## Abstract

UTe_{2} is a promising candidate for spin-triplet superconductors, in which a paramagnetic normal state becomes superconducting due to spin fluctuations. 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 nearly degenerate solutions which may lead to time-reversal breaking multicomponent superconductivity. 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.

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## 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^{1,2,3}. Spin-triplet superconductors are a representative example of TSC candidates^{4,5}. Since ferromagnetic spin fluctuations are regarded as an origin of spin-triplet superconductivity^{6}, uranium (U)-based compounds in which the coexistence of ferromagnetism and superconductivity was observed^{7} 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 UTe_{2}^{8}, 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, UTe_{2} is expected to be a spin-triplet superconductor, evidenced by the temperature independence of the nuclear magnetic resonance (NMR) Knight shift^{8,9} and the large upper critical field^{8,10} 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^{11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}.

Not only does the superconducting state of UTe_{2} but also its normal state exhibits intriguing characteristics such as heavy fermionic behavior with highly enhanced effective mass^{8} and Kondo resonance^{25} arising from strong electron correlation. Moreover, the normal state of UTe_{2} is paramagnetic but is under strong magnetic fluctuations without long-range order^{8}, contrary to other U-based superconductors with robust ferromagnetism^{7}. This indicates that the fermiology of the correlated paramagnetic normal state under strong spin fluctuations can be the quintessential factor governing the unconventional superconductivity of UTe_{2}, which requires solid theoretical verification.

Here we reveal the fermiology of the correlated normal state of UTe_{2} 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 5*f* 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 Γ 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 self-consistent gap 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^{26}. 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 correlated normal-state fermiology we observed provides a unified platform to describe the unconventional superconductivity in UTe_{2}.

## Results

### DFT+DMFT calculation at *T* = 11 K

To date, the band structure of UTe_{2}, with the crystal structure shown in Fig. 1a, has been reported using various DFT-based calculations. For instance, zero-temperature conventional DFT calculations of UTe_{2} 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 5*f* states and conduction electrons (U 6*d* and Te 5*p* electrons) is too strong such that a large gap is opened near the Fermi energy (*E*_{F})^{8,10}. The introduction of magnetism or an on-site Coulomb interaction (*U*)^{29,30,31,32,33,34,35,36} 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. Therefore, the renormalized FS from the Kondo resonance at low temperatures would be different from a quasi-two-dimensional FS obtained in previous DFT + *U* and similar results from some DFT + DMFT calculations at 10 K^{32,33}. Also, in the DFT+DMFT and GW+DMFT calculations^{35,36,37} performed at intermediate temperatures higher than 10 K, the Kondo effect would not take place effectively.

The electronic structure with localized U 5*f* electrons can be described by using open-core DFT calculations in which U 5*f* electrons are pushed into the core states far from *E*_{F}. The resulting band structure in Fig. 1b supports a quasi-two-dimensional (quasi-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 (along the *k*_{x}-axis), Te1 5*p*_{z} states appear to be dominant around *E*_{F} along the Γ-*Y* path. In contrast, the U1-Te2 zigzag chains lie in the *a*-axis direction (along the *k*_{y}-axis); thus, U 6*d*_{z} states are dominant around *E*_{F} along the Γ-*X* path. Such a quasi-2D FS corresponds to the high-temperature phase in which most 5*f* states are localized. The relation between the primitive and conventional Brillouin zones (BZs) is summarized in Fig. 1d.

With decreasing *T*, the emergence of U 5*f* 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 5*f* states) and its *z*-component *m*_{j} to differentiate the orbital characters. Throughout our calculations, the spin-orbit coupling is included for all electrons. The self-energy calculated using the primitive unit cell with two U atoms is obtained at *T* = 11 K. The resulting electronic structure of UTe_{2} at *T* = 11 K is shown in Fig. 2. In comparison to Fig. 1b, the spectral function plot shows that U 5*f* 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}H_{4} states play an important role near the *E*_{F} (the green solid box), while ^{3}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}F_{2} states at − 0.6 eV can be identified with the −0.6 eV signal observed in angle-resolved photoemission spectroscopy (ARPES) measurements^{33,38} and another DFT+DMFT calculation^{33}. Fig. 2b summarizes the atomic multiplet distribution in the energy spectrum near *E*_{F}.

### Emergence of correlated three-dimensional FS at lower temperatures

The transition temperature of UTe_{2} is about 1.6 *K* at ambient pressure^{8}. To estimate the low-*T* (*T* < 11 K) quasiparticle FS from the DFT+DMFT result at *T* = 11 K, we set the imaginary part of the self-energy to zero^{39,40,41}. The corresponding quasiparticle FSs for the primitive and conventional unit cells are plotted in Fig. 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 Γ_{6} (\(=\left\vert j=5/2,{m}_{j}=\pm 1/2\right\rangle\)) orbital originating from the ^{3}H_{4} state near *E*_{F} makes a dominant contribution in the vicinity of *E*_{F} from the *m*_{j}-dependent energy spectrum shown in Fig. 2c. The enhancement of the Γ_{6} state spectral weight at the Fermi level as decreasing *T* is consistent with the results of the GW+DMFT calculation^{35}. The corresponding spectral functions in the conventional unit cell in Fig. 2e show that a hole Fermi pocket encloses the Γ 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. Note that the hole Fermi pocket centered at the Γ point is a discovery of our study, which was absent in previous DFT+DMFT studies that used *m*_{j}-independent self-energies^{32}, and even in a *m*_{j}-resolved GW+DMFT calculation at the temperature above 25K^{35}. This indicates that the appearance of the correlated Fermi surface around the Γ point is a feature of UTe_{2} electronic structure that can be captured only by *m*_{j}-resolved analysis at very low temperatures. There were some studies about a non-interacting three-dimensional (3D) FS^{38} with the DFT method and a pipe-like FS with the DFT+*U* calculation^{30}, while here we explore the correlation-driven isotropic 3D FS.

Unlike the DFT+*U* calculation that achieves the metallic ground state by pushing 5*f* electrons away from *E*_{F}, our DFT+DMFT calculation explains how the Kondo effect induces the metallic behavior of UTe_{2} in which the U Γ_{6} state hybridized with Te *p*_{z} and U \({d}_{{z}^{2}}\) states form 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 Γ point [see Fig. 2g, h]. As *T* decreases, the cylindrical FS at the BZ center develops a more 3D character, eventually forming an ellipsoidal closed surface enclosing the Γ 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^{33}. 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*_{p} point shown in Fig. 2c, which can be considered a precursor of the formation of a 3D correlated FS at very low-*T* (See the calculation of spectral function in the Supplementary Note 3C). Additionally, the recently measured nearly isotropic transport property^{42} and quantum oscillation^{43} further support the presence of the 3D FS. We note that, although recent quantum oscillation measurement^{44,45,46,47} could not identify the 3D Fermi surface, considering that the measurement was done under strong magnetic fields, it is highly probable that the field-induced normal state would be spin-polarized^{48}, thus probing the Fermi surface of the paramagnetic normal state is practically quite challenging.

### Superconducting instability

As the correlated 3D FS consists mainly of U 5*f* 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 UTe_{2}. To explore the superconducting instability of UTe_{2} by considering the spin fluctuation effect on U 5*f* electrons, we solve the linearized self-consistent gap equations given by

where *ξ*, *ξ*_{1}, *ξ*_{2}, *ξ*_{3}, *ξ*_{4} and \(\xi ^{\prime}\) are indices that denote the uranium atomic position and the electron spin. **k** and \({{{{{\bf{k}}}}}}^{\prime}\) indicate the momentum, where \(k =({{{{{\bf{k}}}}}},i{\omega }_{n})\) is an abbreviated notation for the fermionic Matsubara frequency *ω*_{n} = (2*n* + 1)*π**T*. Δ^{ρ} is a gap function that belongs to an irreducible representation (IR) *ρ* of the *D*_{2h} point group, and *λ* is the corresponding gap equation eigenvalue^{30}. For the pairing functions, we use the basis functions listed in Table 1. *V* and \({G}_{{\xi }_{i}{\xi }_{j}}\) are the effective pairing interaction and the Green’s function between *ξ*_{i} and *ξ*_{j}, respectively. *T* and *N* are the temperature and the total number of the momentum mesh. As ferromagnetic spin fluctuation is expected to be the origin of the spin-triplet superconductivity in U-based heavy fermion metals^{49}, we take into account the spin fluctuation effect from the on-site Coulomb interaction *U* of 5*f* electrons within the random phase approximation (RPA) (see Fig. 3d and the Supplementary Note 6). *G* is the normal state Green’s function of U 5*f* Γ_{6} electrons. To obtain *G*, we construct a tight-binding model that reproduces the correlated normal state fermiology from LDA+DMFT by using U Γ_{6}, U \({d}_{{z}^{2}}\) and Te *p*_{z} orbitals (Fig. 3a).

By solving the linearized self-consistent gap equations, we find that the gap functions belonging to *ρ* = *A*_{u}, *B*_{1u}, *B*_{2u}, and *B*_{3u} IRs have nonzero eigenvalues (see Fig. 3b). We note that 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 5*f* electrons give spin-triplet superconductivity. Moreover, the eigenvalues of the four IRs have comparable magnitudes, which is different from previous numerical studies^{29,30}, 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 pipe-like FS considered in other works.

More specifically, our self-consistent gap 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^{50}. (We note that D. Shaffer and D. V. Chichinadze^{50} adopted the model with two-dimensional Fermi surfaces made up of U 6*d* and Te 5*p* orbitals. Due to the emergent *C*_{4} rotation symmetry in the model Hamiltonian, two spin-triplet channels at the renormalization group fixed point appear to be degenerate^{50}. Therefore, the triplet channels belong to the two-dimensional representation *E*_{u} of the emergent point group *D*_{4h}, which descends to the *B*_{2u} and *B*_{3u} IRs of the original point group *D*_{2h}. On the other hand, our calculation based on the Γ point 3D Fermi pocket shows that not only the *B*_{2u} and *B*_{3u} pairings but also the *A*_{u} and *B*_{1u} pairings exhibit comparable transition temperatures, although it still predicts the *B*_{2u} and *B*_{3u} IRs to be more stable than the *A*_{u} and *B*_{1u} IRs. As discussed above, the relatively high transition temperatures of the *A*_{u} and *B*_{1u} IRs in our study are supposed to come from the 3D Fermi pocket that encloses the Γ point in all directions.) 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 5*f* electron wave functions on the 3D FS. That is, as the 3D FS has a relatively small U 5*f* 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). The appearance of almost degenerate pairing states is consistent with recent specific heat measurements^{26} 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} + *i**B*_{3u}, which gives a TRS-breaking Weyl superconductivity (see the Supplementary Note 8).

### Topological superconductivity

The emergence of correlated 3D FS also significantly affects the band topology of the superconducting UTe_{2}. The Bogoliubov-de Gennes Hamiltonian of superconducting UTe_{2} 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 UTe_{2} 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^{3,4,5}. 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} = *π* 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 Γ point (the red dot in Fig. 4a) renders the strong bulk \({{\mathbb{Z}}}_{2}\) index nontrivial. Therefore, UTe_{2} 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. It is noteworthy that the *B*_{2u} pairing is the dominant channel, which is also consistent with the results of the recent ultrasound measurement^{51}.

In contrast, when the nearly degenerate *B*_{2u} and *B*_{3u} pairing channels form a complex order parameter in the form of *α**B*_{2u} + *i*(1 − *α*)*B*_{3u} (0≤*α*≤1), a 4-fold nodal point splits into two 2-fold Weyl points, thus generating a Weyl superconductor. More explicitly, as *α* 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 *α* = *α*_{c}, pairs of Weyl points with the same chiral charge merge on the *k*_{z}-axis. The behavior of Weyl node evolution is similar to the prediction of T. Shishidou et al.^{31}, but the predicted Weyl node positions are different from ours, especially in the *B*_{3u}-dominant regime (0 < *α* < *α*_{c}), due to the different FS geometry. Also, we expect that the electron correlation effect would be more important in our Weyl superconductor dominated by U 5f electrons.

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} = *π* planes (orange and blue planes in Fig. 4b), which are equal to 2 and 4, respectively. When *α* becomes larger than *α*_{c}, the merged Weyl points with chiral charge ± 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 *α* = 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, UTe_{2} 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 UTe_{2}, regardless of whether TRS is broken.

## Discussion

In summary, we have performed the DFT+DMFT calculations and solved the self-consistent gap equations with the tight-binding Hamiltonian to study the strongly correlated normal state of UTe_{2} and the related unconventional spin-triplet superconductivity. Our correlated electronic structure not only explains the spectral features measured in recent ARPES experiments^{33} but also predicts the emergence of correlation-driven 3D FS around the Γ point at low temperatures, which can account for various types of anomalous behaviors in both the normal state and the superconducting state^{29,30}. In particular, the nearly degenerate spin-triplet solution of the self-consistent gap 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 UTe_{2} is a venue to study the intriguing physics of strongly interacting Weyl fermions^{52,53}.

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^{8,11}, anisotropic NMR and dynamical spin susceptibility measurement^{12}, and others^{54,55,56}. We note that although our correlated normal state fermiology shows that ferromagnetic fluctuation plays a dominant role in the superconducting phase transition of UTe_{2}, there are various recent experimental data that support the importance of incommensurate antiferromagnetic fluctuations including the recent neutron scattering measurements^{36,57,58}. Resolving the controversy related to the nature of spin fluctuations is definitely one important issue that should be clarified in future research.

The correlated normal-state fermiology we obtained shows that ferromagnetic fluctuation plays a dominant role in the superconducting phase transition of UTe_{2}. We believe that our theory can provide a unified framework to understand the complex behavior of UTe_{2} and resolve the remaining controversies in this field.

## Method

### Electronic structure calculation

The charge self-consistent version of DFT+DMFT^{59,60} is based on the full-potential linearized augmented plane-wave (FP-LAPW) band method^{61}. The correlated 5*f* electrons are treated dynamically by the DMFT local self-energy (Σ(*ω*)), while all other delocalized *s**p**d* electrons are treated on the DFT level. The DFT+DMFT calculations were performed based on the 5*f*^{2} multiplet, whose validity as the ground state has been experimentally supported by the angle-resolved photoemission, X-ray absorption, and resonant inelastic X-ray spectra^{33,62,63}. The charge and spin fluctuations considered in DMFT enable the description of the Kondo effect correctly. Σ(*ω*) is calculated from the corresponding impurity problem, in which the 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^{64}. 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 (CTQMC)^{65,66}. To avoid numerical noises in the CTQMC steps at low temperatures, the self-energy is directly sampled using the intermediate representation between imaginary-time and real-frequency domains^{67}, and the diagonal elements of hybridization functions are taken into account. The calculated self-energy is analytically continued to the real frequency axis through the maximum entropy method. The one-crossing approximation impurity solver^{60} is used to check the validation of the high-temperature CTQMC calculations. The number of valence electrons from the Fermi surface is calculated using the SKEAF package^{68}. The Fermi surfaces obtained from the DFT and DFT+DMFT calculations are visualized by the XCrySDen package^{69}.

### Linearized self-consistent gap equation

The effective pairing potential used to solve the linearized self-consistent gap equation reads

where \({\hat{\Gamma }}^{0}\) is the bare irreducible vertex that describes the on-site Coulomb interaction (\(U^{\prime}\))^{70}. We note that \(U^{\prime}\) is defined in the renormalized quasiparticle states near the *E*_{F}, which is different from *U* of atomic 5*f* orbitals in the impurity solver in the DMFT loop. The spin susceptibility \(\hat{\chi }({{{{{\bf{q}}}}}})\) is calculated from the bare spin susceptibility (*χ*^{0}) as follows,

within the RPA. To solve the linearized self-consistent gap equations

we introduce *ϕ* defined as

where

Here, \({u}_{n\sigma }^{\mu s}(k)\) is an eigenstate of the given tight-binding model. *μ*, *s*, *n*, *σ* are orbital, spin, band index, and pseudospin degrees of freedom, respectively.

## Data availability

The data that support the findings of this study are available from the authors upon reasonable request.

## Code availability

The charge self-consistent version of the DFT+DMFT package used in this study can be downloaded from http://hauleweb.rutgers.edu/tutorials/. The other codes generated for the current study are available from the authors at reasonable request.

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## Acknowledgements

We thank Seokjin Bae and Yun Suk Eo for the fruitful discussions. H.C.C, 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 Korean government (MSIT) (No. NRF-2021R1A5A1032996).

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B.J.Y. initially conceived the project. H.C.C. and S.H.L. contributed to the theoretical analysis and wrote the manuscript with B.J.Y. H.C.C. did all of the ab initio calculations. B.J.Y. supervised the project. All authors discussed and commented on the manuscript.

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Choi, H.C., Lee, S.H. & Yang, BJ. Correlated normal state fermiology and topological superconductivity in UTe_{2}.
*Commun Phys* **7**, 273 (2024). https://doi.org/10.1038/s42005-024-01708-4

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DOI: https://doi.org/10.1038/s42005-024-01708-4

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