# Multiorbital spin-triplet pairing and spin resonance in the heavy-fermion
superconductor $\mathrm{UTe_{2}}$

Lei Chen⊕ Department of Physics and Astronomy, Rice Center for Quantum
Materials, Rice University, Houston, Texas, 77005, USA  Haoyu Hu⊕ Department
of Physics and Astronomy, Rice Center for Quantum Materials, Rice University,
Houston, Texas, 77005, USA  Christopher Lane  Theoretical Division and Center
for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  Emilian M. Nica  Department of Physics, Arizona State
University, Box 871504, Tempe 85287-1504, AZ, USA  Jian-Xin Zhu  Theoretical
Division and Center for Integrated Nanotechnologies, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Qimiao Si  Department of
Physics and Astronomy, Rice Center for Quantum Materials, Rice University,
Houston, Texas, 77005, USA

###### Abstract

The heavy-fermion system $\mathrm{UTe_{2}}$ is a candidate for spin-triplet
superconductivity, which is of considerable interest to quantum engineering.
Among the outstanding issues is the nature of the pairing state. A recent
surprising discovery is the observation of a resonance in the spin excitation
spectrum at an antiferromagnetic wavevector [C. Duan et al., Nature 600, 636
(2021)], which stands in apparent contrast to the ferromagnetic nature of the
interactions expected in this system. We show how the puzzle can be resolved
by a multiorbital spin-triplet pairing constructed from local degrees of
freedom. Because it does not commute with the kinetic part of the Hamiltonian,
the pairing contains both intra- and inter- band terms in the band basis. We
demonstrate that the intraband pairing component naturally yields a spin
resonance at the antiferromagnetic wavevector. Our work illustrates how
orbital degrees of freedom can enrich the nature and properties of spin-
triplet superconductivity of strongly-correlated quantum materials.

Introduction.  The uranium-based dichalcogenide $\mathrm{UTe_{2}}$ is a prime
candidate for spin-triplet superconductors Ran  _et al._ (2019a); Aoki  _et
al._ (2019); Ran  _et al._ (2019b); Tokunaga  _et al._ (2019); Knebel  _et
al._ (2019); Metz  _et al._ (2019); Sundar  _et al._ (2019); Jiao  _et al._
(2020); Hayes  _et al._ (2021); Nakamine  _et al._ (2021). Because spin-
triplet pairing Rice and Sigrist (1995); Kallin and Berlinsky (2009); Ramires
and Sigrist (2019); Huang  _et al._ (2019) can lead to topological states that
host Majorana fermions, which represent a promising route towards fault
tolerant quantum computing Nayak  _et al._ (2008), this system is of extensive
current interest. $\mathrm{UTe_{2}}$ transitions to a superconducting state
below $T_{c}=1.6$ K. It has a strongly-correlated paramagnetic normal state,
with a Kondo temperature of about $70$ K marking the onset of heavy-fermion
behavior such as a highly enhanced effective mass Kirchner  _et al._ (2020).
Evidence for the spin-triplet superconductivity includes the nearly constant
Knight shift as observed in nuclear magnetic resonance (NMR) measurements Ran
_et al._ (2019a); Tokunaga  _et al._ (2019); Nakamine  _et al._ (2021) across
$T_{c}$, large upper critical fields exceeding the Pauli limit Ran  _et al._
(2019a), and indications of chiral edge states from scanning tunneling
microscopy (STM) experiments Jiao  _et al._ (2020). There is also evidence of
time-reversal symmetry breaking from a non-zero polar Kerr effect Hayes  _et
al._ (2021). Analogy with the uranium-based heavy-fermion superconductors
$\mathrm{UGe_{2}}$, $\mathrm{URhGe}$ and $\mathrm{UCoGe}$ Saxena  _et al._
(2000); Aoki  _et al._ (2001); Huy  _et al._ (2007) suggests the natural
possibility that ferromagnetic (FM) spin correlations are important for
superconductivity in $\mathrm{UTe_{2}}$. Early evidence for the role of FM
correlations in the normal state comes from the enhanced bulk magnetic
susceptibility Ran  _et al._ (2019a) as well as by $\mu$SR measurements Sundar
_et al._ (2019). Still, the nature of the pairing state remains enigmatic
Ishizuka  _et al._ (2019); Xu  _et al._ (2019); Nevidomskyy (2020); Hu and Si
; Shishidou  _et al._ (2021); Kreisel  _et al._ (2021).

Recently, several experiments Thomas  _et al._ ; Duan  _et al._ (2020); Knafo
_et al._ (2021); Butch  _et al._ (2021); Duan  _et al._ (2021); Raymond  _et
al._ (2021) have suggested the presence of antiferromagnetic (AF) spin
correlations in $\mathrm{UTe_{2}}$. The inelastic neutron scattering (INS)
experiments reveal spin fluctuations at an AF wavevector, a feature that has
been attributed to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions
between the $\mathrm{U}$ 5$f$ electrons Duan  _et al._ (2020). Strikingly,
measurements below $T_{c}$ have uncovered a spin resonance near the AF
wavevector Duan  _et al._ (2021); Raymond  _et al._ (2021). The observation of
an AF spin resonance came as a surprise, since it has typically been
associated with spin-singlet pairing Rossat-Mignod  _et al._ (1991); Mook  _et
al._ (1993); Fong  _et al._ (1995); Dai (2015). In the spin-singlet pairing
case, the spin resonance has, quite extensively, been attributed to collective
excitations of a particle-hole spin composite, at a wave vector that connects
parts of the Fermi surfaces with a sign change in the superconducting order
parameter ($\Delta_{k}=-\Delta_{k+q}$) Fong  _et al._ (1995); Liu  _et al._
(1995); Zha  _et al._ (1993); Eschrig (2006). In the case of
$\mathrm{UTe_{2}}$, the central new questions are two-fold: how does spin-
triplet pairing develop when the overall magnetic fluctuations are AF, and how
does the spin-triplet pairing in turn lead to an AF spin resonance?

In this work, we advance a resolution to these seemingly contradictory issues.
The key idea, alluded to by some of us in Ref. Duan  _et al._ , 2021 and
developed here, is that pairing in the strongly-correlated (heavy-fermion)
system $\mathrm{UTe_{2}}$ is to be constructed in terms of its multiple local
degrees of freedom that we refer to as orbitals; these include the sublattice
degrees of freedom associated with different U-sites in a unit cell. As a
result, the spin-triplet pairing is a matrix not only in spin space but also
in orbital space. Reminiscent of a nontrivial class of multiorbital spin-
singlet pairing states Nica  _et al._ (2017); Nica and Si (2021); Pang  _et
al._ (2018); Smidman  _et al._ (2018), the pairing matrix does not commute
with the kinetic part of the Hamiltonian. This in turn implies that, in the
band basis, the pairing matrix contains both intra- and inter-band pairing
components. We show that, while the overall pairing matrix has the spin-
triplet odd-parity form, the intraband pairing component gives rise to a spin
resonance. We achieve an understanding of the spin dynamics at the proof-of-
principle level, leaving more quantitative analyses for future studies. We
note in passing that the interplay between AF correlations and spin-triplet
pairing is potentially relevant to other heavy-fermion superconductors,
including the venerable case of UPt3 Aeppli  _et al._ (1988); Sauls (1994);
Joynt and Taillefer (2002) and the more recently-discussed quantum critical
systems Nguyen  _et al._ (2021); Hu  _et al._ (2021); Pixley  _et al._ (2015).

Competing magnetic interactions and microscopic Hamiltonian.
$\mathrm{UTe_{2}}$ crystallizes in an orthorhombic, centrosymmetric structure,
with space group $71$ ($Immm$) and two $U$ atoms per unit cell. Electronic-
structure calculations Shick  _et al._ (2021); Shick and Pickett (2019); Duan
_et al._ (2020), core-level photoelectron spectroscopy Fujimori  _et al._
(2021, 2019) and the spin size extracted by inelastic neutron scattering Duan
_et al._ (2021) are consistent with the U $5f$ electrons in a doublet of
predominantly $j=5/2$, with $m_{j}=\pm 1/2$.

We thus consider two Kramers doublets per unit cell. The effective
interactions are labeled by $J_{i}$’s ($i=1-4$) in Fig. 1(a). We take the
exchange interactions to be

$\displaystyle
H_{J}=-\sum_{i}I^{\mu}S^{\mu}_{i,A}S^{\mu}_{i,B}+\sum_{ir,mn}J_{r}\bm{S}_{i,m}\cdot\bm{S}_{i+r,n}.$
(1)

Here, $A/B$ labels the two sublattices [Fig. 1(a)], $i$ marks a unit cell and
$S^{\mu}$ is a component of the local spin for $\mu=x,y,z$. Due to the special
role that the intra-“dimer” exchange interaction plays, we refer to
$J_{1}^{\mu}$ as $-I^{\mu}$. In addition, $J_{r}$ ($r\geq 2$) labels the
$r$th-nearest neighbor exchange interaction that connects $m$ and $n$
sublattices [Fig. 1(a)]. We expect to have significant $\mu$-dependence due to
the large spin-orbit coupling (SOC) that is inherent to the $5f$-electrons of
U-ions. The effect of this spin anisotropy is evidenced by magnetic
susceptibility measurements Ran  _et al._ (2019a); Li  _et al._ (2021). In our
analysis, only the spin anisotropy of $J_{1}=-I$ will be important and kept
track of. In practice, we consider $r$ up to $4$ and have

$\displaystyle H_{J}$ $\displaystyle=$
$\displaystyle\sum_{\bm{q}}\begin{pmatrix}\bm{S}_{\bm{q},A}&\bm{S}_{\bm{q},B}\end{pmatrix}{J}(\bm{q})\begin{pmatrix}\bm{S}_{-\bm{q},A}\\\
\bm{S}_{-\bm{q},B}\end{pmatrix}$ (2)

with

$\displaystyle{J}(\bm{q})=$
$\displaystyle[2\,J_{3}\,T_{3}(\bm{q})+2\,J_{4}\,T_{4}(\bm{q})]\,\tau_{0}$ (3)
$\displaystyle+[-I+4\,J_{2}\,T_{2,1}(\bm{q})]\,\tau_{x}+4\,J_{2}\,T_{2,2}(\bm{q})\,\tau_{y},$

where the Pauli matrices $\tau_{i}$ are defined in the orbital space, while
$T_{2,1}(\bm{q})=\cos(q_{x}/2)\cos(q_{y}/2)\cos(q_{z}/2)$,
$T_{2,2}(\bm{q})=\cos(q_{x}/2)\cos(q_{y}/2)\sin(q_{z}/2)$,
$T_{3}(\bm{q})=\cos(q_{x})$, and $T_{4}(\bm{q})=\cos(q_{y})$. The INS
experiments Duan  _et al._ (2021); Knafo  _et al._ (2021) allow for the
inference of FM $I$ and $J_{3}$ and AF $J_{2}$ and $J_{4}$ (see Supplementary
Materials).

Figure 1:  (a) The structure of $\mathrm{UTe_{2}}$. The numbers label the
bonds for the exchange interactions and kinetic parameters. The black thin
lines mark the conventional unit cell, which contains two primitive cells. (b)
The exchange interaction eigenvalue, $J_{-}({\bm{q}})$, plotted within the
$(q_{x},q_{y})$ plane for $q_{z}=0$. It is the most negative around the AF
wavevector, ${\bm{q}}\sim\mathrm{\bf Q}$. The color bar marks the values in
meV.

The U-U distance within a dimer is smaller than all the other U-U distances,
and we expect $I$ to be the strongest. We have performed first-principles
calculations within density functional theory Lane  _et al._ (2021) (see also
Ref. Xu  _et al._ , 2019). Our calculations give average values of
$-J_{1}\equiv I\lesssim 20$ meV and $J_{2}\sim 0.75$ meV. Estimates of
$J_{3,4}$ are more demanding as they require a larger unit cell with five
magnetic configurations considered on an equal footing. Based on these results
and the Kondo temperature scale, we set $I^{x}=15$ $\mathrm{meV}$, $J_{2}=1$
$\mathrm{meV}$, $J_{3}=-0.5$ $\mathrm{meV}$, and $J_{4}=2$ $\mathrm{meV}$. The
largest negative eigenvalue of the exchange interaction matrix is:

$\displaystyle J_{-}(\bm{q})=$
$\displaystyle-I+4J_{2}\cos\frac{q_{x}}{2}\cos\frac{q_{y}}{2}+2J_{3}\cos
q_{x}+2J_{4}\cos q_{y}\,,$ (4)

for $q_{z}=0$. Importantly, the ferromagnetic intra-dimer interaction
$J_{1}\equiv-I$ is featureless in wavevector space. Instead, the inter-unit-
cell AF interactions are responsible for the minimum of $J_{-}(\bm{q})$ near
$\bm{q}\sim\mathrm{\bf Q}=(0,\pi)$, as shown in Fig. 1(b).

The Hamiltonian is that of a Kondo lattice, with the $5f$-electrons occupying
$\Gamma_{5}$ Kramers doublets on the U sites that are Kondo-coupled to the
$spd$-electrons and, in addition, interact with each other through $H_{J}$.
For our proof-of-principle purpose, we adopt an effective description that is
based exclusively on effective $f$-electrons and incorporate $H_{J}$ and a
kinetic part $H_{kin}$ (see below).

Spin-triplet pairing – orbital basis.  The strongly-correlated nature of the
$5f$ electrons suggests the construction of pairing based on the local degrees
of freedom Duan  _et al._ (2021). Given that the FM intra-dimer interaction
$I$ is the strongest, we consider the dominant pairing to be between the $5f$
electrons on a dimer, as introduced in Ref. Hu and Si,  and subsequently in
Ref. Shishidou  _et al._ , 2021. An odd-parity pairing can be constructed in
terms of two fermions with opposite parity [the even and odd parity
combinations of the fermions at the $A$ and $B$ sites, $c_{e/o}=(c_{A}\pm
c_{B})/\sqrt{2}$, where $c_{e}\,(c_{o})$ is even (odd)] and are, equivalently,
written as follows:

$\displaystyle\Delta_{x}$ $\displaystyle=$
$\displaystyle\delta_{x}\tau_{y}\otimes\sigma_{x}\otimes\gamma_{1}$
$\displaystyle\Delta_{y}$ $\displaystyle=$
$\displaystyle\delta_{y}\tau_{y}\otimes\sigma_{y}\otimes\gamma_{1}$
$\displaystyle\Delta_{z}$ $\displaystyle=$
$\displaystyle\delta_{z}\tau_{y}\otimes\sigma_{z}\otimes\gamma_{1}\,.$ (5)

Here, $\delta_{x},\delta_{y},\delta_{z}$ denote the pairing amplitudes whereas
$\sigma_{i}$ and $\gamma_{i}$ are respectively Pauli matrices in spin and
Nambu spaces. A generic spin triplet pairing can be characterized by the
vector $\vec{d}=(\delta_{x},\delta_{y},\delta_{z})$.

In $\mathrm{UTe_{2}}$, the magnetic susceptibility shows extreme anisotropy
Ran  _et al._ (2019a). In the normal state, the magnetic susceptibilities
satisfy $\chi_{x}\gg\chi_{z}>\chi_{y}$, which is consistent with the hierarchy
of exchange interaction, $I^{x}\gg I^{z}>I^{y}$. In a triplet superconductor,
the direction of the dominant exchange interaction promotes superconducting
pairing with a perpendicular spin component Leggett (1975) (see Supplementary
Materials). Accordingly, the energetically favored pairing component is the
$\Delta_{y}$ channel in the $B_{3u}$ irreducible representation of the
$D_{2h}$ point group. We denote its overall amplitude to be $\delta$, and set
$\delta_{x}=\delta_{z}=0$; extension to including these other pairing channels
is discussed below.

Spin-triplet pairing – band basis.  The Bogoliubov de Gennes (BdG) Hamiltonian
is now written as

$H=\sum_{\bm{k}}\Psi_{\bm{k}}^{\dagger}\,[\,H_{kin}(\bm{k})+\Delta(\bm{k})\,]\,\Psi_{\bm{k}}\,,$
(6)

where
$\Psi_{\bm{k}}^{\dagger}=\left(c^{\dagger}_{\bm{k},m,\sigma},c_{-\bm{k},n,\sigma^{\prime}}(i\sigma_{y})\right)$
is the Nambu spinor, and $m,n=A/B$ are the orbital (sublattice) indices.

Figure 2:  (a) The 3D Fermi surface and (b) the corresponding top view, with
the spanning wavevector (red arrow) that is close to $\mathrm{\bf Q}$.

We incorporate the hopping terms $t_{r}$ following the same numbering scheme
of the bonds as for the exchange interactions, and also allow for $t_{5}$
defined on the diagonal counterpart of the bonds $3$ and $4$ [c.f., Fig.
1(a)]. The kinetic part then has the following form

$\displaystyle
H_{kin}=\sum_{i}t_{1}c^{\dagger}_{i,A}c_{i,B}+\sum_{ir,mn}t_{r}c^{\dagger}_{i,m}c_{i+r,n}.$
(7)

In momentum space (see Supplementary Materials for the definition of Fourier
transformation), $H_{kin}$ becomes:

$\displaystyle H_{kin}(\bm{k})=$
$\displaystyle\xi_{0}(\bm{k})\,\tau_{0}+\xi_{x}(\bm{k})\,\tau_{x}+\xi_{y}(\bm{k})\,\tau_{y}$
(8) $\displaystyle=$
$\displaystyle[2\,t_{3}\,T_{3}(\bm{k})+2\,t_{4}\,T_{4}(\bm{k})+4\,t_{5}\,T_{5}(\bm{k})]\,\tau_{0}$
$\displaystyle+[t_{1}+4\,t_{2}\,T_{2,1}(\bm{k})]\,\tau_{x}+4\,t_{2}\,T_{2,2}(\bm{k})\,\tau_{y}\,,$

where $\sigma_{i}$ are the usual Pauli matrices in spin space, and
$T_{5}(\bm{k})=\cos(k_{x})\cos(k_{y})$. We note that
$\xi_{0,x}(\bm{k})=\xi_{0,x}(-\bm{k})$ and
$\xi_{y}(\bm{k})=-\xi_{y}(-\bm{k})$. Guided by the DMFT calculations of the
electronic structure Duan  _et al._ (2020); Miao  _et al._ (2020) and the
Kondo temperature, we take $t_{1}=-0.95$ $\mathrm{meV}$, $4t_{2}=1$
$\mathrm{meV}$, $2t_{3}=-2$ $\mathrm{meV}$, $2t_{4}=2$ $\mathrm{meV}$,
$4t_{5}=6$ $\mathrm{meV}$, and a chemical potential $\mu=3$ $\mathrm{meV}$.
The Fermi surface is shown in Fig. 2(a,b), which has a quasi- two-dimensional
(2D) shape with a Fermi pocket surrounding the $(0,\pi,k_{z})$ line. [A
suitable variation in $\mu$ can lead to the coexistence of another Fermi
pocket surrounding the $(\pi,0,k_{z})$ line.] For these parameters, the total
bandwidth is $W=19$ $\mathrm{meV}$; it is comparable to the strength of the
exchange interaction [c.f., Fig. 1(b)], and each is about the Kondo
temperature up to the typical conversion factor from temperature to energy
(roughly $3$).

We can now project the pairing onto the band basis. Given that the kinetic and
pairing parts do not commute with each other, the two cannot be simultaneously
diagonalized. The pairing part must contain both intra and interband
components. To see this, we first introduce a transformation that diagonalizes
the kinetic energy:

$U^{\dagger}_{\bm{k}}H_{kin}(\bm{k})U_{\bm{k}}=\begin{pmatrix}\epsilon_{-}(\bm{k})&0\\\
0&\epsilon_{+}(\bm{k})\end{pmatrix}\otimes\sigma_{0}$ (9)

where the band energy
$\epsilon_{\pm}(\bm{k})=\xi_{0}\pm\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}$. The
explicit form of $U_{\bm{k}}$ is given in the Supplemental Materials. The
pairing in the band basis takes the form

$\Delta(\bm{k})=U_{\bm{k}}^{\dagger}\Delta_{y}U_{-\bm{k}}=d_{1}(\bm{k})\alpha_{3}\otimes\sigma_{2}+d_{2}(\bm{k})\alpha_{2}\otimes\sigma_{2}\,.$
(10)

Here, $\alpha_{2}$ and $\alpha_{3}$ are Pauli matrices in the band basis, and
the associated inter- and intra- band pairing components are:

$\displaystyle d_{1}(\bm{k})=$
$\displaystyle\delta\,\xi_{y}(\bm{k})/\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})},$
(11) $\displaystyle d_{2}(\bm{k})=$
$\displaystyle\delta\,\xi_{x}(\bm{k})/\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}\,.$

The band-diagonal $\alpha_{3}$ and band-off-diagonal $\alpha_{2}$ pairing
components have $p_{z}$ and $s$ form factors, respectively. As illustrated in
Fig, 3, the intraband gap functions vanish in the $k_{z}=0,\pm 2\pi$ plane.

Figure 3: Schematic illustration of the intraband pairing
($d_{1}(\bm{k})/\delta$) within (a) the $k_{z}=\pi$-plane and (b) the
$k_{x}=0$-plane.

The full BdG spectrum can be obtained by diagonalizing Eq. 6 (see Supplemental
Materials), which yields

$\displaystyle E_{\pm}(\bm{k})$ (12)
$\displaystyle=\sqrt{\left(\sqrt{\xi_{0}^{2}(\bm{k})+\delta^{2}\sin^{2}\phi(\bm{k})}\pm
B(\bm{k})\right)^{2}+\delta^{2}(1-\sin^{2}\phi(\bm{k}))}$

with $B(\bm{k})=\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}$, and
$\sin\phi(\bm{k})=\frac{\xi_{x}(\bm{k})}{B(\bm{k})}$. On the Fermi surface,
$\xi_{0}^{2}(\bm{k})=\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})$, and there is a
full gap. This feature can be modified when the other spin-triplet pairing
components are included (see below), but our conclusion about the AF spin-
resonance will be robust.

Spin-triplet pairing and spin resonance.  The orbital-dependent spin
susceptibility is given by

$\chi_{\alpha\gamma}(\bm{q},i\omega)=\sum_{\lambda}[\chi^{0}(\bm{q},i\omega)]_{\alpha\lambda}[\mathbb{I}+J(\bm{q})[\chi^{0}(\bm{q},i\omega)]]_{\lambda\gamma}^{-1},$
(13)

where $\alpha$, $\beta$, $\gamma$ and $\lambda$ denote the orbitals. For
$q_{z}=0$ of our interest, the total spin susceptibility is given by

$\displaystyle\chi^{\mu}(\bm{q})=2[\chi^{\mu}_{AA}(\bm{q})+\chi^{\mu}_{AB}(\bm{q})]\,.$
(14)

As noted before, $\mu=x,y,z$ denotes the spin directions.

Unlike singlet superconducting states, here $\chi^{\parallel}$ and
$\chi^{\perp}$, the spin susceptibility parallel and perpendicular to the
$\vec{d}$ vector respectively, have different superconducting coherence
factors Morr  _et al._ (2001); Maier and Scalapino (2008) (see Supplementary
Materials). Because of the highly anisotropic nature of the exchange
interaction $I$, the dominating susceptibility is along the $\mu=x$ direction,
for which we can expect a coherence factor favoring a spin resonance to be of
the same sign at $\bm{k}$ and $\bm{k}+\mathrm{\bf Q}$. We will therefore
calculate $\chi^{x}$ in the following (unless otherwise specified).

Figure 4:  (a) The imaginary part of the dynamical spin susceptibility v.s.
$q$ in the normal state (NS) with frequency $\omega=0.25\mathrm{meV}$ and (b)
of the dynamical spin susceptibility in both the superconducting states (SC)
and normal states, at $\mathrm{\bf Q}=(0,\pi)$, as a function of frequency
($\omega$, normalized by $\Delta$, the maximal pairing gap on the Fermi
surface).

We are now in position to present our results on the spin dynamics. We first
consider the results for the normal state (NS), obtained from Eq. 14 with
$\delta=0$. Fig. 4(a) shows the ${\bm{q}}$-dependence of the imaginary part of
the dynamical spin susceptibility at a fixed $\omega=0.21$ $\mathrm{meV}$,
along a momentum path from $(0,\frac{\pi}{2})$ to $(0,\frac{3\pi}{2})$. It
exhibits an AF peak at wavevectors around $\mathrm{\bf Q}=(0,\pi)$, which is
near where $J_{-}(\bm{q})$ is the most negative and also is close to
connecting the Fermi sheets.

In the superconducting phase, we first solve the pairing function self-
consistently to show the development of spin-triplet pairing. This is
described in the Supplementary Materials, where we indeed find the
$\Delta_{y}$ pairing. With the model parameters described earlier, the maximal
value of the gap close to the Fermi surface is found to be
$\Delta=0.44\mathrm{meV}$.

Fig. 4(b) shows the energy dependence of $\mathrm{Im}\chi({\bf Q},\omega)$,
the imaginary part of the $J({\bm{q}})$-enhanced dynamical spin susceptibility
at the wavevector $\mathrm{\bf Q}=(0,\pi)$, in the superconducting state. (The
bare dynamical spin susceptibility is given in the Supplementary Materials.) A
spin resonance is clearly seen around $\omega_{res}\approx 1$ $\mathrm{meV}$.
The resonance can be regarded as a collective excitation from single-particle
and single-hole states, for all the different $k_{z}$, that are located across
the $q_{y}$ direction. The $k_{z}$ variation of the intraband pairing
component [Fig. 3(b)] contributes to the broadening of the resonance.

One of our main conclusions is the existence of a spin resonance at the AF
wavevector. We find the ratio $\omega_{res}/\Delta$ to be larger than what is
typical (that is significantly less than $2$) for the usual quasi-2D (spin-
singlet) superconductors Eschrig (2006). However, the precise value of this
ratio depends on details of the interactions and Fermi surface.

Our results naturally provide an overall understanding of the salient
experimental-motivated issues outline earlier. First, the calculated dynamical
spin susceptibility in the normal state [Fig. 4(a)] realizes the AF
fluctuations that have been observed by INS experiments Duan  _et al._ (2020);
Knafo  _et al._ (2021). This is further illustrated in Fig. 5 (Supplementary
Materials), where the theoretically determined $\bm{q}$-structure is shown to
be comparable with the momentum pattern observed experimentally. Second, by
incorporating the underlying local degrees of freedom, we have shown that such
AF correlations promote the experimentally-evidenced spin-triplet pairing.
Third, we have demonstrated that the spin-triplet pairing gives rise to an AF
spin resonance. Our results [Fig. 4(b)] capture the salient features of the
experimental observation, as illustrated in the Supplementary Materials (Fig.
6). Finally, the SOC-induced spin-anisotropic interactions built into our
model capture the feature that the dominant magnetic fluctuations appear in
the $x$-direction, as observed in the INS measurements Duan  _et al._ (2021).

Discussion.  Several remarks are in order. First, the importance of the local
degrees of freedom and the associated multiorbital form of spin-triplet
pairing reflect the underlying strong correlations of the system and are to be
contrasted with weak-coupling approaches.

Second, a main focus of our work is a proof-of-principle demonstration that,
when multiorbital pairing is considered, the spin-triplet nature of the
superconducting state in $\mathrm{UTe_{2}}$ can naturally be reconciled with
the AF correlations in the normal state and the emergence of an AF spin
resonance in the superconducting state. This sets the stage for extensions of
the model. For example, the pairing function can be generalized to the non-
unitary case, which can i) break the time-reversal symmetry; and ii) lead to a
BdG spectrum that not only depends on the usual $|\vec{d}|^{2}$ gap term, but
also on an additional splitting proportional to $|i\vec{d}^{*}\times\vec{d}|$
Sigrist and Ueda (1991). While the nodal structure of the pairing state will
be affected, the linkage between the spin-triplet pairing and AF spin
resonance is expected to be robust. In the same vein, our work also sets the
stage for addressing the quantitative aspects of the spin resonance, such as
the resonance energy versus the superconducting pairing gap. For the latter,
we note that the multiplicity of the 5$f$ electrons in $\mathrm{UTe_{2}}$ may
well lead to orbital-selective correlations, as has been observed in other
U-based heavy-fermion systems Chen  _et al._ (2019); Giannakis  _et al._
(2019). Thus, there could very well be multiple pairing gaps that remain to be
experimentally probed.

Third, the matrix form for the spin-triplet pairing we have considered bears
some conceptual similarity to its spin-singlet counterpart in the contexts of
both the iron-based superconductors Nica  _et al._ (2017); Nica and Si (2021)
and $\mathrm{CeCu_{2}Si_{2}}$ Pang  _et al._ (2018); Smidman  _et al._ (2018);
Amorese  _et al._ (2020). Our work therefore highlights the notion that the
emerging conceptual framework for multiorbital superconducting pairing, which
is anchored by the iron-based superconductors Yu  _et al._ (2014); Yin  _et
al._ (2014); Ong  _et al._ (2016); Si  _et al._ (2016), may apply to a variety
of correlated systems.

Conclusion.  To summarize, we have highlighted the importance of the matrix
structure associated with the local degrees of freedom of the heavy-fermion
superconductor $\mathrm{UTe_{2}}$. This structure allows us to provide a
natural understanding of some outstanding issues that have been raised for
this system. The puzzling questions include how spin-triplet pairing can
develop when the overall magnetic fluctuations are antiferromagnetic, and how
the spin-triplet pairing can in turn lead to an antiferromagnetic spin
resonance. In our theory, the spin-triplet pairing is enabled by the dominant
intra-dimer ferromagnetic interactions along with the inter-unit-cell
antiferromagnetic interactions. The latter not only are responsible for the
observed antiferromagnetic fluctuations but also make them react to the onset
of superconductivity. We expect that the new insight our work reveals, viz.
the local degrees of freedom can qualitatively enrich the nature and
properties of spin-triplet superconductivity, will be relevant to a variety of
other strongly-correlated quantum materials.

Acknowledgement.— We thank Ryan Baumbach, Nick Butch, Sheng Ran, Brian Maple,
Chandan Setty, Yi-feng Yang and, particularly, Pengcheng Dai and Chunruo Duan
for useful discussions. Work at Rice was primarily supported by the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, under Award
No. DE-SC0018197 and additionally supported by the Robert A. Welch Foundation
Grant No. C-1411. The majority of the computational calculations have been
performed on the Shared University Grid at Rice funded by NSF under Grant
EIA-0216467, a partnership between Rice University, Sun Microsystems, and
Sigma Solutions, Inc., the Big-Data Private-Cloud Research Cyberinfrastructure
MRI-award funded by NSF under Grant No. CNS-1338099 and by Rice University,
and the Extreme Science and Engineering Discovery Environment (XSEDE) by NSF
under Grant No. DMR160057. Work at Los Alamos was supported by LANL LDRD
Program, UC Laboratory Fees Research Program (Grant Number: LFR-20-653926),
and in part by Center for Integrated Nanotechnologies. One of us (Q.S.)
acknowledges the hospitality of the Aspen Center for Physics, which is
supported by NSF grant No. PHY-1607611.

$\oplus$ These authors contributed equally to this work.

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##  I Supplementary Materials

##  II Comparing our results with INS experiments

In this section, we compare our theoretical results with the INS measurements
of Ref. Duan  _et al._ (2021). One comparison is in the momentum domain. As
shown in Fig. 5(a) [i.e., Fig. 1(b) of the main text], the minima of the
exchange interaction are located about the same positions where the AF spin
excitations are observed in the INS experiments: They appear at BZ boundaries
along the $q_{y}$ direction, as marked by the red ellipses in Fig. 5(b) Duan
_et al._ (2021).

The other comparison is in the frequency domain. The spin resonance from our
calculations in the superconducting state, presented in Fig. 6(a) [i.e., Fig.
4(b) of the main text], captures the basic features of the spin resonance
observed in the INS experiments, which is displayed in Fig. 6(b) Duan  _et
al._ (2021).

Figure 5:  (a) The exchange interaction eigenvalue, $J_{-}(\bm{q})$, plotted
in the $(q_{x},q_{y})$ plane for $q_{z}=0$ [Fig. 1(b) of the main text]. (b)
The schematic plot of the INS pattern in the $[H,K,0]$ plane that are marked
as red ellipses [adapted from Ref. Duan  _et al._ , 2021]. The minima of
$J_{-}(\bm{q})$ in (a) are consistent with the spin excitation patterns
observed in (b). In both (a) and (b), the Brillouin zone boundaries are marked
by the solid black lines.  Figure 6:  (a) The imaginary part of the dynamical
spin susceptibility as a function of energy in both the superconducting (SC)
and normal (NS) states [ [Fig. 4(b) of the main text]. (b) The spin
excitations observed at $K=0.59,H=0$ v.s. energy at $T=0.25$ $\mathrm{K}$ (SC)
and $2$ $\mathrm{K}$ (NS) [adapted from Ref. Duan  _et al._ , 2021]. The
theoretical results capture the basic features of the experimental
observations.

##  III Pairing in the band basis

The kinetic term [Eq. 8] in the orbital basis
$\Phi_{\bm{k}}=(A_{\uparrow}(\bm{k}),A_{\downarrow}(\bm{k}),B_{\uparrow}(\bm{k}),B_{\uparrow}(\bm{k}))$
[$j_{\sigma}=c_{j,\sigma}$, with $j=A,B$ for short-hand notation] can be
diagonalized by a unitary matrix

$\displaystyle
U_{\bm{k}}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-u_{\bm{k}}&0&u_{\bm{k}}\\\
-u_{\bm{k}}&0&u_{\bm{k}}&0\\\ 0&1&0&1\\\ 1&0&1&0\end{pmatrix},$ (15)

where
$u_{\bm{k}}=\frac{\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}}{\xi_{x}(\bm{k})+i\xi_{y}(\bm{k})}$.

The corresponding band dispersions are

$\epsilon_{\pm}(\bm{k})=\xi_{0}(\bm{k})\pm\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}.$
(16)

Each band is doubly degenerate due to the spin (of the Kramers doublet). We
define the annihilation operators of lower and upper bands as $a$ and $b$,
respectively, and let
$\phi(\bm{k})=(a_{k,\uparrow},a_{k,\downarrow},b_{k,\uparrow},b_{k,\downarrow})^{T}$

The BdG Hamiltonian can be expressed in the Nambu basis
$\psi(\bm{k})=(a_{k,\sigma},b_{k,\sigma},\mathcal{T}a^{\dagger}_{k,\sigma}{\mathcal{T}^{-1}},\mathcal{T}b^{\dagger}_{k,\sigma}{\mathcal{T}^{-1}})^{T}$,
with $\mathcal{T}$ the time-reversal operator, as

$\displaystyle H_{BdG}(\bm{k})=\begin{pmatrix}E_{k}&\Delta_{k}\\\
\Delta_{k}^{*}&-E_{-k}\end{pmatrix},$ (17)

where

$\displaystyle E_{k}$
$\displaystyle=U_{\bm{k}}^{\dagger}H_{kin}U_{\bm{k}}=\begin{pmatrix}\epsilon_{-}(\bm{k})&0\\\
0&\epsilon_{+}(\bm{k})\end{pmatrix}\otimes\sigma_{0},$
$\displaystyle=\begin{pmatrix}\xi_{0}(\bm{k})-\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}&0\\\
0&\xi_{0}(\bm{k})+\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}\end{pmatrix}\otimes\sigma_{0},$

and

$\displaystyle\Delta_{k}=U^{\dagger}_{\bm{k}}\Delta_{y}U_{-\bm{k}}=\delta\begin{pmatrix}0&-id_{1}&0&d_{2}\\\
id_{1}&0&-d_{2}&0\\\ 0&-d_{2}&0&id_{1}\\\ d_{2}&0&-id_{1}&0\end{pmatrix},$
(19)

with $d_{1}=\frac{\xi_{y}}{\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}$ and
$d_{2}=\frac{\xi_{x}}{\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}}$. The intraband form
factor $d_{1}$ is $p_{z}$-wave, and the interband form factor $d_{2}$ is
$s$-wave. The dispersion of BdG Hamiltonian is

$\displaystyle E_{\pm}(\bm{k})$
$\displaystyle=\sqrt{\delta^{2}+\xi_{0}^{2}(\bm{k})+\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})\pm
2\sqrt{\delta^{2}\xi_{x}^{2}(\bm{k})+\xi_{0}^{2}(\bm{k})\xi_{x}^{2}(\bm{k})+\xi_{0}^{2}(\bm{k})\xi_{y}^{2}(\bm{k})}}$
(20)

where $E_{-}$ ($E_{+}$) corresponds to the band crossing (above) the Fermi
energy. They can be further simplified as

$\displaystyle E_{\pm}(\bm{k})$
$\displaystyle=\sqrt{\delta^{2}+\xi_{0}^{2}(\bm{k})+\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})\pm
2\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}\sqrt{\xi_{0}^{2}(\bm{k})+\delta^{2}\sin^{2}\phi(\bm{k})}}$
(21)
$\displaystyle=\sqrt{\left(\sqrt{\xi_{0}^{2}(\bm{k})+\delta^{2}\sin^{2}\phi(\bm{k})}\pm\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}\right)^{2}+\delta^{2}(1-\sin^{2}\phi(\bm{k}))}$

with
$\sin\phi(\bm{k})=\frac{\xi_{x}(\bm{k})}{\sqrt{\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})}}$.
$E_{-}(\bm{k})=0$ requires both squared terms to be 0. The second term equals
$0$ in $k_{z}=0,2\pi$ planes where $\sin^{2}\phi_{k}=1$. The first term goes
to 0 when

$\delta^{2}=\xi_{x}^{2}(\bm{k})+\xi_{y}^{2}(\bm{k})-\xi_{0}^{2}(\bm{k}).$ (22)

The Fermi surface is specified by $\xi_{0}-\sqrt{\xi_{x}^{2}+\xi_{y}^{2}}=0$.

Finally, we show the explicit expressions that connect the fermion operators
in the orbital and band bases:

$\displaystyle A_{\uparrow}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-u_{k}a_{\downarrow}+u_{k}b_{\downarrow}\right)$
(23) $\displaystyle A_{\uparrow}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-u_{k}a_{\downarrow}+u_{k}b_{\downarrow}\right)$
$\displaystyle B_{\uparrow}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(a_{\downarrow}+b_{\downarrow}\right)$
$\displaystyle B_{\downarrow}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(a_{\downarrow}+b_{\downarrow}\right)\,.$

##  IV self-consistent determination of the spin-triplet pairing function

We start from the anisotropic spin-exchange interactions (see main text),
$I_{x}\gg I_{z}>I_{y}$, and thus consider

$\displaystyle\hat{H}_{ex}=$
$\displaystyle\sum_{i}-I^{x}S^{x}_{i,A}S^{x}_{i,B}-I^{y}S^{y}_{i,A}S^{y}_{i,B}-I^{z}S^{z}_{i,A}S^{z}_{i,B}$
(24) $\displaystyle=\sum_{i}$
$\displaystyle\frac{-I_{x}+I_{y}-I_{z}}{4}\frac{1}{4}\left(\Phi_{i}^{\dagger}[i\tau_{y}\sigma_{0}]\Phi_{i}^{*}\right)\left(\Phi_{i}^{T}[-i\tau_{y}\sigma_{0}]\Phi_{i}\right)$
$\displaystyle+\frac{I_{x}-I_{y}-I_{z}}{4}\frac{1}{4}\left(\Phi_{i}^{\dagger}[-\tau_{y}\sigma_{z}]\Phi_{i}^{*}\right)\left(\Phi_{i}^{T}[\tau_{y}\sigma_{z}]\Phi_{i}\right)$
$\displaystyle+\frac{-I_{x}-I_{y}+I_{z}}{4}\frac{1}{4}\left(\Phi_{i}^{\dagger}[\tau_{y}\sigma_{x}]\Phi_{i}^{*}\right)\left(\Phi_{i}^{T}[-\tau_{y}\sigma_{x}]\Phi_{i}\right)$
$\displaystyle+\frac{I_{x}+I_{y}+I_{z}}{4}\frac{1}{4}\left(\Phi_{i}^{\dagger}[i\tau_{y}\sigma_{y}]\Phi_{i}^{*}\right)\left(\Phi_{i}^{T}[-i\tau_{y}\sigma_{y}]\Phi_{i}\right),$

where
$\Phi_{i}=\left(A_{\uparrow},A_{\downarrow},B_{\uparrow},B_{\downarrow}\right)_{i}^{T}$.
The first three terms correspond to triplet pairing while the last term is the
singlet pairing. The dominance of $I_{x}$ (together with $I_{z}>I_{y}$)
implies that we can focus on the first term. We define

$\delta_{y}=\langle
B_{\uparrow}A_{\uparrow}+B_{\downarrow}A_{\downarrow}-A_{\uparrow}B_{\uparrow}-A_{\downarrow}B_{\downarrow}\rangle\,,$
(25)

and perform Hubbard-Stratonovich transformation to Eq. 24. The effective
Hamiltonian for our self-consistent analysis becomes

$H_{eff}=\sum_{\bm{k}}\frac{1}{2}\begin{pmatrix}H_{kin}(\bm{k})&-\Delta(\bm{k})\\\
-\Delta^{\dagger}(\bm{k})&-H_{kin}(-\bm{k})\end{pmatrix}-\frac{I_{eff}}{16}\delta_{y}^{*}\delta_{y},$
(26)

and the gap function
$\Delta(\bm{k})=\frac{I_{eff}\delta_{y}}{8}i\tau_{y}\sigma_{0}$, with
$I_{eff}=I_{x}-I_{y}+I_{z}$. The pairing amplitude can now be determined self-
consistently. We find $\delta=0.6\mathrm{meV}$. The maximal value of the
corresponding gap close to the Fermi surface is $\Delta=0.44\mathrm{meV}$.

##  V Spin resonance in a single-band spin-triplet superconductor

For a single-band spin-triplet $\Delta_{k}\sigma^{y}$ pairing, the bare spin
susceptibilities along three directions take the forms

$\displaystyle\chi^{0,x}(\bm{q},i\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{k}\frac{1}{2}\left(1-\frac{\epsilon_{k}\epsilon_{k+q}-\Delta_{k}\Delta_{k+q}}{E_{k}E_{k+q}}\right)\left(\frac{1-f(E_{k})-f(E_{k+q})}{i\omega+E_{k}+E_{k+q}}+\frac{1-f(E_{k})-f(E_{k+q})}{-i\omega+E_{k}+E_{k+q}}\right)$
(27) $\displaystyle\chi^{0,y}(\bm{q},i\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{k}\frac{1}{2}\left(1-\frac{\epsilon_{k}\epsilon_{k+q}+\Delta_{k}\Delta_{k+q}}{E_{k}E_{k+q}}\right)\left(\frac{1-f(E_{k})-f(E_{k+q})}{i\omega+E_{k}+E_{k+q}}+\frac{1-f(E_{k})-f(E_{k+q})}{-i\omega+E_{k}+E_{k+q}}\right)$
(28) $\displaystyle\chi^{0,z}(\bm{q},i\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{k}\frac{1}{2}\left(1-\frac{\epsilon_{k}\epsilon_{k+q}-\Delta_{k}\Delta_{k+q}}{E_{k}E_{k+q}}\right)\left(\frac{1-f(E_{k})-f(E_{k+q})}{i\omega+E_{k}+E_{k+q}}+\frac{1-f(E_{k})-f(E_{k+q})}{-i\omega+E_{k}+E_{k+q}}\right)$
(29)

where $\epsilon_{k}$ is the dispersion in the normal state, $\Delta_{k}$ is
the gap function and $E_{k}=\sqrt{\epsilon_{k}^{2}+\Delta_{k}^{2}}$ is the BdG
spectrum. We notice that, the resonance depends on both the direction of the
paring vector $\vec{d}$ and the symmetry of the form factor. If
$\mathrm{sign}(\Delta_{k})=\mathrm{sign}(\Delta_{k+q})$, the resonance appears
in the plane perpendicular to the pairing direction. If
$\mathrm{sign}(\Delta_{k})=-\mathrm{sign}(\Delta_{k+q})$, the resonance
appears in the susceptibility parallel to the pairing direction.

##  VI Notation of Fourier transform

We define primitive vectors of the primitive unit cell as

$\displaystyle\bm{a}_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(1,1,-1),$
(30) $\displaystyle\bm{a}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1,1,1),$ (31) $\displaystyle\bm{a}_{3}$
$\displaystyle=$ $\displaystyle\frac{1}{2}(1,-1,1),$ (32)

where there are two sublattices in the primitive unit cell. The corresponding
reciprocal vectors are

$\displaystyle\bm{b}_{1}$ $\displaystyle=$ $\displaystyle 2\pi(0,1,1)$ (33)
$\displaystyle\bm{b}_{2}$ $\displaystyle=$ $\displaystyle 2\pi(1,0,1)$ (34)
$\displaystyle\bm{b}_{3}$ $\displaystyle=$ $\displaystyle 2\pi(1,1,0).$ (35)

We then discuss the two different definitions of Fourier transformation:

$\displaystyle
c_{\bm{k},\alpha}=\sum_{\bm{R}}c_{i,\alpha}e^{-i\bm{k}\cdot\bm{R}_{i}},$
$\displaystyle\tilde{c}_{\bm{k},\alpha}=\sum_{\bm{R}}c_{i,\alpha}e^{-i\bm{k}\cdot(\bm{R}_{i}+\bm{r}_{\alpha})},$

where $\alpha$ is the sublattice index, $\bm{R}_{i}$ is the position of $i$-th
primitive unit cell and $\bm{r}_{\alpha}$ is the position of sublattice
$\alpha$ in each unit cell
[$\bm{r_{A}}=(0,0,\eta/2),\bm{r_{B}}=(0,0,-\eta/2)$].
$\tilde{c}_{\bm{k},\alpha}$ explicitly includes the additional phase factor
caused by the sublattice displacement and two definitions are connected by the
following relation

$\displaystyle{c}_{\bm{k},\alpha}=\tilde{{c}}_{\bm{k},\alpha}e^{i\bm{k}\cdot\bm{r_{\alpha}}}.$
(36)

Correspondingly, we can define the Fourier transformation of spin operators as

$\displaystyle
S^{\mu}_{\alpha}(\bm{q})=\sum_{\bm{R}}S^{\mu}_{i,\alpha}e^{-i\bm{q}\cdot\bm{R}_{i}}=\sum_{\bm{k}}c_{\bm{k},\alpha}^{\dagger}\sigma^{\mu}c_{\bm{k+q},\alpha},$
$\displaystyle\tilde{S}^{\mu}_{\alpha}(\bm{q})=\sum_{\bm{R}}S^{\mu}_{i,\alpha}e^{-i\bm{q}\cdot(\bm{R}_{i}+\bm{r}_{\alpha})}=\sum_{\bm{k}}\tilde{c}_{\bm{k},\alpha}^{\dagger}\sigma^{\mu}\tilde{c}_{\bm{k+q},\alpha},$

where $\mu=x,y,z$. And

$\displaystyle\bm{S}_{\alpha}(\bm{q})=\tilde{\bm{S}}_{\alpha}(\bm{q})e^{i\bm{q}\cdot\bm{r_{\alpha}}}.$
(37)

The corresponding spin susceptibilities are

$\displaystyle\chi^{\mu}_{\alpha\gamma}(\bm{q},i\omega)=\langle
S^{\mu}_{\alpha}(\bm{q},i\omega)S^{\mu}_{\gamma}(-\bm{q},-i\omega)\rangle$
$\displaystyle\tilde{\chi}^{\mu}_{\alpha\gamma}(\bm{q},i\omega)=\langle\tilde{S}^{\mu}_{\alpha}(\bm{q},i\omega)\tilde{S}^{\mu}_{\gamma}(-\bm{q},-i\omega)\rangle,$

which are related to each other via

$\displaystyle\chi^{\mu}_{\alpha\gamma}(\bm{q},i\omega)=\tilde{\chi}^{\mu}_{\alpha\gamma}(\bm{q},i\omega)e^{i\bm{q}\cdot(\bm{r}_{\alpha}-\bm{r}_{\gamma})}.$
(38)

In the main text, we use the definition of $c_{\bm{k},\alpha}$ and
$\bm{S}_{\alpha}(\bm{q})$ in order to simplify the notation. However, in the
calculation of dynamical spin susceptibility, we transform back to the
$\tilde{\chi}$ via Eq. 38 and sum over all the scattering channels. The final
spin susceptibilities are defined as

$\displaystyle\chi^{\mu}(\bm{q},i\omega)$ $\displaystyle=$
$\displaystyle\sum_{\alpha\gamma}\tilde{\chi}^{\mu}_{\alpha\gamma}(\bm{q},i\omega)=\sum_{\alpha\gamma}\chi_{\alpha\gamma}^{\mu}(\bm{q},i\omega)e^{-i\bm{q}\cdot(\bm{r_{\alpha}}-\bm{r_{\gamma}})}$
(39) $\displaystyle=$
$\displaystyle\chi^{\mu}_{AA}(\bm{q},i\omega)+\chi^{\mu}_{BB}(\bm{q},i\omega)+\chi^{\mu}_{AB}(\bm{q},i\omega)e^{-i\bm{q}\cdot(\bm{r}_{A}-\bm{r}_{B})}+\chi^{\mu}_{BA}(\bm{q},i\omega)e^{-i\bm{q}\cdot(-\bm{r}_{A}+\bm{r}_{B})}.$

We note that for $q_{z}=0$, in Fig. 4, $\tilde{\chi}=\chi$.

##  VII Calculation of spin resonance

We utilize the random phase approximation (RPA) to calculate the dynamical
spin susceptibility of the spin-$x$ component in the spin-triplet $\Delta_{y}$
pairing case. The corresponding equations in matrix form are

$\chi(\bm{q},i\omega)=\chi^{0}(\bm{q},i\omega)[\mathbb{I}+J(\bm{q})\chi^{0}(\bm{q},i\omega)]^{-1},$
(40)

where $\chi^{0}$ is the bare susceptibility with the following matrix:

$\displaystyle\chi^{0}(\bm{q},i\omega)=\begin{pmatrix}\chi^{0}_{AA}(\bm{q},i\omega)&\chi^{0}_{AB}(\bm{q},i\omega)\\\
\chi^{0}_{BA}(\bm{q},i\omega)&\chi^{0}_{BB}(\bm{q},i\omega)\end{pmatrix}.$
(41)

In practice, the dominant contributions to the low frequency spin
susceptibility come from the intraband processes of the lowest band. Using Eq.
23, we approximately calculate bare spin susceptibility via

$\displaystyle\chi^{0}(\bm{q},i\omega)=\sum_{\bm{k}}\begin{pmatrix}\chi^{0}_{aa}(\bm{q},\bm{k},i\omega)&u_{\bm{k}+\bm{q}}^{*}u_{\bm{k}}\chi^{0}_{aa}(\bm{q},\bm{k},i\omega)\\\
u_{\bm{k}+\bm{q}}u_{\bm{k}}^{*}\chi^{0}_{aa}(\bm{q},\bm{k},i\omega)&\chi^{0}_{aa}(\bm{q},\bm{k},i\omega)\end{pmatrix},$
(42)

where $\chi_{aa}^{0}$ denotes the contribution from the lowest bands and

$\chi_{aa}^{0}(\bm{q},\bm{k},i\omega)\sim\sum_{k}\frac{1}{2}\left(1-\frac{\epsilon_{-}(\bm{k})\epsilon_{-}(\bm{k}+\bm{q})-d_{1}(\bm{k})d_{1}(\bm{k}+\bm{q})}{E_{-}(\bm{k})E_{-}(\bm{k}+\bm{q})}\right)\
\frac{1-f(E_{-}(\bm{k}))-f(E_{-}(\bm{k}+\bm{q}))}{-i\omega+E_{-}(\bm{k})+E_{-}(\bm{k}+\bm{q})}.$
(43)

From the above equations, we can calculate
$\chi_{\alpha\gamma}(\bm{q},i\omega)$. In turn, the total spin susceptibility
is given by
$\chi(\bm{q},i\omega)=\sum_{\alpha\gamma}\chi_{\alpha\gamma}(\bm{q},i\omega)e^{-i\bm{q}\cdot(\bm{r_{\alpha}}-\bm{r_{\gamma}})}$.

Figure 7:  (a) The real part and (b) the imaginary part of the bare spin
susceptibility in the normal state (NS) and superconducting state (SC).

##  VIII Bare susceptibilities

Here, we present the bare susceptibility along $x$ direction

$\chi^{0}(Q,i\omega)=\sum_{\alpha,\gamma}\chi^{0}_{\alpha\gamma}(Q,i\omega)e^{-iQ\cdot(\bm{r}_{\alpha}-\bm{r}_{\gamma})}$
(44)

of both the normal state (NS) and the superconducting state (SC) in Fig. 7.

##  IX magnetic exchange interactions inferred from INS experiments

As an application of Eq. 39, we consider the classical limit and suppose the
magnetic form factor of individual $U$ atoms as $f_{\alpha}(\bm{q})$
($\alpha=A/B$) and suppose the coherent factor $u_{k}=1$ everywhere, which
means $\chi_{AA}(\bm{q})=\chi_{BB}(\bm{q})=f^{2}(\bm{q})$ and
$\chi_{AB}(\bm{q})=\chi_{BA}(\bm{q})=f_{A}(\bm{q})f_{B}(\bm{q})$. Eq. 39 can
be reorganized as

$\chi^{\mu}(\bm{q})=2f^{2}(\bm{q})+f_{A}(\bm{q})f_{B}(\bm{q})e^{i\bm{q}(\bm{r}_{A}-\bm{r}_{B})}+f_{A}(\bm{q})f_{B}(\bm{q})e^{-i\bm{q}(\bm{r}_{A}-\bm{r}_{B})}.$
(45)

If the magnetic correlation between $A$ and $B$ sublattices within the same
unit cell is ferromagnetic, i.e. $f_{A}(\bm{q})=f_{B}(\bm{q})=f(\bm{q})$, then
$\chi^{\mu}_{AB}=\chi^{\mu}_{AB}>0$ and we have

$\displaystyle\chi^{\mu}(\bm{q})$ $\displaystyle=f^{2}(\bm{q})\left(2+2\cos
q_{z}\eta\right)$ (46)
$\displaystyle=4f^{2}(\bm{q})\cos^{2}\frac{q_{z}\eta}{2}.$

For the antiferromagnetic
correlation($f_{A}(\bm{q})=-f_{B}(\bm{q})=f(\bm{q})$),
$\chi^{\mu}_{AB}=\chi^{\mu}_{AB}<0$, and we have

$\displaystyle\chi^{\mu}(\bm{q})$ $\displaystyle=f^{2}(\bm{q})\left(2-2\cos
q_{z}\eta\right)$ (47) $\displaystyle=4f^{2}(\bm{q})\sin^{2}\frac{\
q_{z}\eta}{2}.$

Following Eqs. 46, 47, the inelastic neutron-scattering results imply that the
correlation between $A$ and $B$ sublattices are FM, thereby suggesting a
ferromagnetic $I$ Knafo  _et al._ (2021).

Further experimental constraints, on $J_{r\geq 2}$, are less direct, and are
primarily based on the wavevector for the observed AF correlations. The
observed $Q_{y}\gtrsim\pi$ suggests that both $J_{4}$ and $J_{2}$ are AF.
Likewhise, the observed $Q_{x}=0$ suggests that $J_{3}$ is FM.