Introduction

Topological superconductors (TSCs) have received significant attention as a promising platform to achieve stable qubits, using boundary Majorana zero modes with non-Abelian statistics1,2,3. Spin-triplet superconductors are a representative example of TSC candidates4,5. Since ferromagnetic spin fluctuations are regarded as an origin of spin-triplet superconductivity6, uranium (U)-based compounds in which the coexistence of ferromagnetism and superconductivity was observed7 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 UTe28, 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 shift8,9 and the large upper critical field8,10 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 nature11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28.

Not only does the superconducting state of UTe2 but also its normal state exhibits intriguing characteristics such as heavy fermionic behavior with highly enhanced effective mass8 and Kondo resonance25 arising from strong electron correlation. Moreover, the normal state of UTe2 is paramagnetic but is under strong magnetic fluctuations without long-range order8, contrary to other U-based superconductors with robust ferromagnetism7. 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 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 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 symmetry26. 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 UTe2.

Results

DFT+DMFT calculation at T = 11 K

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, zero-temperature 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 (EF)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 K32,33. Also, in the DFT+DMFT and GW+DMFT calculations35,36,37 performed at intermediate temperatures higher than 10 K, the Kondo effect would not take place effectively.

Fig. 1: Crystal structure and open-core density functional theory calculation results.
figure 1

a Schematic crystal structure71 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 density functional theory (DFT) calculation with the conventional unit cell without U 5f electrons near the Fermi energy. 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 Fermi surface (FS) from the open-core DFT calculations plotted using the conventional unit cell. There are two 1D FSs along the kx-axis (U 6d states) and the ky-axis (Te1 5p states). d First Brillouin zones (BZs) of the primitive unit cell and the conventional orthorhombic cell. Blue large letters indicate the time-reversal invariant momentum (TRIM) in the BZ with the conventional cell. Zp, Rp, and Mp are TRIMs defined in the primitive unit cell. b1, b2, and b3 are reciprocal vectors in this primitive unit cell. The arrow from Zp to Γ indicates that the Zp point is folded into the Γ point in the BZ of the conventional unit cell.

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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 EF. 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 kx-axis), Te1 5pz states appear to be dominant around EF along the Γ-Y path. In contrast, the U1-Te2 zigzag chains lie in the a-axis direction (along the ky-axis); thus, U 6dz states are dominant around EF along the Γ-X path. Such a quasi-2D FS corresponds to the high-temperature phase in which most 5f 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 5f electrons near EF 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 mj 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 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 EF (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 f3 states are redistributed around 2 eV above EF (inside the blue solid box in Fig. 2a). Among the f2 multiplets denoted by the symbol 2s+1LJ, where s, L, J indicate the spin, orbital and total angular momentum, respectively, 3H4 states play an important role near the EF (the green solid box), while 3F2 states are located −0.6 eV below EF (the green dashed box) as shown in Fig. 2a. We note that the incoherent spectrum of 3F2 states at − 0.6 eV can be identified with the −0.6 eV signal observed in angle-resolved photoemission spectroscopy (ARPES) measurements33,38 and another DFT+DMFT calculation33. Fig. 2b summarizes the atomic multiplet distribution in the energy spectrum near EF.

Fig. 2: Density functional theory plus dynamical mean field theory calculation results.
figure 2

a Density functional theory plus dynamical mean field theory (DFT+DMFT) spectral functions at T = 11 K plotted using the Brillouin zone (BZ) of the conventional unit cell. b Emergence of atomic states near the Fermi level (EF) 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 (f3) and f2 multiplet states. The atomic multiplet calculation in the DMFT part shows that the 3F2 and 3H4 of the f2 states play important roles, in good agreement with another DFT+DMFT calculation33. c Coherent DFT+DMFT quasiparticle spectrum around EF plotted using the BZ of the primitive unit cell. The Γ6 state (\(\left\vert j=5/2,{m}_{j}=\pm 1/2\right\rangle\)) is dominant around EF. The red elliptical loop indicates the nearly flat bands near EF around the Zp point, which may induce a large spectral weight before the formation of the low-temperature (T) quasiparticle state. d Large quasiparticle Fermi surface (FS) in the BZ of the primitive unit cell. The Zp point is indicated by the red elliptical line. e Coherent DFT+DMFT quasiparticle spectrum around EF plotted using the conventional unit cell. The green- and blue-colored quasiparticle states correspond to the electron FS around the M point and the hole FS around the Γ point, respectively. f, Large quasiparticle FS plotted using the conventional unit cell. The blue hole pocket near the Γ point is the correlated three-dimensional FS and the green electron pocket around the M point is a quasi-two-dimensional 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. In the high-T FS, the red and blue lines represent Fermi pockets originating from the U-Te2 and Te1 chains, respectively. In the low-T FS, the green line represents the two-dimensional electron Fermi pocket, and the blue-colored rectangle means the three-dimensional hole Fermi pocket.

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Emergence of correlated three-dimensional FS at lower temperatures

The transition temperature of UTe2 is about 1.6 K at ambient pressure8. 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 zero39,40,41. 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 Γ6 (\(=\left\vert j=5/2,{m}_{j}=\pm 1/2\right\rangle\)) orbital originating from the 3H4 state near EF makes a dominant contribution in the vicinity of EF from the mj-dependent energy spectrum shown in Fig. 2c. The enhancement of the Γ6 state spectral weight at the Fermi level as decreasing T is consistent with the results of the GW+DMFT calculation35. 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 mj-independent self-energies32, and even in a mj-resolved GW+DMFT calculation at the temperature above 25K35. This indicates that the appearance of the correlated Fermi surface around the Γ point is a feature of UTe2 electronic structure that can be captured only by mj-resolved analysis at very low temperatures. There were some studies about a non-interacting three-dimensional (3D) FS38 with the DFT method and a pipe-like FS with the DFT+U calculation30, while here we explore the correlation-driven isotropic 3D FS.

Unlike the DFT+U calculation that achieves the metallic ground state by pushing 5f electrons away from EF, our DFT+DMFT calculation explains how the Kondo effect induces the metallic behavior of UTe2 in which the U Γ6 state hybridized with Te pz 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 study33. 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 Zp 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 property42 and quantum oscillation43 further support the presence of the 3D FS. We note that, although recent quantum oscillation measurement44,45,46,47 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-polarized48, 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 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 self-consistent gap equations given by

$$\lambda {\Delta }_{{\xi} {\xi} ^{\prime} }^{\rho }({{{\bf{k}}}}) = -\frac{T}{N} \sum_{{k}^{\prime} ,{\xi }_{i = 1,2,3,4}}{V}_{{\xi} {\xi }_{1}{\xi }_{2} {\xi}^ {\prime} }({{{\bf{k}}}}-{{{\bf{k}}}}^{\prime} )\\ \times {G}_{{\xi }_{3}{\xi }_{1}}(-{k}^{\prime} ){\Delta }_{{\xi }_{3}{\xi }_{4}}^{\rho }({{{\bf{k}}}}^{\prime} ){G}_{{\xi }_{4}{\xi }_{2}}({k}^{\prime} )\big),$$
(1)

where ξ, ξ1, ξ2, ξ3, ξ4 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 ωn = (2n + 1)πT. Δρ is a gap function that belongs to an irreducible representation (IR) ρ of the D2h point group, and λ is the corresponding gap equation eigenvalue30. 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 ξi and ξj, 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 metals49, 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 Supplementary Note 6). G is the normal state Green’s function of U 5f Γ6 electrons. To obtain G, we construct a tight-binding model that reproduces the correlated normal state fermiology from LDA+DMFT by using U Γ6, U \({d}_{{z}^{2}}\) and Te pz orbitals (Fig. 3a).

Fig. 3: Superconductivity from the linearized self-consistent gap equation approach.
figure 3

a Fermi surface (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-temperature (T) FS. The region in the black dashed circle shows a relatively low U 5f electron contribution on the three-dimensional (3D) FS. b Eigenvalues of the linearized self-consistent gap 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 direction on the kxky-plane, which shows strong ferromagnetic fluctuations.

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By solving the linearized self-consistent gap equations, we find that the gap functions belonging to ρ = Au, B1u, B2u, and B3u 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 5f electrons give spin-triplet superconductivity. Moreover, the eigenvalues of the four IRs have comparable magnitudes, which is different from previous numerical studies29,30, 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 Au and B1u IRs, and the B2u and B3u IRs appear nearly degenerate, while the B2u and B3u IRs are slightly more favored than the Au and B1u IRs, consistent with a recent renormalization group calculation50. (We note that D. Shaffer and D. V. Chichinadze50 adopted the model with two-dimensional Fermi surfaces made up of U 6d and Te 5p orbitals. Due to the emergent C4 rotation symmetry in the model Hamiltonian, two spin-triplet channels at the renormalization group fixed point appear to be degenerate50. Therefore, the triplet channels belong to the two-dimensional representation Eu of the emergent point group D4h, which descends to the B2u and B3u IRs of the original point group D2h. On the other hand, our calculation based on the Γ point 3D Fermi pocket shows that not only the B2u and B3u pairings but also the Au and B1u pairings exhibit comparable transition temperatures, although it still predicts the B2u and B3u IRs to be more stable than the Au and B1u IRs. As discussed above, the relatively high transition temperatures of the Au and B1u 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 B2u 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 ky-axis (see Fig. 3a), the B2u representation, which has symmetry-protected nodes on the ky-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 measurements26 showing two nearby transition peaks separated by only 80 mK. Since the B2u and B3u IRs have larger eigenvalues than the Au and B1u IRs, we believe the two peaks correspond to the B2u and B3u representations. Moreover, this accidental degeneracy of IRs can lead to a mixed order parameter B2u + iB3u, which gives a TRS-breaking Weyl superconductivity (see the Supplementary Note 8).

Table 1 Transformation properties, basis functions, and gap structures of IRs under the D2h point group symmetry
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Topological superconductivity

The emergence of correlated 3D FS also significantly affects the band topology of the superconducting UTe2. The Bogoliubov-de Gennes 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 Au pairing has a fully gapped spectrum, while the B1u, B2u, and B3u pairings have nodal points on the kz-, ky-, and kx-axes, respectively, as summarized in Table 1.

Let us first consider the fully gapped pairing with the Au 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 points3,4,5. Since a cylindrical quasi-2D FS always simultaneously encloses a pair of TRIM points, one on the kz = 0 plane and the other on the kz = 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, UTe2 with the Au pairing becomes a first-order TSC that hosts gapless Majorana surface states.

Fig. 4: Topological superconductivity in UTe2.
figure 4

a Fermi surfaces (FSs) and time-reversal invariant momentum (TRIM) points in the Brillouin zone of UTe2. b The Chern numbers carried by the occupied states of the Bogoliubov-de Gennes Hamiltonian on the kz = 0 plane (orange) and the kz = π plane (blue). The Chern numbers on the two planes differ by 2, which indicates that the two Weyl nodes (magenta) located between the kz = 0 and kz = π planes have the same charge ( + 1 each), while the other two nodes (yellow) located between the kz = 0 and kz = − π planes both have charge − 1. c Trajectory of Weyl nodes. The blue (red) arrows represent the path of nodes with + ( − ) charge. As α increases from 0 to a critical value αc, the Weyl nodes generated by the αB2u + i(1 − α)B3u pairing move starting from the kx-axis at α = 0 to the kz-axis at α = αc. After the nodes with the same charge accidentally meet on the kz-axis at α = αc, they split and move toward the ky-axis until α > αc reaches 1. d, Zero-energy Majorana arc states (top), and their position expectation values that show their surface localization (bottom).

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In the case of the B1u, B2u, and B3u pairings, they are guaranteed to have nodal points in the presence of the 3D FS that always intersects with the kz-, ky-, and kx-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 B2u pairing is the dominant channel, which is also consistent with the results of the recent ultrasound measurement51.

In contrast, when the nearly degenerate B2u and B3u pairing channels form a complex order parameter in the form of αB2u + i(1 − α)B3u (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 B3u pairing state at the intersection between the 3D FS and the kx-axis split into four 2-fold Weyl points located at the intersection between the 3D FS and the kxkz plane. At α = αc, pairs of Weyl points with the same chiral charge merge on the kz-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 B3u-dominant regime (0 < α < αc), 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 kz > 0 (kz < 0) region have the same monopole charge is further confirmed by computing the Chern number on the kz = 0 and kz = π planes (orange and blue planes in Fig. 4b), which are equal to 2 and 4, respectively. When α becomes larger than αc, 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 kykz plane. Finally, at α = 1, pairs of Weyl points with opposite chiral charges merge and form 4-fold nodal points of the B2u pairing state on the ky-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.

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 UTe2 and the related unconventional spin-triplet superconductivity. Our correlated electronic structure not only explains the spectral features measured in recent ARPES experiments33 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 state29,30. 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 UTe2 is a venue to study the intriguing physics of strongly interacting Weyl fermions52,53.

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 magnetization8,11, anisotropic NMR and dynamical spin susceptibility measurement12, and others54,55,56. We note that although our correlated normal state fermiology shows that ferromagnetic fluctuation plays a dominant role in the superconducting phase transition of UTe2, there are various recent experimental data that support the importance of incommensurate antiferromagnetic fluctuations including the recent neutron scattering measurements36,57,58. 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 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.

Method

Electronic structure calculation

The charge self-consistent version of DFT+DMFT59,60 is based on the full-potential linearized augmented plane-wave (FP-LAPW) band method61. The correlated 5f electrons are treated dynamically by the DMFT local self-energy (Σ(ω)), while all other delocalized spd electrons are treated on the DFT level. The DFT+DMFT calculations were performed based on the 5f2 multiplet, whose validity as the ground state has been experimentally supported by the angle-resolved photoemission, X-ray absorption, and resonant inelastic X-ray spectra33,62,63. 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 (F0 = 8.0 eV, F2 = 7.15317919075 eV, F4 = 4.77832369942 eV, and F6 = 3.53367052023 eV) with U = 8.0 eV and J = 0.6 eV64. 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 domains67, 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 solver60 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 package68. The Fermi surfaces obtained from the DFT and DFT+DMFT calculations are visualized by the XCrySDen package69.

Linearized self-consistent gap equation

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

$$\hat{V}({{{{{\bf{q}}}}}})=-{\hat{\Gamma }}^{0}\hat{\chi }({{{{{\bf{q}}}}}}){\hat{\Gamma }}^{0}-{\hat{\Gamma }}^{0},$$
(2)

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 EF, which is different from U of atomic 5f orbitals in the impurity solver in the DMFT loop. The spin susceptibility \(\hat{\chi }({{{{{\bf{q}}}}}})\) is calculated from the bare spin susceptibility (χ0) as follows,

$$\hat{\chi }({{{{{\bf{q}}}}}})={\left[\hat{1}-{\hat{\chi }}^{0}({{{{{\bf{q}}}}}}){\hat{\Gamma }}^{0}\right]}^{-1}{\hat{\chi }}^{0}({{{{{\bf{q}}}}}}),$$
(3)

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

$$\lambda {\Delta }_{\xi \xi ^{\prime} }^{\rho }({{{{{\bf{k}}}}}})=-\frac{T}{N} \sum_{k^{\prime} ,{\xi }_{j}}{V}_{\xi {\xi }_{1}{\xi }_{2}\xi ^{\prime} }({{{{{\bf{k}}}}}}-{{{{{\bf{k}}}}}}^{\prime} ){G}_{{\xi }_{3}{\xi }_{1}}(-k^{\prime} ){\Delta }_{{\xi }_{3}{\xi }_{4}}^{\rho }({{{{{\bf{k}}}}}}^{\prime} ){G}_{{\xi }_{4}{\xi }_{2}}(k^{\prime} ),$$
(4)

we introduce ϕ defined as

$${[\phi ]}_{{\mu }_{3}{s}_{3}{\mu }_{4}{s}_{4}}^{{\mu }_{1}{s}_{1},{\mu }_{2}{s}_{2}}({{{{{\bf{k}}}}}})= \sum_{{n}_{1}{n}_{2}}{\left[{M}_{{n}_{1}{n}_{2}}\right]}_{{\mu }_{3}{s}_{3}{\mu }_{4}{s}_{4}}^{{\mu }_{1}{s}_{1},{\mu }_{2}{s}_{2}}\frac{f({\bar{\xi }}_{-{{{{{\bf{k}}}}}},{n}_{1},{\sigma }_{1}})-f({\xi }_{{{{{{\bf{k}}}}}},{n}_{2},{\sigma }_{2}})}{{\bar{\xi }}_{-{{{{{\bf{k}}}}}},{n}_{1},{\sigma }_{1}}-{\xi }_{{{{{{\bf{k}}}}}},{n}_{2},{\sigma }_{2}}}$$
(5)
$$={\sum}_{n1}\frac{{\left[{M}_{{n}_{1}{n}_{2}}\right]}_{{\mu }_{3}{s}_{3}{\mu }_{4}{s}_{4}}^{{\mu }_{1}{s}_{1},{\mu }_{2}{s}_{2}}}{2{\xi }_{n1}}\tanh \left(\frac{\xi }{2T}\right),$$
(6)

where

$$\left[{M}_{{n}_{1},{n}_{2}}\right]={\left[{u}_{{n}_{1}{\sigma }_{1}}^{{\mu }_{1}{s}_{1}}(-{{{{{\bf{k}}}}}})\right]}^{* }{\left[{u}_{{n}_{2}{\sigma }_{3}}^{{\mu }_{3}{s}_{3}}({{{{{\bf{k}}}}}})\right]}^{* }\left[{u}_{{n}_{2}{\sigma }_{2}}^{{\mu }_{2}{s}_{2}}({{{{{\bf{k}}}}}})\right]\left[{u}_{{n}_{1}{\sigma }_{4}}^{{\mu }_{4}{s}_{4}}(-{{{{{\bf{k}}}}}})\right].$$
(7)

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.