# Nodal d-wave pairing from spin fluctuations in a thermally disordered anti-
ferromagnet

Nick Bultinck  Department of Physics, Ghent University, Krijgslaan 281, 9000
Gent, Belgium

###### Abstract

We consider electron pairing in a two-dimensional thermally disordered
itinerant anti-ferromagnet. It is shown that transverse spin fluctuations in
such a state can give rise to superconductivity with a sizeable critical
temperature $T_{c}$. Below $T_{c}$ there is quasi-long-range spin-singlet and
$d_{x^{2}-y^{2}}$ superconducting order, together with fluctuating triplet
order at momentum $(\pi,\pi)$. The singlet pairs we find are tightly bound
together, and the pair wavefunction has a purely inter-sublattice structure
due to the U$(1)$ spin rotation symmetry of the anti-ferromagnet.

_Introduction –_ It is well-known that anti-ferromagnetic (AFM) spin
fluctuations generate an attractive interaction between electrons which favors
pairing in the spin-singlet $d_{x^{2}-y^{2}}$-channel [1, 2, 3, 4]. In the
original approach for connecting AFM fluctuations and $d$-wave
superconductivity [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], the normal state is a
conventional Fermi liquid. In the underdoped cuprates, however, the normal
state does not seem to fit the standard Fermi liquid mould. For example, one
of the intriguing properties of the underdoped cuprate normal state is the
pseudogap phenomenon, manifested in the form of a suppressed density of states
at the Fermi energy and Fermi arcs in photoemission spectra. Motivated by
these experimental observations, we consider a magnetic pseudogap state in the
form of a thermally disordered AFM –i.e. an AFM above its critical temperature
(which is $T=0$), but well below the mean-field transition temperature– as the
parent state for superconductivity. This magnetic pseudogap state has a
suppressed density of states at the Fermi energy due to the fluctuating anti-
ferromagnetism. It also produces a spectral weight consisting of $4$ small
Fermi pockets centered at $(\pm\pi/2,\pm\pi/2)$ with faint backsides,
resembling the Fermi arcs seen in photoemission. Inspired by the well-
established connection between AFM spin fluctuations and $d$-wave pairing in
Fermi liquids, our goal here is to investigate the role of spin fluctuations
for mediating superconductivity in such a thermally disordered AFM. In
particular, we focus on spin fluctuations which are the finite-temperature
remnants of the AFM Goldstone modes.

To set up our calculations we start from the effective theory for a thermally
disordered AFM presented in Ref. [11]. The effective action is derived from a
simple mean-field and random phase approximation (RPA). However, one of its
main advantages is that the effective theory contains scattering vertices
between the electrons and Goldstone modes which are not entirely
phenomenological. Instead these vertices are calculated starting from the
microscopic Hamiltonian. And as the tendency of an interaction to promote
$d$-wave pairing crucially relies on its structure in momentum space, having
an explicit expression for the scattering vertices is key for our analysis.

Within the effective theory we find that spin fluctuations indeed give rise to
superconductivity with a critical temperature which is high compared to
conventional phonon-driven superconductors. Crucially, due to the symmetry
properties of the AFM, the gap function we find cannot be a featureless
$s$-wave state but instead needs to have a non-trivial momentum dependence. A
further important property of the effective theory is that the fermion fields
are defined in a `rotating frame' [12, 13, 14, 15], in order to ensure that
long-wavelength Goldstone modes decouple from the electrons [11, 16]. It is
also this rotating frame which ultimately ensures that the superconducting
order parameter we find is spin-singlet. At the end of the manuscript we
discuss a few predictions of our theory which can be used to test whether
superconductivity in a particular model or material indeed originates from
spin fluctuations in a thermally disordered AFM.

Our approach is closely related to both the spin-bag [17, 18] and spin-fermion
[7, 8, 16, 19, 20, 21, 22, 23, 24] models studied in the early days of
high-$T_{c}$ superconductivity. However, it also differs from these pre-
existing theories in some crucial ways. In particular, unlike in the spin-
fermion model, the bare coupling between electrons and spin fluctuations in
our effective theory cannot be described by a purely local and instantaneous
interaction. Our approach also differs from the original spin-bag theory
because we focus on transverse spin fluctuations, previously studied in [25,
26, 27], and use a rotating frame which removes strong inter-band scattering
terms 111See Refs. [52, 53, 54] for a complementary perspective. In these
works, the strong inter-band scattering terms from transverse spin
fluctuations are not eliminated via a rotating frame, but their effect on the
electron Green’s function is explicitly taken into account via an infinite sum
of diagrams in an eikonal approximation. and leads to a spin-singlet
superconductor with nodes on the Fermi surface. Moreover, we consider spin
fluctuations with a non-zero thermal mass to generate the effective
attraction, and use the complete dynamical interaction to calculate $T_{c}$.

Figure 1: (a) Mean-field valence band energy of the AFM insulator. The mean-
field hybridization strength between spin up and down electrons is $\sim
1.93$. The Fermi surface obtained by $\sim 12\%$ hole doping is shown in red.
(b) Goldstone mode dispersion relation
$\tilde{\omega}_{\tilde{{\mathbf{q}}}}$. (c)-(d) Averaged absolute values of
the vertices in Eq. (6) for electrons in the valence band ($\alpha=\beta=0)$
as a function of the pseudo-momentum transfer $\tilde{{\mathbf{q}}}$, where
the averaging is over the incoming pseudo-momentum. Results are obtained on a
$34\times 34$ pseudo-momentum grid.

_Model and results –_ As a microscopic model we consider the Hubbard model on
the square lattice:

$H=-t\sum_{\langle
ij\rangle}\sum_{\sigma}c^{\dagger}_{i,\sigma}c_{j,\sigma}-t^{\prime}\sum_{\langle\langle
ij\rangle\rangle}c^{\dagger}_{i,\sigma}c_{j,\sigma}+h.c.+U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}\,,$
(1)

where the first (second) sum is over nearest (next nearest) neighbors. In the
third term $n_{i,\sigma}$ is the number of electrons with spin $\sigma$ on
site $i$. We take $t\equiv 1$, $t^{\prime}=-0.35$, and $U=5$. At half filling,
the system is an insulating antiferromagnet. We assume that the AFM moments
are ordered in the $x$-direction, and work with following imaginary-time
effective action to describe the interacting electrons and AFM Goldstone modes
at mean-field+RPA level:

$S_{\rm eff}=S_{el}+S_{V}+S_{B}+S_{el-B}$ (2)

The first term is the kinetic energy term for the fermions

$S_{el}=\int\mathrm{d}\tau\sum_{\tilde{{\mathbf{k}}}}\sum_{\alpha}\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}(\partial_{\tau}+E_{\tilde{{\mathbf{k}}},\alpha}-\mu)\psi_{\tilde{{\mathbf{k}}},\alpha}\,,$
(3)

where $\alpha=0,1$ labels the mean-field bands of the AFM insulator – $0$
($1$) is the valence (conduction) band. The mean-field valence band energy
$E_{\tilde{{\mathbf{k}}},0}$ is shown in Fig. 1 (a). The conduction band is
separated by a gap of order $U$. The AFM breaks translation over one site, but
is invariant under the action of
$T^{\prime}_{{\mathbf{r}}}=e^{i{\mathbf{r}}\cdot{\mathbf{Q}}\,\sigma^{z}/2}T_{{\mathbf{r}}}$,
where ${\mathbf{Q}}=(\pi,\pi)$, $\sigma^{i}$ are the Pauli spin matrices, and
$T_{\mathbf{r}}$ implements a translation by lattice vector ${\mathbf{r}}$.
The pseudo-momenta $\tilde{{\mathbf{k}}}$ are the conserved quantum numbers
(modulo reciprocal lattice vectors) associated with the
$T^{\prime}_{\mathbf{r}}$ symmetry. The relation between states with pseudo-
momentum $\tilde{{\mathbf{k}}}$ and crystal momentum ${\mathbf{k}}$ is given
by:

$f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}:=c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}\hskip
2.84544pt\,,\hskip
14.22636ptf^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}:=c^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}\,,$
(4)

where $f^{\dagger}_{\tilde{{\mathbf{k}}},\sigma}$ creates electrons with
pseudo-momentum $\tilde{{\mathbf{k}}}$.

Invariance under spin rotations around the $x$-axis implies that the effective
theory is symmetric under
$f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}\leftrightarrow
f^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}},\downarrow}$. This symmetry
acts on the fermion fields
$\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}=\sum_{\sigma=\uparrow,\downarrow}u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}\bar{\psi}_{\tilde{{\mathbf{k}}},\sigma}$,
where $u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}$ are the coefficients of the
mean-field single-particle states [29], as
$\bar{\psi}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}},\alpha}\leftrightarrow\pm\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}$.
The signs depend on a choice of real gauge for
$u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}$. Here we use a gauge where all signs
are positive.

The second term $S_{V}$ in Eq. (2) contains the instantaneous two-body
interaction for the electrons, which consists of the microscopic Hubbard
interaction, and an interaction which is generated by integrating out the
field conjugate to the Goldstone field $\phi({\mathbf{r}})$ [11]. The
instantaneous interaction is written out in detail in the supplementary
material [29].

The dynamics of the Goldstone field is described by the third term in the
effective action:

$S_{B}=\frac{1}{2}\int\mathrm{d}\tau\,\sum_{\tilde{{\mathbf{q}}}}\left(-\phi_{-\tilde{{\mathbf{q}}}}\partial^{2}_{\tau}\phi_{\tilde{{\mathbf{q}}}}+\omega_{\tilde{{\mathbf{q}}}}^{2}\phi_{\tilde{{\mathbf{q}}}}\phi_{-\tilde{{\mathbf{q}}}}\right)\,,$
(5)

where
$\omega^{2}_{\tilde{{\mathbf{q}}}}=\tilde{\omega}_{\tilde{{\mathbf{q}}}}^{2}+m^{2}$,
with $\tilde{\omega}^{2}_{\tilde{{\mathbf{q}}}}\sim
c^{2}\tilde{{\mathbf{q}}}^{2}$ near $\tilde{{\mathbf{q}}}=0$, and
$\tilde{\omega}^{2}_{\tilde{{\mathbf{q}}}}\sim
c^{2}({\mathbf{Q}}-\tilde{{\mathbf{q}}})^{2}$ near
$\tilde{{\mathbf{q}}}={\mathbf{Q}}$ ($c\approx 1)$. The complete Goldstone
dispersion $\tilde{\omega}_{\tilde{{\mathbf{q}}}}$ is shown in Fig. 1 (b). $m$
is a thermal mass for the Goldstone modes which takes into account that 2D AFM
are disordered at non-zero temperatures. Below we will treat $m$ as a
phenomenological parameter. The final term in Eq. (2) contains the electron-
boson interaction

$\begin{split}S_{el-B}=&\int\mathrm{d}\tau\,\frac{1}{\sqrt{N}}\sum_{\tilde{{\mathbf{q}}},\tilde{{\mathbf{k}}}}\bar{\psi}_{\tilde{{\mathbf{k}}},\alpha}\psi_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}},\beta}\times\\\
&\left(g_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})\,\phi_{-\tilde{{\mathbf{q}}}}+f_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})\,i\partial_{\tau}\phi_{-\tilde{{\mathbf{q}}}}\right)\,,\end{split}$
(6)

where $N$ is the number of lattice sites. Following the procedure explained in
Ref. [11], the vertex functions $g$ and $f$ are constructed from solutions to
the Bethe-Salpeter equation for the Hubbard model (which also determines the
Goldstone mode spectrum $\tilde{\omega}_{\tilde{{\mathbf{q}}}}$). In Figs. 1
(c-d) these vertices are shown for electron scattering within the valence band
($\alpha=\beta=0$). In particular, we show the absolute value of the vertices,
averaged over the incoming pseudo-momentum, as a function of the pseudo-
momentum transfer $\tilde{{\mathbf{q}}}$.

A final crucial property of the effective action is that the fermion fields
are defined in a `rotating frame', which means that the bare/microscopic
fermions $\bar{\psi}^{b}_{{\mathbf{r}},\sigma}$ are related to the fields in
Eq. (3) by
$\bar{\psi}_{{\mathbf{r}},\sigma}=\sum_{\sigma^{\prime}}R_{\sigma\sigma^{\prime}}(\tau,{\mathbf{r}})\bar{\psi}^{b}_{{\mathbf{r}},\sigma^{\prime}}$.
The $2\times 2$ matrix field $R(\tau,{\mathbf{r}})$ is defined as

$R(\tau,{\mathbf{r}})=\exp\left(-i\left[\phi_{z}(\tau,{\mathbf{r}})\sigma^{z}+\phi_{y}(\tau,{\mathbf{r}})\sigma^{y}\right]\right)\,,$
(7)

where $\phi_{z}(\tau,{\mathbf{r}})$ and $\phi_{y}(\tau,{\mathbf{r}})$ contain
momenta $\tilde{{\mathbf{k}}}$ in the first magnetic Brillouin zone, i.e. the
Brillouin zone with reciprocal vectors $(\pi,\pm\pi)$, and are related to the
Goldstone field in Eq. (5) by
$\phi=\phi_{z}+(-1)^{{\mathbf{Q}}\cdot{\mathbf{r}}}\phi_{y}$. The reason for
defining the fermion fields in a rotating frame is that these fermions
decouple from the low-frequency and long-wavelength Goldstone modes [11, 16],
i.e. in this basis we have $\lim_{\tilde{{\mathbf{q}}}\rightarrow
0,{\mathbf{Q}}}g_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})=0$.

We now proceed by changing the chemical potential $\mu$ to hole-dope the AFM.
Upon changing the electron density one should actually redetermine the optimal
mean-field state and the RPA collective modes that go into the construction of
our effective action. However, for metallic systems calculating the scattering
vertices $g$ and $f$ is challenging, as the Goldstone modes are partly hidden
inside the particle-hole continuum. We will therefore keep the effective
action as is, and simply change $\mu$. For our purposes we expect this to be a
reasonable approximation at small doping – in particular, we expect that the
gross features in the momentum dependence of the vertex functions $g$ and $f$
will not change.

To study electron pairing we ignore the empty conduction band, and work
exclusively with the valence band which crosses the Fermi energy. We emphasize
that this step can only be justified by using the rotating frame, which
eliminates inter-band scattering terms of order $U$ [11, 30]. We work at a
hole doping of $\sim 12\%$. The resulting Fermi surface is shown in Fig. 1
(a). To study superconductivity we sum the usual Cooper ladder diagrams, where
each rung consists of both the instantaneous interaction contained in $S_{V}$,
and the interaction generated by tree-level Goldstone mode exchange. The
Cooper ladder sum diverges when following equation has a solution with
$\lambda=-1$:

$\lambda\hat{\Delta}(i\omega^{\prime},\tilde{{\mathbf{k}}}^{\prime})=T\sum_{i\omega}\frac{1}{N}\sum_{\tilde{{\mathbf{k}}}}\frac{V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})\,,$
(8)

where
$V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})$
scatters a pair of electrons with pseudo-momenta
$(\tilde{{\mathbf{k}}},-\tilde{{\mathbf{k}}})$ and frequencies
$(i\omega,-i\omega)$ to a pair of electrons with
$(\tilde{{\mathbf{k}}}^{\prime},-\tilde{{\mathbf{k}}}^{\prime})$ and
$(i\omega^{\prime},-i\omega^{\prime})$, due to both the instantaneous
interaction and Goldstone mode exchange. The explicit expression for $V$ is
given in the supplementary material [29], but let us note here that the
frequency dependence of $V$ comes from both the Goldstone mode propagator and
the direct coupling of the electrons to $\partial_{\tau}\phi$ [see Eq. (6)].
In Eq. (8) we have also introduced the notation
$E_{\tilde{{\mathbf{k}}}}:=E_{\tilde{{\mathbf{k}}},0}$, and used that
$E_{\tilde{{\mathbf{k}}}}=E_{-\tilde{{\mathbf{k}}}}$. To obtain $\lambda$, we
use following ansatz

$\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})=\frac{\Delta(\tilde{{\mathbf{k}}})}{\omega^{2}+\Omega^{2}}\,,$
(9)

where $\Delta(-\tilde{{\mathbf{k}}})=-\Delta(\tilde{{\mathbf{k}}})$ as
required by fermion antisymmetry, and $\Omega$ is a variational parameter
corresponding to an inverse retardation timescale. In Fig. 2 (a) we show
$\lambda+1$ obtained from a variational calculation with this ansatz as a
function of $T$ and $\Omega$, where we have used a thermal Goldstone mass of
$m=0.01$ (for details see [29]). We see that $\lambda$ reaches $-1$ at a
highest temperature of $\sim 0.028$ when $\Omega\approx 0.5$, which is roughly
half the Goldstone bandwidth. Due to the variational nature of our calculation
we have thus obtained a lower bound for $T_{c}$ (within the ladder sum
approximation) which, when using a representative value of $t=0.3$ eV,
corresponds to $\sim 100$ K. In Fig. 2 (b) we show the corresponding gap
function $\Delta(\tilde{{\mathbf{k}}})$. It has the important property
$\Delta(\tilde{{\mathbf{k}}}+{\mathbf{Q}})=-\Delta(\tilde{{\mathbf{k}}})$
(recall that the shift symmetry over momentum ${\mathbf{Q}}$ is a result of
the U$(1)$ spin rotation symmetry). We have also calculated $\lambda$ and
$\Delta(\tilde{{\mathbf{k}}})$ using a thermal mass $m=0.1$ and found that in
this case $T_{c}\gtrsim 0.023$, and the gap function remains essentially
unchanged [29].

Figure 2: (a) $\lambda+1$, with $\lambda$ defined in Eq. (8), as obtained from
a variational calculation with the ansatz in Eq. (9). (b) Corresponding gap
function $\Delta(\tilde{{\mathbf{k}}})$. Results are obtained on a $34\times
34$ pseudo-momentum grid using $m=0.01$.

In the spin basis, the gap function is given by
$\Delta_{\sigma\sigma^{\prime}}(\tilde{{\mathbf{k}}})=u_{0,\sigma}(\tilde{{\mathbf{k}}})u_{0,\sigma^{\prime}}(-\tilde{{\mathbf{k}}})\Delta(\tilde{{\mathbf{k}}})$.
As the AFM breaks spin rotation symmetry, the gap is generically an admixture
of a singlet and a triplet component. To see this, we rewrite the gap function
in the crystal momentum basis, where it takes the form
$\Delta_{\sigma\sigma^{\prime}}({\mathbf{k}})=\Delta^{S}({\mathbf{k}})\sigma^{y}_{\sigma\sigma^{\prime}}+\Delta^{T}_{\mathbf{Q}}({\mathbf{k}})\sigma^{z}_{\sigma\sigma^{\prime}}$
[29]. $\Delta^{S}({\mathbf{k}})$ is a zero-momentum singlet gap, and
$\Delta^{T}_{\mathbf{Q}}({\mathbf{k}})$ is a momentum-${\mathbf{Q}}$ triplet
gap [31, 32, 33, 34, 35]. Written out explicitly, the singlet part is given by

$\Delta^{S}({\mathbf{k}})=u(-{\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}u({\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}\Delta({\mathbf{k}}+{\mathbf{Q}}/2)\,.$
(10)

Note that from $\Delta(-{\mathbf{k}})=-\Delta({\mathbf{k}})$,
$\Delta({\mathbf{k}}+{\mathbf{Q}})=-\Delta({\mathbf{k}})$ and
$u({\mathbf{k}})_{0\downarrow}=u({\mathbf{k}}+{\mathbf{Q}})_{0\uparrow}$ it
follows that $\Delta^{S}(-{\mathbf{k}})=\Delta^{S}({\mathbf{k}})$ as required.

Finally, we calculate the superconducting order parameter in terms of the
microscopic fermions. We start from the anomalous Green's function

$\displaystyle\langle\mathcal{T}c^{\dagger}_{{\mathbf{r}},\sigma}(\tau)c^{\dagger}_{0,\sigma^{\prime}}(0)\rangle$
$\displaystyle=$ $\displaystyle\langle
R^{*}_{\sigma\tilde{\sigma}}(\tau,{\mathbf{r}})R^{*}_{\sigma^{\prime}\tilde{\sigma}^{\prime}}(0,0)\bar{\psi}_{{\mathbf{r}},\tilde{\sigma}}(\tau)\bar{\psi}_{0,\tilde{\sigma}^{\prime}}(0)\rangle$
(11) $\displaystyle\approx$ $\displaystyle\langle
R^{*}_{\sigma\tilde{\sigma}}(\tau,{\mathbf{r}})R^{*}_{\sigma^{\prime}\tilde{\sigma}^{\prime}}(0,0)\rangle\times$
$\displaystyle\langle\bar{\psi}_{{\mathbf{r}},\tilde{\sigma}}(\tau)\bar{\psi}_{0,\tilde{\sigma}^{\prime}}(0)\rangle\,,$

where summation over repeated indices is implied, $\mathcal{T}$ is the time-
ordering operator, and we have used that the bare electrons are related to the
fermion fields $\bar{\psi}_{{\mathbf{r}},\sigma}$ via a rotation with
$R(\tau,{\mathbf{r}})$ as explained above. In the second line we have
approximated the correlation function by a product of the correlation
functions of the matrix field $R$ and the fermions in the rotating frame.
Using $R^{*}=\sigma^{y}R\sigma^{y}$, it immediately follows from the results
of Ref. [36] that a saddle-point approximation of the non-linear sigma model
describing the dynamics of $R$ leads to following expression for the $R$ two-
point function at non-zero temperature:

$\langle
R^{*}_{ss^{\prime}}(\tau,{\mathbf{r}})R^{*}_{\tilde{s}\tilde{s}^{\prime}}(0,0)\rangle=\sigma^{y}_{\tilde{s}s}\sigma^{y}_{s^{\prime}\tilde{s}^{\prime}}D(\tau,{\mathbf{r}})\,,$
(12)

where $D(\tau,{\mathbf{r}})$ is the Fourier transform of
$((i\nu)^{2}-\omega_{\mathbf{q}}^{2})^{-1}$. As
$\langle\bar{\psi}_{{\mathbf{r}},\tilde{\sigma}}(\tau)\bar{\psi}_{0,\tilde{\sigma}^{\prime}}(0)\rangle$
is the Fourier transform of
$\Delta_{\tilde{\sigma}\tilde{\sigma}^{\prime}}({\mathbf{k}})/(\omega^{2}+\Omega^{2})$,
we find that the (equal-time) superconducting order parameter is given by

$\langle
c^{\dagger}_{-{\mathbf{k}},\sigma}c^{\dagger}_{{\mathbf{k}},\sigma^{\prime}}\rangle=\sigma^{y}_{\sigma\sigma^{\prime}}\frac{1}{N}\sum_{\mathbf{q}}\frac{n(\omega_{\mathbf{q}})-n(-\omega_{\mathbf{q}})}{\omega_{\mathbf{q}}}\Delta^{S}({\mathbf{k}}-{\mathbf{q}})\,,$
(13)

where $n(\omega)=1/(\exp(\omega/T)-1)$ is the Bose-Einstein distribution
function, and a momentum-independent prefactor has been dropped. Note that the
contraction of $\Delta_{\sigma\sigma^{\prime}}({\mathbf{k}})$ with the $R$
two-point function annihilates the momentum-${\mathbf{Q}}$ triplet component,
such that the superconducting order parameter is translation invariant and
pure singlet. This is similar to how a spin-rotation invariant Green's
function was obtained in Refs. [36, 11, 37, 38, 39]. In Fig. 3 we plot the
order parameter (13) both in momentum and real space. It clearly has a nodal
$d$-wave structure, whose origin can be traced back to the single-band nature
of the pairing function, which imposes
$\Delta(-\tilde{{\mathbf{k}}})=-\Delta(\tilde{{\mathbf{k}}})$, together with
the $C_{4}$ and time-reversal symmetries. From Fig. 3 (b) we also see that the
electron pairs are tightly bound together: the pair wavefunction becomes
negligibly small when the electrons are separated by more than $\sim 5$
lattice sites. From the same figure we also see that the pair wavefunction is
only non-zero when the electrons occupy different sublattices. This is a
direct consequence of the U$(1)$ spin rotation symmetry of the AFM, which
imprints on the gap function the property
$\Delta({\mathbf{k}}+{\mathbf{Q}})=-\Delta({\mathbf{k}})$. From Eq. (11) it is
also clear that the pair wavefunction will decay faster if the spin moments
become more disordered. If we use a thermal mass $m=0.1$ such that the spin
correlations become more short-ranged, we find that the pair wavefunction is
essentially zero if the electrons are separated by more than one lattice site
[29].

Figure 3: (a) Plot of the superconducting order parameter $\langle
c^{\dagger}_{-{\mathbf{k}}}\sigma^{y}c^{\dagger}_{{\mathbf{k}}}\rangle$ as a
function of momentum ${\mathbf{k}}$. (b) Real-space order parameter/pair
wavefunction $\langle
c^{\dagger}_{\mathbf{r}}\sigma^{y}c^{\dagger}_{0}\rangle$. Results are
obtained for a $34\times 34$ system using $m=0.01$.

_Discussion –_ We have shown that with microscopically calculated scattering
vertices, finite-temperature remnants of AFM Goldstone modes can mediate a
sizeable attractive interaction between electrons which leads to spin-singlet
$d_{x^{2}-y^{2}}$ superconducting order. The fact that $T_{c}$ drops only
moderately upon increasing the thermal mass of the Goldstone modes by an order
of magnitude is consistent with the expectation that pairing is predominantly
mediated by short-wavelength and high-energy transverse spin fluctuations, as
these couple most strongly to the electrons. The effective theory used in this
work is obtained from a simple mean-field+RPA analysis, which makes it unclear
how reliable our estimate for $T_{c}$ is. An interesting route for future
research would therefore be to improve our results by using the two-particle
self-consistent approach [40] applied to broken-symmetry systems [41], or by
combining the functional renormalization group with mean-field theory [34, 35,
42, 43].

In this work we have ignored Landau damping of the Goldstone modes. We expect
that this will only quantitatively change our results, and here we only aim to
get an order of magnitude estimate for $T_{c}$. Our analysis does reveal a few
distinct universal features of electron pairing in fluctuating itinerant AFM
which can be used to test the theory in more controlled numerical studies, or
perhaps even in experiment. In particular, our key predictions are: (1) the
presence of fluctuating triplet order at momentum ${\mathbf{Q}}$, (2) a pair
wavefunction which decays faster than the spin correlation function, and (3) a
purely inter-sublattice pair wavefunction. The latter was also found in the
spin-bag approach [17, 18], which is not surprising since it is a consequence
of the symmetries of the AFM.

The theory presented here is undoubtedly too simplified to capture the rich
physics of the underdoped cuprates. For example, it does not take charge
stripes, nematicity and pair density-wave orders into account, all of which
can intertwine with each other, the AFM, and the uniform superconductor in
intricate ways [44, 45]. Nevertheless, we believe that our approach can shed
light on the role of the higher-temperature pseudogap state as a parent state
for superconductivity. Of course, this requires one to assume that the
pseudogap state observed in experiment is indeed a fluctuating AFM, which is
still a matter of current debate. But at least for the Hubbard model the
magnetic pseudogap scenario has received considerable numerical support [46,
47, 48, 49, 37, 50, 51].

_Ackowledgements –_ The author would like to thank Andrey Chubukov, Walter
Metzner and Pietro Bonetti for helpful discussions, and Rafael Fernandes for
pointing out Ref. [31]. This research is supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 101076597 - SIESS).

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— Supplementary Material —

##  Appendix A Appendix A: Definition of
$u(\tilde{{\mathbf{k}}})_{\alpha,\sigma}$ and $S_{V}$

Underlying the effective action in the main text is the mean-field Hamiltonian
in the pseudo-momentum basis:

$H_{MF}=\sum_{\tilde{{\mathbf{k}}}}f_{\tilde{{\mathbf{k}}}}^{\dagger}h(\tilde{{\mathbf{k}}})f_{\tilde{{\mathbf{k}}}}\,,$
(14)

where
$f_{\tilde{{\mathbf{k}}}}=(f_{\tilde{{\mathbf{k}}},\uparrow},f_{\tilde{{\mathbf{k}}},\downarrow})^{T}$,
with $f^{\dagger}_{\tilde{{\mathbf{k}}},\sigma}$ defined in Eq. (4) in the
main text. The precise form of the mean-field Hamiltonian is

$h(\tilde{{\mathbf{k}}})=\left(\begin{matrix}\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}&M\\\
M&\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}\end{matrix}\right)\,,$ (15)

where $M\sim 1.93$ is the self-consistently determined (using the parameters
for the Hubbard model given in the main text) hybridization between the spin
up and down electrons which produces the AFM order, and

$\displaystyle\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}$ $\displaystyle=$
$\displaystyle\varepsilon(\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2)\,,$ (16)
$\displaystyle\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}$ $\displaystyle=$
$\displaystyle\varepsilon(\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2)\,.$ (17)

Here, $\varepsilon({\mathbf{k}})=-2t(\cos k_{x}+\cos
k_{y})-2t^{\prime}(\cos(k_{x}+k_{y})+\cos(k_{x}-k_{y})$ is the dispersion of
the Hubbard model in Eq. (1). The mean-field energies are defined as the
eigenvalues of $h(\tilde{{\mathbf{k}}})$, and are given by

$E_{\tilde{{\mathbf{k}}},\alpha}=\frac{1}{2}\left(\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}+\varepsilon_{\tilde{{\mathbf{k}}},\downarrow}\pm\sqrt{(\varepsilon_{\tilde{{\mathbf{k}}},\uparrow}-\varepsilon_{\tilde{{\mathbf{k}}},\downarrow})^{2}+4M^{2}}\right)\,.$
(18)

The states $|u(\tilde{{\mathbf{k}}})_{\alpha}\rangle$ are the corresponding
eigenstates of $h(\tilde{{\mathbf{k}}})$. Note that the mean-field Hamiltonian
satisfies
$h(\tilde{{\mathbf{k}}}+{\mathbf{Q}})=\sigma^{x}h(\tilde{{\mathbf{k}}})\sigma^{x}$,
which is a manifestation of the U$(1)$ spin rotation symmetry around the
$x$-axis. The eigenstates of $h(\tilde{{\mathbf{k}}})$ (in a real gauge) thus
satisfy
$|u(\tilde{{\mathbf{k}}}+{\mathbf{Q}})_{\alpha}\rangle=\pm\sigma^{x}|u(\tilde{{\mathbf{k}}})_{\alpha}\rangle$.
We partially fix the gauge by requiring these signs to all be positive.

Written out explicitly, the instantaneous interaction in the effective theory
used in the main text is given by

$S_{V}=\int\mathrm{d}\tau\frac{1}{N}\sum_{\tilde{{\mathbf{q}}},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}\left[\frac{U}{2}\left(\bar{\psi}_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}}}\Lambda_{\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}})\psi_{\tilde{{\mathbf{k}}}}\right)\left(\bar{\psi}_{\tilde{{\mathbf{k}}}^{\prime}+\tilde{{\mathbf{q}}}}\Lambda_{-\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}}^{\prime})\psi_{\tilde{{\mathbf{k}}}^{\prime}}\right)-\frac{c}{2aw}\big{(}\bar{\psi}_{\tilde{{\mathbf{k}}}}f_{\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}})\psi_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}}}\big{)}\left(\bar{\psi}_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{q}}}}f_{\tilde{{\mathbf{q}}}}^{\dagger}(\tilde{{\mathbf{k}}}^{\prime})\psi_{\tilde{{\mathbf{k}}}^{\prime}}\right)\right]\,.$
(19)

The first term in Eq. (19) is simply the microscopic Hubbard interaction
$\frac{U}{2}\sum_{\mathbf{r}}:n_{\mathbf{r}}^{2}:$ rewritten in the mean-field
basis. The unitary basis transformation from the orbital basis to the mean-
field basis gives rise to the form factors:

$\left[\Lambda_{{\mathbf{q}}}({\mathbf{k}})\right]_{\alpha\beta}=\langle
u(\tilde{{\mathbf{k}}}-\tilde{{\mathbf{q}}})_{\alpha}|u(\tilde{{\mathbf{k}}})_{\beta}\rangle\,.$
(20)

The second term in Eq. (19) is obtained by integrating out the field conjugate
to the Goldstone field $\phi({\mathbf{r}})$ [11]. As in the main text
$c\approx 1$ is the Goldstone mode velocity, $a$ is the lattice constant, and
$w\approx 1$ is a dimensionless number which is fixed by the requirement that
in the rotating frame it should hold that
$\lim_{\tilde{{\mathbf{q}}}\rightarrow
0,{\mathbf{Q}}}g_{\tilde{{\mathbf{q}}}}=0$ [11]. The matrices
$\left[f_{\tilde{{\mathbf{q}}}}(\tilde{{\mathbf{k}}})\right]_{\alpha\beta}$
contain the scattering vertices
$f_{\tilde{{\mathbf{q}}},\alpha\beta}(\tilde{{\mathbf{k}}})$ used in Eq. (6).

##  Appendix B Appendix B: Definition of
$V(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})$

The interaction
$V(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})$
which scatters a pair of electrons with frequencies $(i\omega,-i\omega)$ and
pseudo-momenta $(\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}})$ to a pair of
electrons with $(i\omega^{\prime},-i\omega^{\prime})$ and pseudo-momenta
$(\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}}^{\prime})$ can be written
as

$V(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})=V_{I}(\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})+V_{G}(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})\,.$
(21)

The first term comes from the instantaneous interaction contained in $S_{V}$
and is given by

$V_{I}(\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})=U\Lambda_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(\tilde{{\mathbf{k}}})\Lambda_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(-\tilde{{\mathbf{k}}})-\frac{c}{aw}f_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(\tilde{{\mathbf{k}}}^{\prime})f_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(-\tilde{{\mathbf{k}}}^{\prime})\,,$
(22)

where we have put the band indices equal to zero because we only consider
scattering within the valence band which crosses the Fermi energy. The second
term in Eq. (21) is generated by tree-level Goldstone mode exchange and is
given by

$V_{G}(i\omega-i\omega^{\prime},\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime})=\frac{\left[g_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(\tilde{{\mathbf{k}}}^{\prime})+(\omega-\omega^{\prime})f_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}},00}(\tilde{{\mathbf{k}}}^{\prime})\right]\left[g_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(-\tilde{{\mathbf{k}}}^{\prime})-(\omega-\omega^{\prime})f_{\tilde{{\mathbf{k}}}-\tilde{{\mathbf{k}}}^{\prime},00}(-\tilde{{\mathbf{k}}}^{\prime})\right]}{(i\omega-i\omega^{\prime})^{2}-\omega_{\tilde{{\mathbf{k}}}^{\prime}-\tilde{{\mathbf{k}}}}^{2}}\,.$
(23)

Figure 4: Results for $\lambda$ on a $34\times 34$ system using a thermal mass
$m=0.01$ and keeping the $N_{\omega}=1000$ (left) and $N_{\omega}=2000$
(right) smallest Matsubara frequencies.

##  Appendix C Appendix C: Variational solution of the gap equation

In this appendix we explain in more detail how we solve the gap equation

$\lambda\hat{\Delta}(i\omega^{\prime},\tilde{{\mathbf{k}}}^{\prime})=T\sum_{i\omega}\frac{1}{N}\sum_{\tilde{{\mathbf{k}}}}\frac{V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})$
(24)

variationally with the ansatz

$\hat{\Delta}(i\omega,\tilde{{\mathbf{k}}})=\frac{\Delta(\tilde{{\mathbf{k}}})}{\omega^{2}+\Omega^{2}}\,.$
(25)

As a first step we multiply both sides of Eq. (24) with
$\hat{\Delta}^{*}(i\omega^{\prime},\tilde{{\mathbf{k}}}^{\prime})/(\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2})$, and sum over both
$i\omega^{\prime}$ and $\tilde{{\mathbf{k}}}^{\prime}$. We obtain

$\displaystyle\lambda\sum_{\tilde{{\mathbf{k}}}}\Delta^{*}(\tilde{{\mathbf{k}}}^{\prime})\Delta(\tilde{{\mathbf{k}}}^{\prime})\sum_{i\omega^{\prime}}\frac{1}{(\omega^{\prime
2}+\Omega^{2})^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}=$ (26)
$\displaystyle\sum_{\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}\Delta^{*}(\tilde{{\mathbf{k}}}^{\prime})\Delta(\tilde{{\mathbf{k}}})\frac{T}{N}\sum_{i\omega,i\omega^{\prime}}\frac{1}{\omega^{\prime
2}+\Omega^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})\frac{1}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\frac{1}{\omega^{2}+\Omega^{2}}\,,$

where we have filled in the ansatz for the gap function (25). By defining the
matrices

$\displaystyle D(\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})$
$\displaystyle=$
$\displaystyle\delta_{\tilde{{\mathbf{k}}},\tilde{{\mathbf{k}}}^{\prime}}\sum_{i\omega^{\prime}}\frac{1}{(\omega^{\prime
2}+\Omega^{2})^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}$ (27) $\displaystyle
M(\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})$ $\displaystyle=$
$\displaystyle\frac{T}{N}\sum_{i\omega,i\omega^{\prime}}\frac{1}{\omega^{\prime
2}+\Omega^{2}}\frac{1}{\omega^{\prime
2}+(E_{\tilde{{\mathbf{k}}}^{\prime}}-\mu)^{2}}V(i\omega^{\prime}-i\omega,\tilde{{\mathbf{k}}}^{\prime},\tilde{{\mathbf{k}}})\frac{1}{\omega^{2}+(E_{\tilde{{\mathbf{k}}}}-\mu)^{2}}\frac{1}{\omega^{2}+\Omega^{2}}$
(28)

we can bring Eq. (26) in the form

$\lambda\langle\Delta|D|\Delta\rangle=\langle\Delta|M|\Delta\rangle\,,$ (29)

where $|\Delta\rangle$ is the vector with components
$\Delta(\tilde{{\mathbf{k}}})$. Via following trivial rewriting of this
equation

$\lambda\left(\langle\Delta|D^{1/2}\right)\left(D^{1/2}|\Delta\rangle\right)=\left(\langle\Delta|D^{1/2}\right)D^{-1/2}MD^{-1/2}\left(D^{1/2}|\Delta\rangle\right)$
(30)

it becomes clear that the optimal $\lambda$ corresponds to the smallest
eigenvalue of $D^{-1/2}MD^{-1/2}$, and the gap function is obtained from the
corresponding eigenvector. In practice we evaluate the sums over Matsubara
frequencies in Eqs. (27) and (28) with a frequency cutoff. For the results
presented in the main text we have kept the 1000 smallest Matsubara
frequencies. We have checked that our results are converged upon changing the
cutoff. For example, in Fig. 4 we compare results obtained by using 1000 and
2000 Matsubara frequencies. We find that the change in $\lambda$ is
negligible.

##  Appendix D Appendix D: Singlet and triplet components of the gap function

By solving the gap equation we find a gap function which is defined in pseudo-
momentum space and in the mean-field band basis. To interpret the gap function
physically it is easier to go back to the crystal momentum and spin basis. In
this appendix we work out this transformation for the spin-singlet and triplet
components of the gap function.

###  D.1 Singlet part

To obtain the singlet component we start from the equations

$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\downarrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}\rangle$
(31) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\downarrow}u(\tilde{{\mathbf{k}}})_{0\uparrow}\,\Delta(\tilde{{\mathbf{k}}})\,,$
(32)

where the first line follows from the definition of
$f^{\dagger}_{\tilde{{\mathbf{k}}},\sigma}$, and the second line follows from
the definition of $\Delta(\tilde{{\mathbf{k}}})$ as the gap function in the
mean-field basis. Similarly we also have the equalities

$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\uparrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}\rangle$
(33) $\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2+{\mathbf{Q}},\uparrow}c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2+{\mathbf{Q}},\downarrow}\rangle$
(34) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\uparrow}u(\tilde{{\mathbf{k}}})_{0\downarrow}\,\Delta(\tilde{{\mathbf{k}}})\,.$
(35)

Combining Eqs. (31), (32), (33), (34) and (35) we find

$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}})_{0\downarrow}u({\mathbf{k}})_{0\uparrow}\,\Delta({\mathbf{k}})$
(36) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}})_{0\uparrow}u({\mathbf{k}}+{\mathbf{Q}})_{0\downarrow}\,\Delta({\mathbf{k}}+{\mathbf{Q}})$
(37)

A trivial shift in momentum then gives

$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}},\downarrow}c^{\dagger}_{{\mathbf{k}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}u({\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}\,\Delta({\mathbf{k}}+{\mathbf{Q}}/2)$
(38) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}},\uparrow}c^{\dagger}_{{\mathbf{k}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}u({\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}\,\Delta({\mathbf{k}}-{\mathbf{Q}}/2)\,.$
(39)

As $\Delta({\mathbf{k}}+{\mathbf{Q}})=-\Delta({\mathbf{k}})$ and
$u({\mathbf{k}}+{\mathbf{Q}})_{0,\uparrow}=u({\mathbf{k}})_{0,\downarrow}$ the
right-hand side Eq. (38) is equal to minus the right-hand side of Eq. (39).
Eqs. (38) and (39) thus constitute the singlet-component of the gap.

###  D.2 Triplet part

To obtain the triplet component we follow essentially the same steps as to
obtain the singlet component. Starting from the equalities

$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\uparrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{\tilde{{\mathbf{k}}}-{\mathbf{Q}}/2,\uparrow}\rangle$
(40) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\uparrow}u(\tilde{{\mathbf{k}}})_{0\uparrow}\,\Delta(\tilde{{\mathbf{k}}})$
(41)

and

$\displaystyle\langle
f^{\dagger}_{-\tilde{{\mathbf{k}}},\downarrow}f^{\dagger}_{\tilde{{\mathbf{k}}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{-\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{\tilde{{\mathbf{k}}}+{\mathbf{Q}}/2,\downarrow}\rangle$
(42) $\displaystyle=$ $\displaystyle
u(-\tilde{{\mathbf{k}}})_{0\downarrow}u(\tilde{{\mathbf{k}}})_{0\downarrow}\,\Delta(\tilde{{\mathbf{k}}})$
(43)

we find

$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}})_{0\uparrow}u({\mathbf{k}})_{0\uparrow}\,\Delta({\mathbf{k}})$
(44) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}-{\mathbf{Q}}/2,\downarrow}c^{\dagger}_{{\mathbf{k}}-{\mathbf{Q}}/2,\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}})_{0\downarrow}u({\mathbf{k}}+{\mathbf{Q}})_{0\downarrow}\,\Delta({\mathbf{k}}+{\mathbf{Q}})$
(45)

Shifting the momentum then gives

$\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}},\uparrow}c^{\dagger}_{{\mathbf{k}},\uparrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}-{\mathbf{Q}}/2)_{0\uparrow}u({\mathbf{k}}+{\mathbf{Q}}/2)_{0\uparrow}\Delta({\mathbf{k}}+{\mathbf{Q}}/2)$
(46) $\displaystyle\langle
c^{\dagger}_{-{\mathbf{k}}+{\mathbf{Q}},\downarrow}c^{\dagger}_{{\mathbf{k}},\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
u(-{\mathbf{k}}+{\mathbf{Q}}/2)_{0\downarrow}u({\mathbf{k}}-{\mathbf{Q}}/2)_{0\downarrow}\,\Delta({\mathbf{k}}-{\mathbf{Q}}/2)\,,$
(47)

which corresponds to the triplet component with crystal momentum
${\mathbf{Q}}$.

##  Appendix E Appendix E: Additional numerical results

In this final appendix we present additional numerical results obtained using
a thermal mass $m=0.1$. Other parameters are identical to the ones used in the
main text. The system size is $34\times 34$.

In Fig. 5 (a) we plot $\lambda+1$ as a function of $T$ and $\Omega$. We see
that the highest $T_{c}\approx 0.023$ occurs for $\Omega\approx 0.5$. The
corresponding gap function shown in Fig. 5 (b) is identical to the one
obtained in the main text using a smaller thermal mass $m=0.01$.

In Fig. 6 we show the superconducting order parameter defined in Eqs. (11) and
(13) in the main text. Compared to the results in the main text obtained with
a smaller thermal mass, we see that the order parameter in Fig. 6 is much
smoother in momentum space, and hence more short-ranged in real-space. This
illustrates the statement made in the main text that our theory predicts a
system with shorter-range AFM fluctuations to also have a shorter-range pair
wavefunction.

Figure 5: (a) $\lambda+1$ as a function of $T$ and $\Omega$ using a thermal
mass $m=0.1$. (b) Gap function giving rise to the highest $T_{c}$ at
$\Omega=0.5$. Results are obtained on a $34\times 34$ system. Figure 6:
Superconducing order parameter and corresponding pair wavefunction obtained
with $m=0.1$. (a) Superconducting order parameter as a function of crystal
momentum. (b) Pair wavefunction in real-space. Results are obtained on a
$34\times 34$ system.