ for
$\lambda\to\infty$. The curves end short before $T_{c}=0$ due to convergence
problems of our numeric procedure for $T_{c}\to 0$.

Models. Our first example is
$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\lambda\Omega^{2}/[\Omega^{2}+(\omega_{n}-\omega_{m})^{2}]$,
which is like a phonon potential with one optical phonon mode but with the
wrong sign (this is obtained from (4) by setting $\mu^{*}=0$,
$\alpha^{2}F(\Omega)=\lambda(\Omega_{0}/2)\delta(\Omega-\Omega_{0})$ (Dirac
delta), renaming $\Omega_{0}\to\Omega$ after the integration, and changing
$\lambda$ to $-\lambda$); this corresponds to (44) with $f(x)=1/(1+x^{2})$,
and we refer to it as inverted phonon interaction. Our second example is
$f(x)=\log(1+1/x)$ which, as we show below following [16], describes a
dynamically screened Coulomb potential close to a quantum critical point; we
refer to this as screened Coulomb interaction. Our third example is
$f(x)=1/x^{\gamma}$ inspired by [18, 16], as discussed in the main text; we
refer to it as inverted $\gamma$-model.

To motivate our second example, we start with the potential in (13) in three
dimensions (3D), restrict the momenta to the Fermi surface:
$|\mathbf{k}|=|\mathbf{k}^{\prime}|=k_{F}$ (Fermi momentum) so that
$(\mathbf{k}-\mathbf{k}^{\prime})^{2}=2k_{F}^{2}-2k_{F}^{2}\cos(\theta)$ with
$\theta$ the angle between $\mathbf{k}$ and $\mathbf{k}^{\prime}$, and average
over $\theta$:

$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=N(0)\frac{1}{2}\int_{0}^{\pi}\frac{g_{\text{eff}}}{[2k_{F}^{2}-2k_{F}^{2}\cos(\theta)]+\Gamma\frac{|\omega_{n}-\omega_{m}|}{[2k_{F}^{2}-2k_{F}^{2}\cos(\theta)]^{1/2}}}\sin(\theta)d\theta=\frac{g_{\text{eff}}N(0)}{6k_{F}^{2}}\log\left(1+\frac{8k_{F}^{3}}{\Gamma|\omega_{n}-\omega_{m}|}\right)$
(45)

where, in the second step, we computed an exact integral (the reason for
including the DOS is explained in [10], for example). This proves a claim
after (13) in the main text. Clearly this has the form of a potential in (44)
with $f(x)=\log(1+1/x)$, $\lambda=g_{\text{eff}}N(0)/6k_{F}^{2}$, and
$\Omega=8k_{F}^{3}/\Gamma$. One can make a similar computation in 2D
considered in [16] but, in this case, the resulting function $f(x)$ does not
have such a simple explicit formula; thus, for simplicity, we restricted our
numeric investigations to the 3D case.

Method. For the potential $V$ in (44),

$V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})=\lambda
f(|\omega_{n}-\omega_{m}|/\Omega)-\lambda f(|\omega_{n}+\omega_{m}|/\Omega).$
(46)

One can easily check that, if the function $f(x)$ of $x\geq 0$ is
monotonically decreasing, then $V^{-}$ is non-negative; thus, in such a case,
odd-f SC cannot be ruled out by (8).

To show that such a model indeed describes odd-f SC, we compute $T_{c}$ using
(7). For that, we write (7) in the following way,

$v_{n}=\lambda\sum_{m\geq 0}M_{n,m}(v_{n}-v_{m}),\quad M_{n,m}\equiv\frac{\pi
T}{\lambda\omega_{n}}V^{-}(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m}),\quad
v_{n}\equiv\frac{\Delta^{-}(\mathrm{i}\omega_{n})}{\omega_{n}};$ (47)

we find it convenient to define the matrix $M=(M_{n,m})_{n,m\geq 0}$ so that
it is independent of the coupling parameter $\lambda$. Inserting (46), we find

$M_{n,m}=\frac{1}{2n+1}\left[f(|n-m|X)-f(|n+m+1|X)\right],\quad
X\equiv\frac{2\pi T}{\Omega},$ (48)

i.e., $M$ depends only on one parameter $X$. Thus, the $T_{c}$-equation (7)
can be written as eigenvalue equation for the matrix $A=(A_{n,m})_{n,m\geq 0}$
with the matrix elements

$A_{n,m}\coloneqq\delta_{n,m}\sum_{k\geq 0}M_{n,k}-M_{n,m},$ (49)

i.e.,

$\sum_{m\geq 0}A_{n,m}v_{m}=\frac{1}{\lambda}v_{n}.$ (50)

Denoting the largest eigenvalue of the matrix $A$ as $g(X)$, we get the
following odd-f $T_{c}$-equation,

$\frac{1}{\lambda}=g(2\pi T_{c}/\Omega).$ (51)

We recall that, for even-f SC, $T_{c}$ is given by the McMillan formula [29]
with has the form $T_{c}=\Omega\exp(-1/\lambda)$ with
$\lambda=V(\mathrm{i}\omega_{0},\mathrm{i}\omega_{0})$ and $\Omega$ determined
by the function $V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})$ in (4); this
has exactly the form (51) with $g(X)=-\log(X)$. Thus, our odd-f
$T_{c}$-equation is similar to a conventional one, except that the function
$-\log(X)$ is replaced by the function $g(X)$ obtained by diagonalizing a
$X$-dependent infinite matrix $A$.

To compute $g(X)$, we approximate $M$ by a $N\times N$-matrix,
$M_{N}=(M_{n,m})_{n,m=0}^{N-1}$, compute the corresponding finite matrix
$A_{N}$ from $M_{N}$ using (49) with $M$ replaced by $M_{N}$, compute the
larges eigenvalue, $g(X)$, of $A_{N}$ numerically, and check convergence by
plotting $g(X)$ for different values of $N$. Thus, we made sure that all plots
in Figs. 1 are numerically accurate.

It is worth noting that the solution method described above can be used for
any potential of the form (44); we did not make any assumption on the sign of
$f(x)$.

Analytic result for inverted $\gamma$-model. The inverted $\gamma$-model
corresponds to (44) with $f(x)=1/x^{\gamma}$. In this case, the potential in
(44) is invariant under the scaling transformations
$(\lambda,\Omega)\to(s^{-\gamma}\lambda,s\Omega)$ for arbitrary $s>0$. This
and our $T_{c}$-equation (51) imply $g(X/s)/g(X)=s^{\gamma}$ for all $s>0$ and
$X=2\pi T_{c}/\Omega>0$; the latter equation has a unique solution depending
on one constant $C\geq 0$: $g(X)=CX^{-\gamma}$ for all $X>0$. Inserting this
in (51) we obtain

$T_{c}=K\Omega\lambda^{1/\gamma}\quad(\lambda>0)$ (52)

with $K=C^{1/\gamma}/2\pi$.

Numeric results. We computed the function $g(X)$ numerically for various
examples. We found that the convergence of our numeric procedure improves with
increasing values of $X$. In the limit $X\to 0$, we encountered convergence
problems, i.e., for $X<0.005$ or so the required values of $N$ to reach
convergence were too large for our limited computation resources. A related
numeric issue for our second and third examples, $f(x)=\log(1+1/x)$ and
$1/x^{\gamma}$, is that $f(x)$ diverges as $x\to 0$; to solve this problem, we
replaced $1/x$ by $1/\sqrt{x^{2}+\varepsilon^{2}}$ and varied $\varepsilon$ to
make sure that $\varepsilon$ is small enough that results do not change within
the desired accuracy if $\varepsilon$ is decreased further. We found that
changes in the parameter $\varepsilon$ mainly affect the function $g(X)$ in
the regime $X\to 0^{+}$. Due to these numeric limitations of our method, we
were not able to compute $g(X)$ reliably in the limit $X\to 0^{+}$ using our
numeric method. However, for the inverted $\gamma$-model, our analytic result
shows that $g(0)=+\infty$, which implies that odd-f SC is possible for
arbitrarily small $\lambda>0$ for this model.

We also checked the $T_{c}$-equation (52) for the inverted $\gamma$-model
numerically and found perfect agreement for some value of $K$ we computed
numerically; for example, we obtained $K=0.1822$, $0.1911$ and $0.1829$ for
$\gamma=0.5$, $1$ and $2$, respectively (numerical values we give are accurate
until and including the last digit).

We also found that, for $\lambda>10$ or so, the numerically computed
$T_{c}$-curves in Fig. 1 can be well approximated by $T_{c}\approx
0.18\,\Omega\lambda^{1/2}$ (inverted phonon interaction) and $T_{c}\approx
0.19\,\Omega\lambda$ (screened Coulomb interaction), respectively; note that,
in the former and latter cases, $f(x)\approx 1/x^{2}$ and $f(x)\approx 1/x$
for large $x$, respectively.

Discussion. We find it interesting that the $T_{c}$-curves for odd-f SC and
repulsive interactions are qualitatively similar to $T_{c}$-curves for
conventional even-f SC: $T_{c}$ grows monotonically with the coupling constant
$\lambda$. However, for conventional even-f SC, the $T_{c}$-curve is
determined by the universal function $g(X)=-\log(X)$, while for odd-f SC this
function $g(X)$ is model dependent. For the inverted Coulomb potential, our
numerical results suggest that odd-f SC is only possible for
$\lambda>\lambda_{c}$ with $\lambda_{c}\approx 1>0$ (see Fig. 1(a)), but for
the the inverted $\gamma$-model, $\lambda_{c}=0$; the latter and our numerics
suggest to us that $\lambda_{c}=0$ also for the screened Coulomb potential
(see Fig. 1(b)). We believe that $\lambda_{c}>0$ for all potentials
$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})$ which remain finite as
$|\omega_{n}-\omega_{m}|\to 0$.

Our numeric results suggest that the exact $T_{c}$-formula (52) for the
inverted $\gamma$-model captures the leading large-$\lambda$ behaviour of many
models: if
$V(\mathrm{i}\omega_{n},\mathrm{i}\omega_{m})\approx\lambda(\Omega/|\omega_{n}-\omega_{m}|)^{\gamma}$
for $|\omega_{n}-\omega_{m}|\to\infty$, then $T_{c}\approx
K\Omega\lambda^{1/\gamma}$ for $\lambda\to\infty$.

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