Orbital selective Kondo effect in heavy fermion superconductor UTe2

Heavy fermion systems emerge from the collective Kondo effect, and their superconductivity can serve as a promising platform for realizing next-generation quantum technologies. However, it has been a great challenge to explore many-body effects in heavy fermion systems with ab-initio approaches. We computed the electronic structure of UTe2 without purposive judgements, such as intentional selection of on-site Coulomb interaction and disregarding spin-orbit coupling. We show that U-5f electrons are highly localized in the paramagnetic normal state, giving rise to the Kondo effect. It is also found that the hybridization between U-5f and U-6d predominantly in the orthorhombic ab-plane is responsible for the high-temperature Kondo effect. In contrast, the hybridization between U-5f and Te-5p along the c-axis manifests the Kondo scattering at a much lower temperature, which could be responsible for the low-temperature upturn of the c-axis resistivity. Our results show that the electron correlation in UTe2 is orbital selective, which naturally elucidates the recent experimental observations of anomalous temperature dependence of resistivity. Furthermore, we suggest that the Kondo effect is suppressed at high pressure owing to weak localization of magnetic moments, which results from enhanced U-5f electron hopping. Our discovery provides significant insight toward understanding anisotropic quantum behavior including selective re-entrant superconductivity in heavy fermion UTe2.


INTRODUCTION
Heavy fermion superconductors, arising from the Kondo lattice, opened up the possibility of magnetically mediated unconventional superconductivity. Such kind is now believed to be the foundation of realizing topological superconductivity. Prominent evidence for unconventional superconductivity in the heavyfermion system includes upper critical field limited by Pauli paramagnetism 1,2 , a spatially varying order parameter 3 , and reentrant superconductivity that depends on the orientation of the applied magnetic field 4 . These exotic superconducting properties originate from the strong electronic correlation including the Kondo effect in the vicinity of a magnetic quantum critical point (QCP). QCP manifests non-Fermi liquid behavior in the heavyfermion compounds over a wide temperature range. Both superconducting and normal state properties are closely tied to the symmetry of crystal structure which is the backbone of the directional electronic correlation. The strong electronic correlation often leads to a exotic superconducting state such as a spintriplet pairing.
The uranium-based superconductors are promising candidates for the realization of the spin-triplet superconductivity 4 . The prominent examples include URhGe 5 and UCoGe 6 with the critical temperature T c = 0.25 K and 0.8 K, respectively. Both compounds undergo the superconducting phase transition from the ferromagnetically ordered state, making the equal spin pairing plausible 4 . Recently, superconductivity was discovered in Curie paramagnetic UTe 2 with T c ≈ 2 K 7-9 that is one of the highest among the known uranium-based superconductors. T c can be further enhanced nearly twofold at~1 GPa 10,11 . However, the mechanism behind this celebrated high T c has not been resolved. The main obstacle is insufficient knowledge of the electronic structure of the normal state in UTe 2 , which is a prerequisite for the understanding of superconductivity.
UTe 2 represents the most recent case of exotic heavy-fermion superconductivity. It exhibits the chiral in-gap state 12 and superconducting Weyl nodes 13 , and both properties are consistent with topological superconductivity that harbors Majorana zero energy states. Furthermore, UTe 2 exhibits unusual re-entrant behavior that depends on the orientation of the applied magnetic field 10,14 . In the normal state, UTe 2 exhibits the iconic incoherent-tocoherent crossover around T = 50 K in the resistivity with an electrical current along the orthorhombic a-direction 7,15 , which is reminiscent of prototypical Kondo lattice YbRh 2 Si 2 16 . Both UTe 2 and YbRh 2 Si 2 exhibit a negative slope in electrical resistivity at room temperature, implying that inelastic scattering dominates over the electron-phonon scattering. The magnetic contribution of resistivity is suggestive of À ln T dependence observed in most of the Kondo lattice systems above the coherent temperature T * , for instance, in-plane resistivity in CeCoIn 5 between T = 40 K and 180 K 17 . While the Kondo scattering is not the only possibility for the resistivity upturn, recent spectroscopy experiments successfully observed the formation of a hybridized band well above the coherent temperature in CeCoIn 5 18,19 . Most notably a combined study of ARPES and DMFT identified the occurrence of Kondo resonance up to 200 K in CeCoIn 5 20 . In contrast, no hybridization gap has been observed in UTe 2 at any temperature, leaving the scattering mechanism in the normal state elusive in this heavy fermion compound.
The c-axis transport property in UTe 2 is qualitatively different where it is metallic between 50 K and at least 300 K. The c-axis resistivity shows a rapid upturn below 50 K before exhibiting coherent-like behavior at the onset around 13 K 21 . Moreover, magnetic susceptibility below~100 K shows distinct temperature-dependence in all three symmetry directions 7,15 . The anisotropic transport and magnetic properties suggest that the orbitaldependent electron correlation needs to be taken into account. These outstanding issues put heavy fermion UTe 2 in a league of its own, and thus the understanding of the electronic structure of its normal state is the most pressing issue. While several ARPES studies were reported 22,23 , a complete picture of the band structure is still awaited. Therefore, theoretical determination of the accurate band structure is highly desired.
The normal state of UTe 2 can be best explained within a framework of Kondo lattice where the periodic local magnetic moments are screened by spins of conduction electrons 24 . The Ce-based Kondo lattice has been most widely studied by the dynamical mean field theory combined with density functional theory (DFT+DMFT) 20,[25][26][27][28][29] . In a recent study by Choi et al. 25 , the temperature evolution of the Kondo effect was illustrated by analyzing the spectral function A(k, ω) for CeCoGe 2 . The incoherent Ce-4f state is hybridized with conduction electrons (f-c hybridization), and the stronger contribution of Ce-4f state near the Fermi level distorts conduction electron bands, resulting in kinks in the bands at the Fermi level. At low temperatures, the incoherent Ce-4f state forms coherent bands with the renormalized carriers, initiating the coherent Kondo lattice states. The f-c hybridization, which is an indispensable element in Kondo physics, occurs selectively in the type of orbitals. In CeCoIn 5 , the selective f-c hybridization of two d bands and three crystalline electric-field split f levels gives arise to different dispersion of d bands in the vicinity of the Fermi level 20 . In PuCoGa 5 , the Fermiliquid behavior appears differently depending on Pu-5f states in j = 5/2 or j = 7/2 multiplet, which are induced by spin-orbit coupling (SOC), resulting in larger Kondo scale for f electrons in j = 5/2 than j = 7/2 multiplet 30 .
The role of SOC in the Kondo effect is a long standing issue 31 , for which UTe 2 , comprising heavy elements, is a necessary avenue. Obviously, SOC has a significant influence on the band structure in UTe 2 as well as its superconductivity. To date, there are two DFT+DMFT studies on UTe 2 without considering SOC 23,32 . Both show a flat peak hybridized with conduction electrons at 10 K and the peak was suppressed at 200 K, suggesting Kondo coherence at low temperature.
While 4f electrons in the lanthanide elements are usually localized, 5f electrons in the actinide can be either itinerant or localized depending on the crystal structure, crystal electric field and SOC, giving arise to a highly anisotropic electronic structure. For instance, 5f electrons in US, USe and UTe are respectively itinerant, intermediate and localized 33 . This characteristic of 5f electrons leads to a rich phase diagram with various ground states that can be chemically tuned. Canonical examples include the coexistence of Kondo behavior and a ferromagnetic order in UTe 34 , UCu 0.9 Sb 2 35 and UCo 0 . 5Sb 2 35,36 with the relatively high Curie temperatures of T C = 102 K (UTe), 113 K (UCu 0.9 Sb 2 ), and 64.5 K (UCo 0.5 Sb 2 ). The interplay between the Kondo effect and magnetically ordered phases is described by underscreened Kondo lattice model which describes the exchange coupling between conduction electron and partially screened impurity's magnetic moment 37 . While the Kondo effect is apparent in UTe 2 , unlike aforementioned uranium compounds, a long range magnetic ordering is not observed. It signals that the localized magnetic moments are sufficiently screened to prevent a long range magnetic ordering. Instead, it undergoes a superconducting phase transition at about T = 2 K. Other superconductors such as UCoGe 38 and URhGe 39 that exhibit Kondo-like behavior at high temperature orders ferromagnetically with T C = 3 K (UCoGe) 40 and 9.5 K (URhGe) 39 .
In this work, we ascertain the electronic structure of UTe 2 without adjusting parameters and its temperature evolution with/ without the inclusion of SOC. By inclusion of SOC, we found a multi-scaled Kondo effect. The SOC causes degenerated U-5f states of the non-SOC system to split into partially occupied states in j = 5/2 and unoccupied states in j = 7/2 multiplet. Within the j = 5/2 subspace, we found two groups of U-5f states, which selectively hybridize with conduction elections p or d, giving rise to the orbital-dependent Kondo effect.

Coulomb interaction tensor and electron occupation
First, we calculate on-site Coulomb interaction U C and exchange interaction J H within the constrained random phase approximation (cRPA) 41 . The calculated U C and J H for both U-5f and U-6d orbitals with inclusion of SOC are shown in Fig. 1 (see Supplementary Fig. 2 for non-SOC results). Both U C and J H increase and saturate to the bare unscreened value at high frequency.
Using the Coulomb interaction tensor, the electron occupancy of U-5f orbitals at 300 K are subsequently calculated including SOC, which is presented in Table 1. The total occupation in U-5f orbitals is 2.27. The SOC split U-5f into j = 5/2 and j = 7/2 multiplet, resulting in pushing U-5f states of non-SOC system, which are centered around 0.3 eV above the Fermi level (see Fig. 2d). This interaction gives rise to partially occupied j = 5/2 states and unoccupied j = 7/2 states. On the contrary, the U atom is strongly oxidized in the non-SOC simulation where the corresponding electron is transferred to Te-5p orbitals, and the resultant occupancy is 1.17 which is significantly smaller than that from the SOC simulation. The occupation numbers exhibit no sizable temperature variation from 25 to 2000 K. The calculated U-5f occupation with SOC is in good agreement with the measured 5f 2 configuration 42 .

Kondo scattering driven by SOC
To learn the Kondo effect due to the localized U-5f in UTe 2 , we calculated the spectral function at various temperatures along the high symmetry orientations depicted in Fig. 2a. The calculated spectral functions with and without SOC at 300 K are respectively shown in Fig. 2b and c. Within SOC simulation, the flat heavy electron and hole bands are formed in the vicinity of the Fermi level along every high symmetry line. (see also Fig. 3a). Within the non-SOC simulation, the coherent bands near the Fermi energy are comprised of U-5f, U-6d, and Te-5p orbitals, and the flat bands do not form down to 100 K.
Figures 2d and e show the projected density of states (DOS) for U-5f and U-6d orbitals, respectively. The results with and without SOC are shown in red and blue lines, respectively. In the SOC system, a sharp U-5f peak emerges about 0.02 eV above the Fermi level, whereas a sharp U-5f peak is centered around 0.3 eV above the Fermi level in the non-SOC system. In comparison to U-5f occupation of 1.17 in the non-SOC system, the SOC system manifests a higher U-5f occupation of 2.27 in the larger U-5f DOS below the Fermi level (see inset of Fig. 2d). In the non-SOC system, three coherent peaks appear between −1 eV and 0 eV, which are hybridized with U-6d as shown in Fig. 2e. In contrast, the U-5f and U-6d DOSs below the Fermi level are broad in the SOC system, and the shape of the U-5f spectral function is consistent with incoherent states. Figure 2f shows the temperature dependence of U-5f projected DOS at the Fermi level, which is determined by [Eq. 1] where ω 0 is the first Matsubara frequency and G is the calculated local Green's function. D(iω 0 ) for both SOC f 2 and non-SOC f −3 (l = 3, m = −3) gradually increase upon cooling down to 700 K because of formation of quasiparticle peak where the spectral weight at the Fermi level is transferred from the upper and lower Hubbard bands 25,43 . However, the SOC f 2 and non-SOC f −3 exhibit qualitatively different behavior below 700 K. Whereas D(iω 0 ) of f −3 in the non-SOC system is rapidly reduced, that of f 2 in the SOC system soars. {f 1 , f 3 , f 4 , f 5 , f 6 } in j = 5/2 multiplet show the same temperature evolution with f 2 down to 300 K. We attribute the  Table 1. Calculated electron occupation of U-5f orbitals in UTe 2 at T = 300 K. U-5f orbitals are labelled for convenience in this work. Label abruptly enhanced D(iω 0 ) to the emergence of the Kondo scattering in the SOC system where abundant f DOS near the Fermi level facilitates the f-c hybridization. The six partially occupied U-5f orbitals in j = 5/2 multiplet are characterized with − 1/2 spin by Hund's rule, and the total spin moment of the impurity increases, resulting in a relatively higher onset temperature of the Kondo screening process 44 .
To investigate the relevant energy scales of Kondo physics, we calculate the local total angular momentum susceptibility given as [Eq. 2] Figure 2 (g) shows the product of temperature and χ JZ loc of U-5f as a function of temperature. The plot reveals two characteristic temperatures of Kondo lattice in UTe 2 : Kondo scattering temperatures (T KS ) and lattice coherence temperature (T * ). The high temperature T KS below which χ JZ loc deviates from the Curie-Weiss behaviors indicates the onset of the Kondo scattering process. The low temperature T * with a local maximum of χ JZ loc suggests the crossover of incoherent-to-coherent scattering. Our estimation of T * ≃ 50 K and T KS > 300 K is consistent with the negative slope of ab-plane resistivity between 50 and 300 K 21 . T KS in UTe 2 is much higher than~200 K in CeCoIn 5 20 and CeCoGe 2 25 , and we attribute the higher T KS to the larger total f valence electrons in U-5f than Ce-4f and strong SOC in UTe 2 .

Orbital selective Kondo scattering
To elucidate the transport properties in UTe 2 between T = 25 K and 300 K, we analyze the SOC system in depth. The crystal electric field and SOC lift the degeneracy of the U-5f orbitals, grossly forming two groups, f α = {f 2 , f 3 , f 4 , f 5 } and f β = {f 1 , f 6 } (see Table 1), with an energy separation of~0.12 eV. These two groups show distinct electronic structures in this temperature range in terms of DOS, spectral functions, and charge susceptibility. We found that f α and f β are hybridized with U-6d and Te2-p {3, 6} states, respectively. Here, Te2-p {3, 6} includes Te2-5p states of (j = 3/2, j z = −3/2) and (j = 3/2, j z = 3/2).
We computed orbital-resolved spectral functions, and the hybridization between f α and U-6d is presented in Fig. 3a. The kink-like structure appears at the intersection of the two bands, which is consistent with the Kondo effect 25 . The abrupt change in the U-6d conduction bands dispersion at the Fermi level is clearly visible along the Γ-X, X 1 -Z, S-W, and L 1 -Y symmetry lines, all of which belong to the ab-plane. Thus, these hybridizations will affect electrical transport in the ab-plane. We define the onset temperature of Kondo scattering due to hybridization between f α and U-6d, T d k KS % 500 K at which the kink-like structure starts appearing at the Fermi level. Note that T d k KS % T KS , indicating f α -orbitals are responsible for the high temperature Kondo effect. Below T = 400 K, the heavy mass dispersion appears, and the first excited state is disconnected, indicating the progression of the active Kondo scattering. At T = 25 K, the incoherent f α -bands below the Fermi level are absent, and the coherent f α peak above the Fermi level is enhanced near the Fermi level, resulting in a single coherent peak (see Fig. 3a, b, and Supplementary Fig. 3). The formation of coherent f bands near the Fermi level signals renormalization of the carriers 25 and indicates the coherent Kondo lattice in UTe 2 20 . We determine the crossover temperature T d k cor (often denoted by T * ) from the incoherent scattering to the coherent Kondo lattice by employing the charge susceptibility [Eq. 3], where n i is the occupation of i-th orbital, which evaluates charge fluctuation 45 . Figure 3c shows the temperature dependent χ charge of f α (green) and U-6d (blue) which abruptly surge below show dρ/dT < 0 between 50 and 300 K and start to drop rapidly around 50 K 21 as shown in Fig. 3f. Te1-5p orbitals, hybridized with U-5f and U-6d, constitute the conduction bands along the X-L, W-R, Γ-Y, and Y 1 -Z symmetry lines. However, we did not find any evidence of the Kondo effect, and the hybridized bands remain metallic at high temperatures.
Below we focus on f β -orbitals. Figure 3d shows the f β orbitalresolved spectral functions where f β is hybridized with Te2-p {3, 6} along Z-Γ in the vicinity of the Fermi level (see also Supplementary Fig. 4). This result indicates that the conduction electron along the c-axis is originated from Te2-p {3, 6} . The DOS of Te2-p {3, 6} (data not shown) consists of spectral weight at the Fermi level over the entire temperature range and shows a small but notable increase as lowering the temperature. Figure 3e shows f β -projected DOS where a single coherent peak is present slightly above the Fermi level at high temperatures. The peak position approaches the Fermi level upon cooling, accompanied by a gradual reduction of the peak height. At T = 50 K, another peak appears below the Fermi level. The two peaks merge at T < 50 K, forming a single, enhanced peak at the Fermi level (see also Supplementary Fig. 3). The temperature evolution of DOS of f β at the Fermi level can shed light on the transport properties, and the inset of Fig. 3e shows temperature evolution of D(iω 0 ) of f β . Whereas D(iω 0 ) of f α monotonically increases upon cooling (see inset of Fig. 3b), D(iω 0 ) of f β is nearly temperatureindependent between 150 and 300 K. It exhibits a local minimum at 50 K before sharply increasing at lower temperatures. We attribute our observations to the Kondo scattering involving f β and Te2-p {3, 6} at T = 50 K (see Supplementary Fig. 4). We thus define T p ? KS ¼ 50 K as the onset temperature of the Kondo effect along the c-aixs, and, χ charge of f β manifests a sudden drop below T = 30 K as shown in Fig. 3c. The hybridization between f β and Te2-p {3, 6} is responsible for the c-axis transport, and our result is consistent with the recent experiment where ρ c is metallic in~50 K < T < 300 K and exhibits an upturn below~50 K 21 as shown in Fig. 3f.

ARPES
We compare our calculated spectral function at T = 25 K to recent ARPES measurements at T = 20 K 22,23 . As shown in Fig. 4, the calculated spectral functions along the Γ-X and Z-X 1 consist of one parabolic band and incoherent spectral weights, which agree with both measured ARPES spectra in the binding energy of E B < 1.0 eV along the same symmetry lines. Particularly, dispersive Kondo resonance peaks near the Fermi level (which is not presented in the non-SOC spectral function.) appear in the ARPES 22 . There are incoherent f β bands in − 0.2 eV > E − E F > − 0.4 eV. The corresponding bands appear in − 0.4 eV > E − E F > − 0.6 eV in both ARPES data. As shown in Fig. 3d, the spectral weight is more pronounced at lower energy down to 25 K from 300 K. Also, transport measurement shows abrupt change on the resistivity below 50 K (see Fig. 3f), indicating the rapid change on the electronic structure. Thus, we suggest that the variance may partly be caused by the temperature difference between simulation and ARPES measurement.

DISCUSSION
We have shown distinct U-5f peaks in the vicinity of the Fermi level for both SOC and non-SOC systems at 300 K in Fig. 2d.
The origin of characteristic f peak features and its subsequent temperature evolution can be identified with the self-energy and hybridization function. In Fig. 5a for SOC system, the f 3 selfenergies exhibit strong poles at the Fermi level below 300 K due to the Kondo screening process. While this gives rise to the prominent f peak at the Fermi level, the hybridization functions did not show notable changes at the Fermi level with respect to temperatures. In contrary, as shown in Fig. 5b for non-SOC system, self-energies do not have a pole at the Fermi level. Instead, the broader f bands in the vicinity of the Fermi level are developed by orbital hybridization as indicated by pronounced hybridization functions. The diverging hybridization functions as lowering temperature indicates enhanced electron or hole hopping between f and adjacent orbitals. The imaginary part of the local self-energy ImΣ(iω n ) for a Fermi liquid varies as iω n at a small Matsubara frequency ω n at low temperatures. The deviation from the linear variation, i.e., non-Fermi liquid, may occur due to freezing of localized spin moments 46 . The imaginary part of self-energy at T = 100 K (shown in Fig. 5) was fit with, ImΣðiω n Þ ' ÀΓ þ Aðiω n Þ α , and we obtained α ≈ 0.22 (0.99) and Γ ≈ 0.55 (0) for f 3 (f −3 ) in the SOC (non-SOC) system. Our fitting results indicate that SOC boosts the Kondo screening and is responsible for non-Fermi liquid behavior in the normal state of UTe 2 before the coherence of the Kondo lattice develops.
The understanding of the electronic structure at high pressure may provide important clues about the origin of enhanced T c at high pressure 10,11 . We suggest that U-5f electron is less localized at high pressure, resulting in a suppressed Kondo effect which is reminiscent of the non-SOC system. In this work, UTe 2 at high pressure is studied with simulations, where structures were constructed by adopting reduced experimental lattice constants without geometry relaxation. The geometry optimization using many-body methods (with proper electronic structure) is not yet feasible. Our simulations do not show quantitative result, but reveal distinct features due to reduced atomic distance. The modeled structures are denoted by 100, 098, and 096 for 0, 2 and 4 percent reduced experimental lattice constants, respectively. As shown in Fig. 6a, pole strengths in the self-energy decreased in association with a reduced lattice constants, unlike enhanced divergence of the hybridization functions. As shown in Fig. 6b, we do not find a local minimum in χ JZ loc , which indicates the onset of the Kondo scattering process, down to 50 K for 098 and 096 simulations. Reducing lattice constants causes more overlap between f and adjacent orbitals. It enhances f electron hopping between the orbitals, resulting in decreased local magnetic moments. Our results imply the Kondo effect is weakened at high pressure. Therefore, it is predicted that the onset of resistivity upturn by the Kondo effect may shift down to the lower temperature with respect to increasing pressure. According to the mutichannel Kondo model 47,48 , the impurity spin S can be screened by multiple spins of conduction electrons. Depending on the local magnetic moment and the number of conduction electron channel n, Kondo systems can be classified into three types: (i) Fully screened case (n = 2S) where S is completely screened by the sufficient number of conduction electron channels leading to the Fermi-liquid behavior at low temperature. (ii) Underscreened case (n < 2S) where S is partially screened at low temperature. In this case, the reduced effective spin can give rise to a long-range magnetic ordering due to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. (iii) Overscreened case (n > 2S) where a critical phenomenon may occur due to overcompensated S in the zero-temperature limit. The canonical example of the underscreened Anderson lattice model includes uranium monochalcogenides. Particularly, the Kondo effect in UTe is modeled with S = 1 and n < 2 in which the reduced effective spins interact through  the RKKY exchange interaction. As a consequence, the ferromagnetic order and the Kondo behavior due to the partially screened impurity spins can coexist in the ferromagnetic phase. The underscreened Anderson lattice model was extended by implementing the spinorbit interaction by Yuan et al. 49 . They show that the degeneracy of U-5f bands is lifted by spin-orbit interaction. In a non-cubic structure, the spin-orbit interaction is not isotropic due to the anharmonic hybridization and electronic correlations, which is fundamentally responsible for the anisotropic electronic structure. In UTe 2 , we show that the U-5f orbitals split into f α and f β with distinct characters due to the crystal electric field and SOC. f α is responsible for the Kondo effect along the c-axis, while f β governs the Kondo effect in ab-plane.
In terms of the multichennel Kondo model, it is therefore reasonable to apply this model to f α and f β , separately. For f β , we found that the coherent Kondo lattice gives rise to the Fermi-liquid behavior below T d k cor ¼ 50 K, which is consistent with the experimental observation of rapid downturn of electrical resistivity at low temperature 21 . Furthermore, the absence of a long range magnetic order suggests both anisotropic Kond effects of f α and f β are consistent with the sufficient screening of the impurity spins.
In summary, we performed first-principles simulations to investigate the electronic structure of heavy fermion superconductor UTe 2 . We found that the inclusion of SOC is necessary to reproduce the observed occupancy of the 5f orbital-configuration, anomalous temperature-dependence of electrical resistivity by Kondo effect, and band structure near the Fermi level measured by ARPES. In regard to electron correlations in heavy fermion compounds, we note a number of DFT+DMFT studies for Ce-and Pu-based compounds 20,25,26,30,50 , where conduction electron (d or p) dependent correlations is not found. As such, our discovery suggests that UTe 2 has complex electronic structure, which manifests orbitalselective hybridization among f {1, 6} , f {2, 3, 4, 5} , d, and p orbitals. Our work shed light on the understanding of anisotropic H-T phase diagram with multiple phases 51 and the selective re-entrant superconducting properties 14 , both of which are the cornerstone of understanding of possible p-wave topological superconductivity in UTe 2 .

METHODS LQSGW and DMFT calculations
We use ab-initio linearized quasiparticle self-consistent GW (LQSGW) and dynamical mean field theory (DMFT) method [52][53][54] to calculated the electronic structure of UTe 2 which crystallizes into orthorhombic space group Immm (No. 71) 55,56 . The LQSGW+DMFT is designed as a simplified version of the full GW+EDMFT approach [57][58][59] . It calculates electronic structure by using LQSGW approaches 60,61 . Then, it corrects the local part of GW self-energy within DMFT [62][63][64] . Within the methodology, the only parameters we adopted from experiments are lattice constants (a = 4.1611, b = 6.1222, c = 13.955 Å) 56 , and we explicitly calculate all other quantities such as double-counting energy and Coulomb interaction tensor. Then, local self-energies for U-5f and U-6d are obtained by solving two different single impurity models using continuous time quantum Monte Carlo method. The local self-energy is presumed to be diagonal in spherical harmonics basis by eliminating the off-diagonal elements in the hybridization functions. We show relatively small and negligible offdiagonal elements in Supplementary Fig. 1. For the LQSGW+DMFT scheme, the code ComDMFT 54 was used. For the LQSGW part of the LQSGW+DMFT scheme, the code FlapwMBPT 61 was used. For the details, please see the Supplementary Methods section.