††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Correlated normal state fermiology and topological superconductivity in UTe2

Hong Chul Choi  Center for Correlated Electron Systems, Institute for Basic
Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Seung Hun Lee  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826,
Korea  Bohm-Jung Yang<EMAIL_ADDRESS>Center for Correlated Electron
Systems, Institute for Basic Science, Seoul 08826, Korea  Department of
Physics and Astronomy, Seoul National University, Seoul 08826, Korea  Center
for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea

###### Abstract

UTe2 is a promising candidate for spin-triplet superconductors, in which a
paramagnetic normal state becomes superconducting due to spin fluctuations.
The subsequent discovery of various unusual superconducting properties has
promoted the use of UTe2 as an exciting playground to study unconventional
superconductivity, but fathoming the normal state fermiology and its influence
on the superconductivity still requires further investigation. Here, we
theoretically show that electron correlation induces a dramatic change in the
normal state fermiology with an emergent correlated Fermi surface (FS) driven
by Kondo resonance at low temperatures. This emergent correlated FS can
account for various unconventional superconducting properties in a unified
way. In particular, the geometry of the correlated FS can naturally host
topological superconductivity in the presence of odd-parity pairings, which
become the leading instability due to strong ferromagnetic spin fluctuations.
Moreover, two pairs of odd-parity channels appear as accidentally degenerate
solutions, which can naturally explain the multicomponent superconductivity
with broken time-reversal symmetry. Interestingly, the resulting time-reversal
breaking superconducting state is a Weyl superconductor in which Weyl points
migrate along the correlated FS as the relative magnitude of nearly degenerate
pairing solutions varies. We believe that the correlated normal state
fermiology we discovered provides a unified platform to describe the
unconventional superconductivity in UTe2.

##  I introduction

Topological superconductors (TSCs) have received significant attention as a
promising platform to achieve stable qubits, using boundary Majorana zero
modes with non-Abelian statistics Sarma  _et al._ (2015); Sato and Fujimoto
(2016); Sato and Ando (2017). Spin-triplet superconductors are a
representative example of TSC candidates Sato (2010); Fu and Berg (2010).
Since ferromagnetic spin fluctuation is regarded as an origin of spin-triplet
superconductivity Sigrist (2005), uranium (U)-based compounds in which the
coexistence of ferromagnetism and superconductivity was observed Aoki  _et
al._ (2019a), have been considered as a promising playground to investigate
the physics of spin-triplet superconductivity and related topological
properties. In this context, the recent discovery of a new U-based
superconductor UTe2 Ran  _et al._ (2019), for which various pieces of evidence
of unconventional spin-triplet superconductivity have been observed, has
immediately come to the forefront of TSC research. More specifically, UTe2 is
expected to be a spin-triplet superconductor, evidenced by the temperature
independence of the nuclear magnetic resonance (NMR) Knight shift Ran  _et
al._ (2019); Nakamine  _et al._ (2019) and the large upper critical field Ran
_et al._ (2019); Aoki  _et al._ (2019b) above the Pauli limit. Additionally,
follow-up studies have further revealed the unconventional nature of the spin-
triplet superconductivity including gaplessness, topological properties, time-
reversal symmetry (TRS) breaking, and a multicomponent nature Sundar  _et al._
(2019); Tokunaga  _et al._ (2019); Kittaka  _et al._ (2020); Knebel  _et al._
(2019); Miyake  _et al._ (2019); Knafo  _et al._ (2019); Aoki  _et al._
(2020); Lin  _et al._ (2020); Braithwaite  _et al._ (2019); Thomas  _et al._
(2020); Knebel  _et al._ (2020); Ran  _et al._ (2020); Metz  _et al._ (2019);
Bae  _et al._ (2021); Jiao  _et al._ (2020); Hayes  _et al._ (2021); Hillier
_et al._ (2012); Machida and Ohmi (2001).

Not only the superconducting state of UTe2 but also its normal state exhibits
intriguing characteristics such as heavy fermionic behavior with highly
enhanced effective mass Ran  _et al._ (2019) and Kondo resonance Jiao  _et
al._ (2020) arising from strong electron correlation. Moreover, the normal
state of UTe2 is paramagnetic but is proximate to a ferromagnetic quantum
critical point, contrary to other U-based superconductors with robust
ferromagnetism. This indicates that the fermiology of the correlated
paramagnetic normal state under strong ferromagnetic spin fluctuations can be
the quintessential factor governing the unconventional superconductivity of
UTe2, which requires solid theoretical verification.

Here we reveal the fermiology of the correlated normal state of UTe2 and the
resulting spin-triplet superconductivity with nontrivial topological
properties. Using density functional theory (DFT) plus dynamical mean field
theory (DMFT) calculations with angular momentum-dependent self-energy
corrections, we show that the Kondo effect drives the formation of hybridized
bands between U $5f$ and conduction electrons, leading to a drastic change in
the Fermi surface (FS). Namely, at low temperature $T$, we obtain a large
correlated FS enclosing the $\Gamma$ point arising from the Kondo effect. The
emergence of the correlated FS can not only explain the observed heavy fermion
physics but also reconcile various types of unconventional superconducting
behavior as follows. First, the geometry of the correlated FS can host
topological superconducting phases when odd-parity pairing is developed.
Second, by solving the linearized Eliashberg equations with the random phase
approximation, we show that odd-parity spin-triplet superconducting channels
become the leading instability due to strong ferromagnetic spin fluctuations.
Moreover, two pairs of odd-parity channels appear as accidentally degenerate
solutions, which can naturally explain the multicomponent superconductivity
with broken time-reversal symmetry Hayes  _et al._ (2021). Interestingly, we
find that the time-reversal breaking superconducting state is a Weyl
superconductor in which the positions of Weyl points vary depending on the
relative magnitude of nearly degenerate pairing solutions with the
trajectories bounded by the correlated FS. We believe that the fermiology of
the correlated normal state we discovered provides a unified way of
understanding the unconventional superconductivity of UTe2.

##  II Band structure calculations

To date, the band structure of UTe2, with the crystal structure shown in Fig.
1a, has been reported using various DFT-based calculations. For instance,
conventional DFT calculations of UTe2 predicted a paramagnetic insulating
ground state, and thus failed to reproduce its metallic phase at low
temperatures. This occurred because in DFT calculations, the hybridization
between U $5f$ states and conduction electrons (U $6d$ and Te $5p$ electrons)
is too strong such that a large gap is opened near the Fermi energy ($E_{F}$)
Ran  _et al._ (2019); Aoki  _et al._ (2019b). Introduction of magnetism or an
on-site Coulomb interaction ($U$) Ishizuka  _et al._ (2019); Ishizuka and
Yanase (2021); Shishidou  _et al._ (2021); Xu  _et al._ (2019); Miao  _et al._
(2020); Shick and Pickett (2019) can partially resolve this issue and restore
the metallic ground state. However, as the DFT+$U$ method generally suppresses
charge fluctuations, the Kondo effect is not properly described. Similarly,
the renormalized FS from the Kondo resonance at low temperatures was not
successfully described even in previous DFT plus DMFT calculations Xu  _et
al._ (2019); Miao  _et al._ (2020); Duan  _et al._ (2020).

Figure 1: Crystal structure and open-core DFT calculation results. a,
Schematic crystal structure Momma and Izumi (2011) of UTe2 in the conventional
orthorhombic body-centered unit cell. There are two types of Te atoms (Te1 and
Te2) as well as U atoms. The blue and red-green solid lines denote the Te1
chains along the $b$-axis and the U-Te2 chains along the $a$-axis,
respectively. The red vertical dashed lines indicate the nearest neighbor U
atoms along the $c$-axis. b, Band structure obtained from the open-core DFT
calculation with the conventional unit cell without U $5f$ electrons near
$E_{F}$. The red arrows indicate the hole Fermi pocket around $X$ with a U
$5d$ orbital character, and the blue arrows indicate the electron Fermi pocket
around $Y$ with a Te1 $5p$ orbital character. c, FS from the open-core DFT
calculations plotted using the conventional unit cell. There are two 1D FSs
along the $k_{x}$-axis (U $6d$ states) and the $k_{y}$ axis (Te1 5p states).
d, First Brillouin zones (BZs) of the primitive unitcell and the conventional
orthorhombic cell. Blue large letters indicate the time-reversal invariant
momentum (TRIM) in the BZ with the conventional cell. $Z_{p}$,$R_{p}$, and
$M_{p}$ are TRIMs defined in the primitive unit cell. $b_{1}$,$b_{2}$, and
$b_{3}$ are reciprocal vectors in this primitive unit cell. The arrow from
$Z_{p}$ to $\Gamma$ indicates that the $Z_{p}$ point is folded into the
$\Gamma$ point in the BZ of the conventional unit cell.

The electronic structure with localized U $5f$ electrons can be described by
using open-core DFT calculations in which U $5f$ electrons are pushed into the
core states far from $E_{F}$. The resulting band structure in Fig. 1b supports
a quasi-two-dimensional (2D) FS as shown in Fig. 1c. The origin of this FS is
the intrachain and interchain interactions of the two orthogonal quasi-1D
atomic chain structures in Fig. 1a represented by blue solid lines and red-
green zigzag lines. As the Te1 chains lie in the $b$-axis direction, Te1
5$p_{z}$ states appear to be dominant around $E_{F}$ along the
$\Gamma$-$Y$-$M$ path. In contrast, the U1-Te2 zigzag chains lie in the
$a$-axis direction; thus, U 6$d_{z}$ states are dominant around $E_{F}$ along
the $\Gamma$-$X$-$M$ path. Such a quasi-2D FS corresponds to the high
temperature phase in which most $5f$ states are localized.

Figure 2: DFT+DMFT calculation results. a, DFT+DMFT spectral functions at $T$
= 11 K plotted using the BZ of the conventional unit cell. b, Emergence of
atomic states near $E_{F}$ in the DFT+DMFT calculations. The f-electron
spectral function is plotted between -3 eV and 3 eV. The correlation effect
drives additional upper Hubbard band ($f^{3}$) and $f^{2}$ multiplet states.
The atomic multiplet calculation in the DMFT part shows that the
${}^{3}\mathrm{F}_{2}$ and ${}^{3}\mathrm{H}_{4}$ of the $f^{2}$ states play
important roles, in good agreement with another DFT+DMFT calculation Miao  _et
al._ (2020). c, Coherent DFT+DMFT quasiparticle spectrum around $E_{F}$
plotted using the BZ of the primitive unit cell. The $\Gamma_{6}$ state
($|j=5/2,j_{m}=\pm 1/2\rangle$) is dominant around $E_{F}$. The red elliptical
loop indicates the nearly flat bands near $E_{F}$ around the $Z_{p}$ point,
which may induce a large spectral weight before the formation of the low $T$
quasiparticle state. d, Large quasiparticle FS in the BZ of the primitive unit
cell. The $Z_{p}$ point is indicated by the red elliptical line. e, Coherent
DFT+DMFT quasiparticle spectrum around $E_{F}$ plotted using the conventional
unit cell. The green- and blue- colored quasiparticle states correspond to the
FSs in Fig. 2f with the same color. f, Large quasiparticle FS plotted using
the conventional unit cell. The blue hole pocket near the $\Gamma$ point is
the correlated 3D FS and the green electron pocket around the $M$ point is a
quasi-2D FS, which are indicated by the green and blue lines in Fig. 2e. g, h,
Schematic figure describing the Lifshitz transition from the high-T FS to the
low-T correlated FS.

With decreasing $T$, the emergence of U $5f$ electrons near $E_{F}$ gives rise
to a Lifshitz transition in the FS. To capture the Liftshitz transition
induced by the Kondo resonance, we perform DFT+DMFT calculations. Contrary to
the preceding DFT+DMFT calculations, we allow the electronic self-energy to
vary depending on the angular momentum $j$ ($j=5/2$ and $7/2$ for U $5f$
states) and its $z$-component $m_{j}$ to differentiate the orbital characters.
The self-energy calculated using the primitive unit cell with two U atoms is
obtained at T= 11 K. The resulting electronic structure of UTe2 at $T=$11 K is
shown in Fig. 2. In comparison to Fig. 1b, the spectral function plot shows
that U $5f$ electrons arise around $E_{F}$ (inside the green solid box in Fig.
2a), but are not yet fully hybridized with conduction electrons at this
temperature. Meanwhile, the upper Hubbard bands of $f^{3}$ states are
redistributed around 2 eV above $E_{F}$ (inside the blue solid box in Fig.
2a). Among the $f^{2}$ multiplets denoted by the symbol ${}^{2s+1}L_{J}$,
where $s$, $L$, $J$ indicate the spin, orbital, and total angular momentum,
respectively, ${}^{3}\mathrm{H}_{4}$ states play an important role near the
$E_{F}$ (the green solid box), while ${}^{3}\mathrm{F}_{2}$ states are located
-0.6 eV below $E_{F}$ (the green dashed box) as shown in Fig. 2a. We note that
the incoherent spectrum of ${}^{3}\mathrm{F}_{2}$ states at $-0.6$ eV can be
identified with the $-0.6$ eV signal observed in angle-resolved photoemission
spectroscopy (ARPES) measurements Miao  _et al._ (2020); Fujimori  _et al._
(2019) and another DFT+DMFT calculation Miao  _et al._ (2020). Fig. 2b
summarizes the atomic multiplet distribution in the energy spectrum near
$E_{F}$.

To obtain the quasiparticle FS from LDA+DMFT, we set the imaginary part of the
self-energy to zero Kang and Kotliar (2021); Nomoto and Ikeda (2014); Choi
_et al._ (2012). The corresponding quasiparticle FSs for the primitive and
conventional unit cells are plotted in Figs. 2d and f, respectively. The
formation of the Kondo resonance state represents the Lifshitz transition to
the low-T state with a large FS. We find that the $\Gamma_{6}$
($=|j=5/2,m_{j}=\pm 1/2\rangle$) orbital originating from the
${}^{3}\mathrm{H}_{4}$ state near $E_{F}$ makes a dominant contribution in the
vicinity of $E_{F}$ through the $m_{j}$-dependent energy spectrum as shown in
Fig. 2c. The corresponding spectral functions in the conventional unit cell
show that a hole Fermi pocket encloses the $\Gamma$ point and the electron
pocket forms a quasi-2D green Fermi sheet around the $M$ point, as shown in
Fig. 2f, which is equivalent to the FS topology of the primitive cell in Fig.
2d.

Unlike the DFT+$U$ calculation that achieves the metallic ground state by
pushing $5f$ electrons away from $E_{F}$, our DFT+DMFT calculation explains
how the Kondo effect induces the metallic behavior of UTe2 in which the U
$\Gamma_{6}$ state hybridized with Te $p_{z}$ and U $d_{z^{2}}$ states forms
an emergent FS at very low $T$.

Through the Kondo hybridization, the quasi-2D FS surrounding the $X$ or $Y$
point morphs into another quasi-2D FS surrounding the BZ corners and a
cylindrical FS closing the $\Gamma$ point [see Fig. 2g, h]. As $T$ decreases,
the cylindrical FS at the BZ center develops a more three-dimensional (3D)
character, eventually forming an ellipsoidal closed surface enclosing the
$\Gamma$ point. The 3D nature of the correlated FS is also consistent with the
fact that a strong hopping parameter between U atoms along the $c$-axis (see
the red dashed lines in Fig. 1a) comparable to that along the $a$-axis is
required to construct a tight binding model describing the very low-$T$ FS. We
note that a strong spectral weight around the $Z$ point of the BZ of the
primitive unit cell was measured in a recent high resolution ARPES study Miao
_et al._ (2020). We speculate that in the course of the FS evolution from the
cylindrical FS to the 3D one, a strong spectral feature can appear from the
nearly flat quasiparticle bands around the $Z$ point shown in Fig. 2c, which
can be considered a precursor of the formation of a $3D$ correlated FS at very
low $T$. Additionally, the recently measured nearly isotropic transport
property Eo  _et al._ (2021) further supports the presence of the 3D FS.

##  III Superconducting instability

As the correlated 3D FS consists mainly of U $5f$ electrons with strong local
Coulomb interaction $U$, and is thus susceptible to the related spin
fluctuations, its emergence at low $T$ should have a crucial impact on the
superconductivity of UTe2. To explore the superconducting instability of UTe2
by considering the spin fluctuation effect on U $5f$ electrons, we solve the
linearized Eliashberg equations given by

where $\xi$, $\xi_{1}$, $\xi_{2}$, and $\xi^{\prime}$ are indices that denote
the uranium atomic position and the electron spin, and $k$ and $k^{\prime}$
indicate the momentum. $\Delta^{\rho}$ is a gap function that belongs to an
irreducible representation (IR) $\rho$ of the $D_{2h}$ point group. For the
pairing functions, we use the basis functions listed in Table. 1. $V$ is the
effective pairing interaction. As ferromagnetic spin fluctuation is expected
to be the origin of the spin-triplet superconductivity in U-based heavy
fermion metals Sigrist and Ueda (1991), we take into account the spin
fluctuation effect from the on-site Coulomb interaction $U$ of $5f$ electrons
within the random phase approximation (RPA) (see Fig. 3d and the SI sup ). $G$
is the normal state Green’s function of U $5f$ $\Gamma_{6}$ electrons. To
obtain $G$, we construct a tight-binding model that reproduces the correlated
normal state fermiology from LDA+DMFT by using U $\Gamma_{6}$, U $d_{z^{2}}$
and Te $p_{z}$ orbitals (Fig. 3a).

Table 1: Transformation properties, basis functions, and gap structures of IRs under the $D_{2h}$ point group symmetry. Here, PN denotes point nodes. IR | $E$ | $C_{2z}$ | $C_{2y}$ | $C_{2x}$ | $I$ | Basis functions | Gap structure
---|---|---|---|---|---|---|---
$A_{u}$ | 1 | 1 | 1 | 1 | -1 |  $k_{x}\hat{x}$, $k_{y}\hat{y}$, $k_{z}\hat{z}$ | Fully gapped
$B_{1u}$ | 1 | 1 | -1 | -1 | -1 |  $k_{y}\hat{x}$, $k_{x}\hat{y}$ | PN ($k_{z}$-axis)
$B_{2u}$ | 1 | -1 | 1 | -1 | -1 |  $k_{x}\hat{z}$, $k_{z}\hat{x}$ | PN ($k_{y}$-axis)
$B_{3u}$ | 1 | -1 | -1 | 1 | -1 |  $k_{z}\hat{y}$, $k_{y}\hat{z}$ | PN ($k_{x}$-axis)
Figure 3: Superconductivity from the linearized Eliashberg equation approach.
a, FS reproduced by the tight-binding model. The colors represent the weight
of the U $5f$ electron component in the quasiparticle wavefunction on the low
$T$ FS. The region in the black dashed circle shows a relatively low U $5f$
electron contribution on the 3D FS. b, Eigenvalues of the linearized
Eliashberg equations. The pair of IRs in each red box are almost degenerate.
c, Schematic superconducting gap structures of four IRs on the correlated 3D
FS. d, RPA spin susceptibility in the $a$-axis (easy axis) direction on the
$k_{x}k_{y}$-plane, which shows strong ferromagnetic fluctuations.

By solving the linearized Eliashberg equations, we find that the gap functions
belonging to $\rho=A_{u},B_{1u},B_{2u}$, and $B_{3u}$ IRs have nonzero
eigenvalues (see Fig. 3b). Interestingly, all four 1D IRs belong to the odd-
parity spin-triplet channels, which is consistent with the fact that the
strong ferromagnetic fluctuations of U $5f$ electrons give spin-triplet
superconductivity. Moreover, the eigenvalues of the four IRs have comparable
magnitudes, which is different from previous numerical studies Ishizuka  _et
al._ (2019); Ishizuka and Yanase (2021), in which some of these four IRs were
significantly more favored than the others. This might be the direct outcome
of the isotropic nature of our correlated 3D FS, which is distinct from the
anisotropic cylindrical FS considered in other works.

More specifically, our Eliashberg equation calculations predict two pairs of
almost degenerate IRs. Namely, the $A_{u}$ and $B_{1u}$ IRs, and the $B_{2u}$
and $B_{3u}$ IRs appear nearly degenerate, while the $B_{2u}$ and $B_{3u}$ IRs
are slightly more favored than the $A_{u}$ and $B_{1u}$ IRs, consistent with a
recent renormalization group calculation Shaffer and Chichinadze (2022). We
note that the IRs of each pair become the same IR when the system’s symmetry
is lowered by applying an external magnetic field along the $z$-direction.
Among the four IRs, the largest eigenvalue appears in the $B_{2u}$ channel
which is favored due to the inhomogeneous distribution of U $5f$ electron wave
functions on the $3D$ FS. That is, as the $3D$ FS has a relatively small U
$5f$ electron weight on the $k_{y}$-axis (see Fig. 3a), the $B_{2u}$
representation, which has symmetry protected nodes on the $k_{y}$-axis, has an
advantage in lowering the total free energy (see Fig. 3c and Table 1).
Notably, the appearance of almost degenerate pairing states is consistent with
recent specific heat measurements Hayes  _et al._ (2021) showing two nearby
transition peaks separated by only 80 mK. Since the $B_{2u}$ and $B_{3u}$ IRs
have larger eigenvalues than the $A_{u}$ and $B_{1u}$ IRs, we believe the two
peaks correspond to the $B_{2u}$ and $B_{3u}$ representations. Moreover, this
accidental degeneracy of IRs can lead to a mixed order parameter
$B_{2u}+iB_{3u}$, which gives a TRS breaking Weyl superconductivity (see the
SI sup ).

Figure 4: Topological superconductivity in UTe2. a, FSs and TRIM points in the
BZ of UTe2. b, The Chern numbers carried by the occupied states of the BdG
Hamiltonian on the $k_{z}=0$ plane (orange) and the $k_{z}=\pi$ plane (blue).
The Chern numbers on the two planes differ by 2, which indicates that the two
Weyl nodes (magenta) located between the $k_{z}=0$ and $k_{z}=\pi$ planes have
the same charge ($+1$ each), while the other two nodes (yellow) located
between the $k_{z}=0$ and $k_{z}=-\pi$ planes both have charge $-1$. c,
Trajectory of Weyl nodes. The blue (red) arrows represent the path of nodes
with $+$ ($-$) charge. As $\alpha$ increases from 0 to a critical value
$\alpha_{c}$, the Weyl nodes generated by the $\alpha
B_{2u}+i(1-\alpha)B_{3u}$ pairing move starting from the $k_{x}$-axis at
$\alpha=0$ to the $k_{z}$-axis at $\alpha=\alpha_{c}$. After the nodes with
the same charge accidentally meet on the $k_{z}$-axis at $\alpha=\alpha_{c}$,
they split and move toward the $k_{y}$-axis until $\alpha>\alpha_{c}$ reaches
1. d, Zero-energy Majorana arc states (top), and their position expectation
values that show their surface localization (bottom).

##  IV Topological superconductivity

The emergent correlated 3D FS also significantly affects the band topology of
the superconducting UTe2. The Bogoliubov-de Gennes (BdG) Hamiltonian of
superconducting UTe2 can have various gap structures depending on the symmetry
of the pairing function. From symmetry-based analysis, one can show that the
$A_{u}$ pairing has a fully gapped spectrum, while the $B_{1u}$, $B_{2u}$, and
$B_{3u}$ pairings have nodal points on the $k_{z}$-, $k_{y}$-, and
$k_{x}$-axes, respectively, as summarized in Table 1.

Let us first consider the fully gapped pairing with the $A_{u}$ IR. When TRS
is preserved, a $3D$ spinful superconductor such as UTe2 belongs to the
Altland-Zirnbauer class DIII whose topological invariant is the $3D$
$\mathbb{Z}$ winding number. In this case, the parity of this $3D$ winding
number (a $\mathbb{Z}_{2}$ invariant) can be captured by counting the numbers
of Kramers-degenerate FSs surrounding the TRIM points Fu and Berg (2010); Sato
(2010); Sato and Ando (2017). Since a cylindrical quasi-2D FS always
simultaneously encloses a pair of TRIM points, one on the $k_{z}=0$ plane and
the other on the $k_{z}=\pi$ plane (blue dots in Fig. 4a), it does not
contribute to the nontrivial $\mathbb{Z}_{2}$ invariant. In contrast, the
correlated $3D$ FS enclosing only the $\Gamma$ point (the red dot in Fig. 4a)
renders the strong bulk $\mathbb{Z}_{2}$ index nontrivial. Therefore, UTe2
with the $A_{u}$ pairing becomes a first order TSC that hosts gapless Majorana
surface states.

In the case of the $B_{1u}$, $B_{2u}$, and $B_{3u}$ pairings, they are
guaranteed to have nodal points in the presence of the 3D FS that always
intersects with the $k_{z}$-, $k_{y}$-, and $k_{x}$-axes. When TRS is present,
these pairing channels support fourfold degenerate gapless points along their
high-symmetry lines as summarized in Table 1.

In contrast, when the nearly degenerate $B_{2u}$ and $B_{3u}$ pairing channels
form a complex order parameter in the form of $\alpha
B_{2u}+i(1-\alpha)B_{3u}$ ($0\leq\alpha\leq 1$), a 4-fold nodal point splits
into two 2-fold Weyl points, thus generating a Weyl superconductor. More
explicitly, as $\alpha$ becomes slightly larger than zero, two 4-fold nodal
points of the $B_{3u}$ pairing state at the intersection between the 3D FS and
the $k_{x}$-axis split into four 2-fold Weyl points located at the
intersection between the 3D FS and the $k_{x}k_{z}$ plane. At
$\alpha=\alpha_{c}$, pairs of Weyl points with the same chiral charge merge on
the $k_{z}$-axis. The fact that the two Weyl nodes in the $k_{z}>0$
($k_{z}<0$) region have the same monopole charge is further confirmed by
computing the Chern number on the $k_{z}=0$ and $k_{z}=\pi$ planes (orange and
blue planes in Fig. 4b), which is equal to 2 and 4, respectively. When
$\alpha$ becomes larger than $\alpha_{c}$, the merged Weyl points with chiral
charge $\pm 2$ again split into 4 Weyl points, which are located at the
intersection between the FS and the $k_{y}k_{z}$ plane. Finally, at
$\alpha=1$, pairs of Weyl points with opposite chiral charges merge and form
4-fold nodal points of the $B_{2u}$ pairing state on the $k_{y}$-axis. In the
Weyl superconductor phase, UTe2 hosts surface Majorana arcs connecting pairs
of Weyl nodes with opposite chiral charges projected on the surface BZ as
shown in Fig. 4d. The geometry of the correlated 3D FS promises nontrivial
topology of superconducting UTe2, regardless of whether TRS is broken.

##  V Discussion

In summary, we have performed DFT+DMFT calculations and solved the Eliashberg
equations with the tight-binding Hamiltonian to study the strongly correlated
normal state of UTe2 and the related unconventional spin-triplet
superconductivity. Our correlated electronic structure not only explains the
spectral features measured in recent ARPES experiments Hayes  _et al._ (2021)
but also predicts the emergence of correlation-driven $3D$ FS around the
$\Gamma$ point at low temperatures, which can account for various types of
anomalous behaviors in both the normal state and the superconducting state
Ishizuka  _et al._ (2019); Ishizuka and Yanase (2021). In particular, the
nearly degenerate spin-triplet solution of the Eliashberg equations naturally
predicts a TRS breaking Weyl superconductor in which the positions of Weyl
points vary depending on the relative magnitude of the nearly degenerate
pairing solutions, which can be verified in future experiments. Additionally,
the fact that the correlated 3D FS supports the emergence of Weyl points
indicates the strongly correlated nature of the Weyl fermions in this system.
Thus we propose that UTe2 is a venue to study the intriguing physics of
strongly interacting Weyl fermions Roy  _et al._ (2017); Lai  _et al._ (2018).

Finally, we note that our correlated FS supports the strong ferromagnetic
fluctuation, which is consistent with recent experimental results such as the
scaling behavior of magnetization Ran  _et al._ (2019); Sundar  _et al._
(2019), ferromagnetic fluctuation signals observed in NMR experiments Tokunaga
_et al._ (2019), and others Willa  _et al._ (2021); Butch  _et al._ (2022).
Even though incommensurate antiferromagnetic fluctuations have also been
detected in recent neutron scattering measurements Duan  _et al._ (2020);
Knafo  _et al._ (2021); Duan  _et al._ (2021), the correlated normal state
fermiology we obtained shows that ferromagnetic fluctuation plays a dominant
role in the superconducting phase transition of UTe2. We believe that our
theory can provide a unified framework to understand the complex behavior of
UTe2 and resolve the remaining controversies in this field.

Acknowledgements We thanks for fruitful discussions with Seokjin Bae and Yun
Suk Eo. H.C.C, S.H.L., B.J.Y. were supported by the Institute for Basic
Science in Korea (Grant No. IBS-R009-D1). S.H.L., B.J.Y. were supported by
Samsung Science and Technology Foundation under Project Number SSTF-BA2002-06,
and the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (No. 2021R1A2C4002773, and No. NRF-2021R1A5A1032996).

Author contributions BJY initially conceived the project. HCC and SHL
contributed to the theoretical analysis and wrote the manuscript with BJY. HCC
did all of the ab initio calculations. BJY supervised the project. All authors
discussed and commented on the manuscript.

Competing financial interest statement The authors have no competing financial
interests to declare.

Supplementary Information for “Correlated normal state fermiology and
topological superconductivity in UTe2”

##  S1 Computational details

The charge self-consistent version of DFT+DMFT Kotliar  _et al._ (2006), as
implemented in Ref. Haule  _et al._ , 2010, is based on the full-potential
linearized augmented plane-wave (FP-LAPW) band method Blaha  _et al._ (2001).
The correlated $5f$ electrons are treated dynamically by the DMFT local self-
energy ($\Sigma(\omega)$), while all other delocalized $spd$ electrons are
treated on the DFT level. The charge and spin fluctuations considered in DMFT
enable the description of the Kondo effect correctly. $\Sigma(\omega)$ is
calculated from the corresponding impurity problem, in which full atomic
interaction matrix is taken into account ($F^{0}=8.0$ eV,
$F^{2}=7.15317919075$ eV, $F^{4}=4.77832369942$ eV, and $F^{6}=3.53367052023$
eV) with $U=8.0$ eV and $J=0.6$ eV Cowan (1981). A temperature of 1.1 meV (11
K) is used in the calculations. To solve the impurity problem, we used the
continuous quantum Monte Carlo approximation (Gull  _et al._ (2011); Haule
(2007)).

##  S2 Tight binding Hamiltonian

Using U $\Gamma_{6}$, U $d_{z^{2}}$, and Te $p_{z}$ orbitals, the tight-
binding model was constructed to reproduce a 3D FS enclosing the $\Gamma$
point, and 2D FS sheets shown in Fig. 2d,f.

$\displaystyle\varepsilon_{d}$ $\displaystyle=1.574276$ (S1)
$\displaystyle\varepsilon_{f}$ $\displaystyle=-0.17$
$\displaystyle\varepsilon_{p}$ $\displaystyle=-1.9952828$ $\displaystyle
t^{\textrm{intra}}_{d1}$ $\displaystyle=-0.7513$ $\displaystyle
t^{\textrm{intra}}_{d2}$ $\displaystyle=0.0610$ $\displaystyle
t^{\textrm{intra}}_{f1}$ $\displaystyle=0.1$ $\displaystyle
t^{\textrm{intra}}_{f2}$ $\displaystyle=0.01$ $\displaystyle
t^{\textrm{intra}}_{df1}$ $\displaystyle=-0.06$ $\displaystyle
t^{\textrm{intra}}_{df2}$ $\displaystyle=-0.01$ $\displaystyle
t^{\textrm{intra}}_{p1}$ $\displaystyle=0.1043321$ $\displaystyle
t^{\textrm{intra}}_{d1}$ $\displaystyle=-1.097552$ $\displaystyle
t^{\textrm{intra}}_{d2}$ $\displaystyle=0.095234$ $\displaystyle
t^{\textrm{inter}}_{f1}$ $\displaystyle=-0.1.$ $\displaystyle
t^{\textrm{inter}}_{f2}$ $\displaystyle=0.04$ $\displaystyle
t^{\textrm{inter}}_{df1}$ $\displaystyle=-0.25$ $\displaystyle
t^{\textrm{inter}}_{df2}$ $\displaystyle=-0.01$ $\displaystyle
t^{\textrm{inter}}_{p1}$ $\displaystyle=-1.970172$ $\displaystyle t_{dp}$
$\displaystyle=0.25$ $\displaystyle t_{fp}$ $\displaystyle=0.13$

The above hopping terms are explained in Fig. S1.

Figure S1: TB hopping parameters in the conventional orthorhombic unit cell.

The tight-binding Hamiltonian reads

$\displaystyle H$ $\displaystyle=\begin{pmatrix}H_{ff}&H_{fd}&H_{fp}\\\
H_{df}&H_{dd}&H_{dp}\\\ H_{pf}&H_{pd}&H_{pp}\\\ \end{pmatrix},$ (S2)

where $H_{ij}$’s are $8\times 8$ matrices that represent hopping from
$i$-orbital to $j$-orbital.

The below spin-orbit couplingIshizuka and Yanase (2021) with $\alpha_{1}=0.2$
and $\alpha_{2}=0.2$ is added to $H_{ff}$ so that the anisotropic spin
susceptibility along the $a$-axis is realized.

$\displaystyle H_{\textrm{ASOC}}$ $\displaystyle=\big{(}\alpha_{1}\sin
k_{y}\sigma_{x}-\alpha_{2}\sin
k_{x}\sigma_{y}\big{)}\otimes\tau^{\textrm{intra}}_{z}\otimes\tau^{\textrm{inter}}_{0}$
(S3)

where, $\sigma_{i}$, $\tau^{\textrm{intra}}_{j}$, and
$\tau^{\textrm{inter}}_{k}$ are the Pauli matrices for the spin, intra rung
sublattice, and inter rung sublattice degrees of freedoms, respectively.

##  S3 Linearized Eliashberg equation

The effective pairing potential used to solve the linearized Eliashberg
equation reads

$\hat{V}(q)=-\hat{\Gamma}^{0}\hat{\chi}(q)\hat{\Gamma}^{0}-\hat{\Gamma}^{0},$
(S4)

where $\hat{\Gamma}^{0}$ is the bare irreducible vertex that describes the on-
site Coulomb interaction ($U^{\prime}$) Kanasugi and Yanase (2020). We note
that $U^{\prime}$ is defined in the renormalized quasiparticle states near the
$E_{F}$, which is different from $U$ of atomic $5f$ orbitals in the impurity
solver in the DMFT loop. The spin susceptibility $\hat{\chi}(q)$ is calculated
from the bare spin susceptibility ($\chi_{0}$) as follows,

$\displaystyle\hat{\chi}(q)$
$\displaystyle=\big{[}\hat{1}-\hat{\chi}^{0}(q)\hat{\Gamma}^{0}\big{]}^{-1}\hat{\chi}^{0}(q),$
(S5)

within the RPA. $\hat{\chi}(q)$ shows a ferromagnetic peak which diverges as
$U^{\prime}$ increases (Fig S3). When spin-orbit coupling is introduced, the
$a$-axis is favored as an easy axis. The ferromagnetic instability was checked
through the full Brillouin zone as shown in Fig S2.

Figure S2: Spin susceptibility in the full Brillouin zone. The ferromagnetic
instability could be seen clearly. We used a $30\times 30\times 30$ mesh in
the full Brillouin zone. The secondary peaks are located along the
$k_{z}$-axis around the $\Gamma$ point. On-Site Coulomb interaction is set at
$U^{\prime}$= 2.83 eV.  Figure S3: Spin susceptibility as a function of
$U^{\prime}$ at $q=\Gamma$ (ferromagnetic fluctuation) with and without spin
orbit coupling (SOC) a, Without the SOC, isotropic ferromagnetic fluctuation
occurs. b, With the SOC, the easy axis is clearly recognized as the a-axis
with the substantial divergence of $\chi_{xx}$.

To solve the linearized Eliashberg equations

$\displaystyle\lambda\Delta_{\xi\xi^{\prime}}^{\rho}=-\frac{T}{N}\tsum\slimits@_{k^{\prime},\xi_{j}}V_{\xi\xi_{1}\xi_{2}\xi^{\prime}}(k-k^{\prime})G_{\xi_{3}\xi_{1}}(-k^{\prime})\Delta_{\xi_{3}\xi_{4}}^{\rho}(k^{\prime})G_{\xi_{4}\xi_{2}}(k^{\prime}),$
(S7)

we introduce $\phi$ defined as

$\displaystyle[\phi]^{\mu_{1}s_{1},\mu_{2}s_{2}}_{\mu_{3}s_{3}\mu_{4}s_{4}}(\mathbf{k},i\omega_{n}=0)$
$\displaystyle=\tsum\slimits@_{n_{1}n_{2}}\big{[}M_{n_{1}n_{2}}\big{]}^{\mu_{1}s_{1},\mu_{2}s_{2}}_{\mu_{3}s_{3}\mu_{4}s_{4}}\frac{f(\bar{\xi}_{-k,n_{1},\sigma_{1}})-f(\xi_{k,n_{2},\sigma_{2}})}{\bar{\xi}_{-k,n_{1},\sigma_{1}}-\xi_{k,n_{2},\sigma_{2}}}$
(S8)
$\displaystyle=\tsum\slimits@_{n1}\frac{\big{[}M_{n_{1}n_{2}}\big{]}^{\mu_{1}s_{1},\mu_{2}s_{2}}_{\mu_{3}s_{3}\mu_{4}s_{4}}}{2\xi_{n1}}\tanh(\frac{\xi}{2T}),$
(S9)

where

$\big{[}M_{n_{1},n_{2}}\big{]}=\big{[}u^{\mu_{1}s_{1}}_{n_{1}\sigma_{1}}(-k)\big{]}^{*}\big{[}u^{\mu_{3}s_{3}}_{n_{2}\sigma_{3}}(k)\big{]}^{*}\big{[}u^{\mu_{2}s_{2}}_{n_{2}\sigma_{2}}(k)\big{]}\big{[}u^{\mu_{4}s_{4}}_{n_{1}\sigma_{4}}(-k)\big{]}.$
(S10)

Here, $u^{\mu s}_{n\sigma}(k)$ is an eigenstate of the given tight binding
model. $\mu$, $s$, $n$, $\sigma$ are orbital, spin, band index, pseudospin
degrees of freedom, respectively. Solving the linearized Eliashberg equations
at $T=0.003$ eV, we obtain eigenvalues as a function of U’ shown in Fig. S4.
As U’ is increased, B2u and B3u pairings become degenerated and Their
eigenvalues is larger than the other degenerated Au and B1u pairings.

Figure S4: Linearized Eliashberg equation as a function of $U^{\prime}$.
Eigenvalues are ploted as a function of $U^{\prime}$.

##  S4 Topological invariant

Under TRS and spatial inversion symmetry, the (strong) topological invariant
($\omega$) in an odd-parity superconductor can be defined as below,

$\displaystyle\omega=\frac{1}{2}\tsum\slimits@_{K_{i}}n(K_{i})~{}~{}(mod~{}2),$
(S11)

where $\nu_{1}$, $\nu_{2}$ and $\nu_{3}$ are the weak invariants defined on
the $kx=0$, $k_{y}=0$, and $k_{z}=0$ planes. The topological invariants for
Fig. 1c, Fig. 2f are summarized in Table S1.

Table S1: Topological invariants in $D_{2h}$. case | $\Gamma$ | $X$ | $M$ | $Y$ | $Z$ | $S$ | $T$ | $R$ | ($\omega$,$\nu_{1}$,$\nu_{1}$,$\nu_{1}$)
---|---|---|---|---|---|---|---|---|---
Open-core | 0 | 4 | 0 | -4 | 0 | 4 | -4 | 0 | (0,0,0,0)
DFT+DMFT | 2 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | (1,1,1,1)
Figure S5: The Wilson loop eigenvalue spectra The Wilson loop eigenvalue
spectra of the occupied states of the BdG Hamiltonian of UTe2 with arbitrary
pairing function in $B_{2u}+iB_{3u}$ symmetry on the $k_{z}=0$ plane (orange)
and the $k_{z}=\pi$ plane (blue). The winding numbers of the two Wilson loop
shows that the Chern numbers carried by the two planes differ by 2. It
indicates that two Weyl nodes located between the $k_{z}=0$ and $k_{z}=\pi$
planes have the same charge.

When TRS is broken by $B_{2u}+iB_{3u}$ pairing, the 2D BdG Hamiltonians on the
$k_{z}=0$ plane and the $k_{z}=\pi$ plane are fully gapped. Thus we can
calculate Chern number carried by the occupied bands for each plane. The
difference between Chern numbers of the two planes is equal to the total
monopole charge of Weyl nodes located between the two planes. As shown in Fig.
S5, the two Weyl nodes in $0<k_{z}<\pi$ region has $+1$ charge for each, and
the other two in $-\pi<k_{z}<0$ region has $-1$ charge for each.

##  S5 Spontaneous TRS breaking

We consider the most general form of Ginzburg-Landau free energy density that
describes a system with two almost degenerate complex order parameters
$\eta_{1}$ and $\eta_{2}$. The free energy density reads

$\displaystyle\mathcal{F}=$
$\displaystyle\frac{r_{0}}{2}(\eta_{1}^{*}\eta_{1}+\eta_{2}^{*}\eta_{2})-\frac{x}{2}(\eta_{1}^{*}\eta_{1}-\eta_{2}^{*}\eta_{2})+\frac{u_{+}}{8}(\eta_{1}^{*}\eta_{1}+\eta_{2}^{*}\eta_{2})^{2}$
(S15)
$\displaystyle+\frac{u_{-}}{8}(\eta_{1}^{*}\eta_{1}-\eta_{2}^{*}\eta_{2})^{2}-\frac{g}{8}[-i(\eta_{1}^{*}\eta_{2}-\eta_{1}\eta_{2}^{*})]^{2},$

where $u$, $g$, $\lambda$ are phenomenological coupling coefficients and
$u\pm=u\pm(g+\lambda)$. Let $\eta_{1}=r_{1}e^{i}\theta_{1}$ and
$\eta_{2}=r_{2}e^{i}\theta_{2}$. By differentiating Eq. S15 by $r_{1}$,
$r_{2}$, $\theta_{1}$, and $\theta_{2}$, we obtain the extremum conditions for
$\mathcal{F}$. Particularly, when differentiated by $\theta_{1}$ or
$\theta_{2}$, the last term in Eq. S15 gives
$\sin{2(\theta_{2}-\theta_{1})}=0$. Thus, when $u>0$ and $g>0$, $\mathcal{F}$
is minimized if $\theta_{2}-\theta_{1}=n\pi/2$ where $n$ is an arbitrary
integer. In such a case, $\eta_{1}+i\eta_{2}$ becomes the ground state order
parameter of this system.

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