Insulator-metal transition and topological superconductivity in UTe2 from a first-principles calculation

TL;DR

This paper uses first-principles calculations with electron correlation effects to reveal the insulator-metal transition and topological superconductivity in UTe$_2$, predicting its gap structure and low-energy excitations.

Contribution

It introduces a GGA+U approach to accurately model UTe$_2$, demonstrating topological superconductivity and analyzing its gap node symmetry and topology.

Findings

Insulator-metal transition driven by Coulomb interaction.

Topological superconductivity at intermediate U values.

Predicted gap structure and low-energy excitations.

Abstract

We theoretically study superconductivity in UTe$_2$, which is a recently-discovered strong candidate for an odd-parity spin-triplet superconductor. Theoretical studies for this compound faced difficulty because first-principles calculations predict an insulating electronic state, incompatible with superconducting instability. To overcome this problem, we take into account electron correlation effects by a GGA$+U$ method and show the insulator-metal transition by Coulomb interaction. Using Fermi surfaces obtained as a function of $U$, we clarify topological properties of possible superconducting states. Fermi surface formulas for the three-dimensional winding number and three two-dimensional $\mathbb{Z}_2$ numbers indicate topological superconductivity at an intermediate $U$ for all the odd-parity pairing symmetry in the $Immm$ space group. Symmetry and topology of superconducting gap node are analyzed and the gap structure of UTe$_2$ is predicted. Topologically protected low-energy excitations are highlighted, and experiments by bulk and surface probes are proposed to link Fermi surfaces and pairing symmetry. Based on the results, we also discuss multiple superconducting phases under magnetic fields, which were implied by recent experiments.

Tables

Table 1. Table 1: Correspondence between high-symmetry points in the original BZ and folded BZ.

	Original	Folded
	$\Gamma$, $X$	${\Gamma }_p:(0,0,0)$
	$R$	$U_p:[\left(\pi /a\right),0,\left(\pi /c\right)]$
	$S$	$T_p:[0,\left(\pi /b\right),\left(\pi /c\right)]$
	$T$	$S_p:[\left(\pi /a\right),\left(\pi /b\right),0]$
	$W$	$R_p:[\left(\pi /a\right),\left(\pi /b\right),\left(\pi /c\right)]$

Table 2. Table 2: Occupation number n ​ ( K i ) 𝑛 subscript 𝐾 𝑖 n(K_{i}) at high symmetry points in the folded BZ and topological invariants (modulo two) corresponding to each topology of FSs. The values n ​ ( K i ) − 180 𝑛 subscript 𝐾 𝑖 180 n(K_{i})-180 are shown below, with X p = ( π / a , 0 , 0 ) subscript 𝑋 𝑝 𝜋 𝑎 0 0 X_{p}=(\pi/a,0,0) , Y p = ( 0 , π / b , 0 ) subscript 𝑌 𝑝 0 𝜋 𝑏 0 Y_{p}=(0,\pi/b,0) , and Z p = ( 0 , 0 , π / c ) subscript 𝑍 𝑝 0 0 𝜋 𝑐 Z_{p}=(0,0,\pi/c) .

FSs		${\Gamma }_p$	$X_p$	$S_p$	$Y_p$	$Z_p$	$U_p$	$R_p$	$T_p$		$(\omega ,{\nu }_1,{\nu }_2,{\nu }_3)$
(i)		$6$	$4$	$4$	$8$	$4$	$0$	$4$	$4$		$(1,1,1,1)$
(ii)		$6$	$0$	$4$	$8$	$4$	$0$	$4$	$8$		$(1,1,1,1)$
(iii)		$4$	$0$	$4$	$8$	$4$	$0$	$4$	$8$		$(0,0,0,0)$

Table 3. Table 3: Gap structures, nontrivial topological indices, and surfaces hosting stable Majorana states for odd-parity pairing states.

FSs(i,ii)				FSs(iii)
IR	Gap structure	Topological index	Surfaces	IR	Gap structure	Topological index	Surfaces
$A_u$	Full gap	$(\omega ,{\nu }_1,{\nu }_2,{\nu }_3)$	(100), (010), (001)	$A_u$	Full gap	$\omega \in 2\mathbb{Z}$	Unpredicted
$B_{1u}$	Point node ($\Lambda$)	${\nu }_3$	(100), (010)	$B_{1u}$	Full gap	Trivial	None
$B_{2u}$	Point node ($\Delta$)	${\nu }_2$	(100), (001)	$B_{2u}$	Point node ($\Delta ,U$)	Trivial	None
$B_{3u}$	Point node ($\Sigma ,F)$	${\nu }_1$	(010), (001)	$B_{3u}$	Point node ($\Sigma ,F)$	Trivial	None

Table 4. Table 4: Classification of odd-parity superconducting order parameters for point groups (a) D 2 ​ h subscript 𝐷 2 ℎ D_{2h} and (b) C 2 ​ h y superscript subscript 𝐶 2 ℎ 𝑦 C_{2h}^{y} .

(a) $D_{2h}$ (zero magnetic field)
IR		$E$	$C_{2z}$	$C_{2y}$	$C_{2x}$	$I$	${\sigma }_z$	${\sigma }_y$	${\sigma }_x$		Basis functions
$A_u$		$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$		$k_x\hat{x},k_y\hat{y},k_z\hat{z}$
$B_{1u}$		$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$		$k_y\hat{x},k_x\hat{y}$
$B_{2u}$		$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$		$k_x\hat{z},k_z\hat{x}$
$B_{3u}$		$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$		$k_z\hat{y},k_y\hat{z}$

Table 5. Table 5: Expected anisotropy in thermal conductivity at low temperatures for each FSs and pairing symmetry. Anisotropy of fully gapped states cannot be predicted.

FSs	$A_u$	$B_{1u}$	$B_{2u}$	$B_{3u}$
(i)	Unpredicted	${\kappa }_c>{\kappa }_{a,b}$	${\kappa }_b>{\kappa }_{a,c}$	${\kappa }_a>{\kappa }_{b,c}$
(ii)	Unpredicted	${\kappa }_c>{\kappa }_{a,b}$	${\kappa }_b>{\kappa }_{a,c}$	${\kappa }_a>{\kappa }_{b,c}$
(iii)	Unpredicted	Unpredicted	${\kappa }_b>{\kappa }_{a,c}$	${\kappa }_a>{\kappa }_{b,c}$

Table 6. Table S1: Atomic coordinates of UTe 2 . Lattice constants are a = 4.1617 𝑎 4.1617 a=4.1617 , b = 6.1276 𝑏 6.1276 b=6.1276 , and c = 13.9650 𝑐 13.9650 c=13.9650 Å.

Atom	$x$	$y$	$z$
U ($4i$)	$0$	$0$	$0.13480$
Te1 ($4j$)	$0.5$	$0$	$0.29770$
Te2 ($4h$)	$0$	$0.2510$	$0.5$

Table 7. Table S2: Classification of superconducting gap structures on mirror planes and rotational axes under (a) zero magnetic field and (b) magnetic fields along the b 𝑏 b axis. In the tables, we show the EAZ class for each pairing symmetry. Letters in parenthesis represent the absence ( 0 0 ) or presence ( ℤ ℤ \mathbb{Z} or ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} ) of a topological invariant, and gap structures: (G) full gap, (P) point nodes, or (L) line nodes.

(a) Zero magnetic field
$D_{2h}$	$k_z=0,\mathrm{\hspace{0.17em}2}\pi /c$	$k_y=0$	$k_x=0$	$k_z$ axis ($\Lambda$)	$k_y$ axes ($\Delta ,U$)	$k_x$ axes ($\Sigma ,F$)
$A_u$	C ($0$, G)	C ($0$, G)	C ($0$, G)	CI ($0$, G)	CI ($0$, G)	CI ($0$, G)
$B_{1u}$	C ($0$, G)	AIII ($0$, G)	AIII ($0$, G)	BDI (${\mathbb{Z}}_2$, P)	CI ($0$, G)	CI ($0$, G)
$B_{2u}$	AIII ($0$, G)	C ($0$, G)	AIII ($0$, G)	CI ($0$, G)	BDI (${\mathbb{Z}}_2$, P)	CI ($0$, G)
$B_{3u}$	AIII ($0$, G)	AIII ($0$, G)	C ($0$, G)	CI ($0$, G)	CI ($0$, G)	BDI (${\mathbb{Z}}_2$, P)

Equations

$$\omega =\frac{1}{2}\sum _{K_i}n(K_i)(\mathrm{mod}2).$$

$${\bar{\mathfrak{G}}}^{\mathbf{k}}={\bar{\mathcal{G}}}^{\mathbf{k}}+\mathfrak{T}{\bar{\mathcal{G}}}^{\mathbf{k}}+\mathfrak{C}{\bar{\mathcal{G}}}^{\mathbf{k}}+\Gamma {\bar{\mathcal{G}}}^{\mathbf{k}}.$$

$$W_{\alpha }^{\mathfrak{T}}$$

$$W_{\alpha }^{\mathfrak{C}}$$

$$W_{\alpha }^{\Gamma }$$

Full text

A recent discovery of superconductivity in UTe2 Ran et al. (2019a) is attracting much attention. Distinct differences of UTe2 from other uranium-based ferromagnetic superconductors Aoki and Flouquet (2014); Aoki et al. (2019a) are a rather high superconducting transition temperature $T_{c}\sim 1.6$ K and a nonmagnetic behavior down to $25$ mK Sundar et al. (2019). Uniform magnetic susceptibility, magnetization, NMR Knight shift, and $1/T_{1}T$ support the proximity of metallic ferromagnetic quantum criticality Ran et al. (2019a); Aoki et al. (2019b); Tokunaga et al. (2019). An extremely large upper critical field and re-entrant superconductivity have been observed by high-field experiments Ran et al. (2019a, b); Knebel et al. (2019). These observations strongly suggest odd-parity superconductivity induced by ferromagnetic fluctuations. A large specific heat coefficient $\gamma=117$ mJ K*-2mol-1* indicates itinerant heavy $f$ electrons Aoki et al. (2019b); Knafo et al. (2019); Miyake et al. (2019). A large residual value of $\gamma$ in the superconducting state Ran et al. (2019a); Aoki et al. (2019b) suggests a time-reversal symmetry breaking nonunitary pairing, which is known to exist in ferromagnetic superconducting states. However, a direct transition from a normal to a nonunitary superconducting state is prohibited in the presence of spin-orbit coupling by symmetry because the orthorhombic $D_{2h}$ point group of UTe2 includes only one-dimensional (1D) representations. Experimental studies examining this issue are in progress.

Identifying the topological nature of quantum states has been one of the central issues in modern condensed matter physics. Because odd-parity superconductors are a strong candidate of topological superconductors accompanied by Majorana quasiparticles Sato (2009, 2010); Fu and Berg (2010), many studies have been focused on the odd-parity superconductivity Qi and Zhang (2011); Sato and Fujimoto (2016); Sato and Ando (2017). However, odd-parity superconducting materials are rare. Therefore, identifying topological properties of a fresh and good platform UTe2 is awaited. A nonmagnetic behavior of UTe2 enables time-reversal invariant (class DIII) topological superconductivity, and a relatively high transition temperature at ambient pressure allows many experimental tools, which were hard to use for ferromagnetic superconductors Aoki and Flouquet (2014); Aoki et al. (2019a). Theoretically, it is important to specify Fermi surfaces (FSs) to identify topological superconductivity. Topological invariants depend on FSs and some of them can be obtained by FS formulas Sato (2009, 2010); Fu and Berg (2010).

Information on FSs can also be linked to gap structures of unconventional superconductors, and therefore, it enables us to study pairing symmetry by measurements of low energy excitations Sigrist and Ueda (1991). Recent progress in topological theory attached a renewed attention to the gap node. According to modern classification of gapless superconductors Kobayashi et al. (2014, 2016, 2018); Sumita et al. (2019), all the superconducting gap nodes are topologically protected. Thus, the criterion of topological superconductivity and gap nodes based on FSs provides a prediction of bulk and surface measurements, revealing the pairing symmetry and Majorana surface states.

Theoretically, a band structure has been studied for UTe2 from first-principles Aoki et al. (2019b). However, the previously obtained result shows an insulating state with a small gap of $13$ meV, which contradicts metallic behaviors in electric resistivity Ran et al. (2019a); Aoki et al. (2019b). On the other hand, small FSs appear in another first-principles band calculation using the relativistic linearized augmented plane wave method Fujimori et al. (2019). This is also incompatible with transport measurements Niu et al. indicating a large carrier density, as well as their angle-resolved photoemission spectroscopy (ARPES) detecting large intensities around the $R$ point at the Fermi level Fujimori et al. (2019). These discrepancies between naive band structure calculations and experiments imply that the Coulomb interaction is crucially important for UTe2.

In this Letter, we provide the first report of a microscopic analysis linking the electronic state and superconductivity in UTe2. We show that the insulator-metal transition is induced by Coulomb interaction. For empirically reasonable values of $U$, a metallic state is realized, and FSs promise the topological superconductivity for all possible odd-parity pairings. The superconducting gap node ensured by crystal symmetry is predicted by the group theoretical classification combined with topological arguments. In addition, we discuss multiple superconducting phases under magnetic fields along the $b$ axis. A phase transition inside the superconducting phase is proposed.

GGA+U calculation — The topology of the FS is crucially important for unconventional superconductors, in particular, for the gap structure and topological superconductivity. As we introduced previously, two band structures have been reported. One is insulating Aoki et al. (2019b) and the other is metallic with small FSs Fujimori et al. (2019). They are clearly contradicting each other and incompatible with experiments. We therefore carry out the density functional theory (DFT) electronic structure calculations in the paramagnetic state using the wien2k package bla . We use the relativistic full-potential linearized augmented plane wave + local orbitals method within the generalized gradient approximation (GGA). In addition to the DFT calculation providing a noninteracting band structure, we introduce the correlation effect of $f$ electrons by the GGA$+U$ method Anisimov et al. (1991). Details of our band calculations are given in the Supplemental Material sup .

The numerical results are given in Figs. 1-2. The DFT band structure is insulating, and the band gap is $14$ meV. Thus, the results are consistent with Ref. Aoki et al. (2019b). On the other hand, the GGA$+U$ calculation shows the closing of band gap (Fig. 1), and we observe metallic FSs for $U>1.0$ eV. It turns out that the correlation effect of $f$ electrons causes an insulator-metal transition. A moderate value of $U$ in the GGA$+U$ calculation may be reliable at low temperatures, where the itinerant $f$ electrons form a heavy fermion state consistent with specific heat measurements Ran et al. (2019a); Aoki et al. (2019b). A larger $U$ may be appropriate above the Kondo temperature, where the $f$ electrons are localized. Although we cannot determine the value of $U$ in the framework of the GGA$+U$ method, we can deduce which FS is more appropriate using the comparison with future experiments such as ARPES or quantum oscillations. Furthermore, the GGA$+U$ calculation should be compared with other methods such as GGA+DMFT, which is left as a future work.

The obtained FSs are illustrated in Figs. 2(b)-2(d), each of which shows different topologies labeled by (i)-(iii) in Fig. 1(b). For $U=1.0$ eV, a tiny electron sheet appears at the Brillouin zone (BZ) boundary around the $X$ point and a tiny hole sheet around the $R$ point. The FSs dramatically increase their volume by an increase of $U$, involving a topological Lifshitz transition from (i) to (ii). For $U=1.1$ eV, there appears a ringlike sheet with a heavy effective mass dominated by $j_{z}=\pm 5/2$ and $\pm 1/2$ components sup . We also see a cylindrical sheet dominated by $j_{z}=\pm 5/2$ and $\pm 3/2$ components. The ringlike sheet changes to a two-dimensional (2D) cylindrical sheet at $U=2.0$ eV, which is shown in Fig. 2(d), with a topological transition at $U\simeq 1.6$ eV. These Fermi sheets consist of $j_{z}=\pm 5/2$ and $\pm 3/2$ components mixed with $d_{3z^{2}-r^{2}}$ and $p_{y}$ orbitals having a light effective mass sup . The electron sheet has a large carrier density $n\sim 0.2$ per spin which is compensated by the hole sheet. Thus, we see large FSs occupying $40\%$ of the BZ, in accordance with transport measurements Niu et al. . On the other hand, low carrier density shown in Fig. 2(b) or that obtained by a slight carrier doping to the DFT band structures is incompatible with experiments for UTe2. We also confirmed the insulator-metal transition by the density of states sup . Then, we notice that the incipient hybridization gap is shifted upwards in energy by $10$ meV ($U=1.0$ eV). The $f$-electron states with $j=5/2$ multiplet are dominant around the Fermi level.

Topological superconductivity — Here, we discuss topological properties of UTe2, assuming odd-parity superconductivity. We also assume that time-reversal symmetry is preserved in the superconducting phase since this is natural from the group-theoretical perspective as discussed previously. Thus, we focus on the zero-magnetic-field phase and the putative nonunitary pairing state Ran et al. (2019a) is beyond our scope.

Topology of odd-parity superconductors is closely related to the topology of FSs. Actually, the parity of various topological invariants is determined by the occupation number at high-symmetry points in the BZ Sato (2009, 2010); Fu and Berg (2010). For example, we can identify the parity of the three-dimensional (3D) winding number $\omega$ by the following formula:

[TABLE]

Here, $K_{i}$ runs over eight time-reversal invariant momenta (TRIM) in the 3D BZ, and the occupation number $n(K_{i})$ is an even integer due to Kramers degeneracy. Similar formulas are also known for 1D and 2D $\mathbb{Z}_{2}$ invariants Sato (2009, 2010); Fu and Berg (2010).

In this Letter, we concentrate on topological invariants related to the $(100)$, $(010)$, and $(001)$ surface states. It should be noted that the size of the primitive cell is doubled because the translation symmetry on these surfaces is compatible with the doubled unit cell. For this reason, we apply the formulas to the folded BZ, instead of the original one. To be specific, the former and the latter BZ correspond to the unit cell formed by $\{a\hat{x},b\hat{y},c\hat{z}\}$ and $\{(-a\hat{x}+b\hat{y}+c\hat{z})/2,(a\hat{x}-b\hat{y}+c\hat{z})/2,(a\hat{x}+b\hat{y}-c\hat{z})/2\}$, respectively. The correspondence between high-symmetry points in the two BZs is shown in Table 1.

We obtained from the GGA$+U$ results the occupation number at TRIM and corresponding topological invariants (Table 2). Here, $\nu_{1}$, $\nu_{2}$, and $\nu_{3}$ are the $\mathbb{Z}_{2}$ invariants defined on the $k_{x}=0$, $k_{y}=0$, and $k_{z}=0$ planes, respectively. $\mathbb{Z}_{2}$ invariants defined on the other time-reversal invariant planes are obtained from $(\omega,\nu_{1},\nu_{2},\nu_{3})$, as is the case for topological insulators, and they are trivial in our results.

According to Table 2, superconductivity in UTe2 is topologically nontrivial for moderate values of $U$ in the regions (i) and (ii) when the bulk state is gapped. Majorana states appear on the $(100)$, $(010)$, and $(001)$ surfaces. This is one of the central results of this Letter.

Here we comment on the effect of excitation nodes on the topological superconductivity. Although the winding number $\omega$ is ill-defined in the gapless states, some of the 2D $\mathbb{Z}_{2}$ invariants $\nu_{i}$ may still be well defined and the corresponding surface Majorana states may appear. Well-defined topological invariants and surfaces hosting Majorana states are summarized in Table 3 for each pairing symmetry and FSs. For example, $B_{1u}$ superconducting state has point nodes on the $k_{z}$ axis, meaning $\nu_{1}$ and $\nu_{2}$ are ill-defined. However, $\nu_{3}$ is well defined and $(100)$ and $(010)$ surface states are robust. Thus, topological superconductivity can be detected with scanning tunneling microscopy or ARPES for clean surfaces even in gapless superconducting states.

It should be noticed that our results do not exclude the possibility of topological superconductivity for FSs (iii): Indeed, 3D winding number $\omega\in\mathbb{Z}$ can be a finite even integer. By symmetry, this is allowed only for the $A_{u}$ pairing state Yoshida et al. (2019). We do not discuss this case because $\omega$ depends on detailed properties of superconducting gap function. We also leave the possibility of topological crystalline superconductivity as a future issue.

Topological gap node — Now we discuss gap structures in the superconducting state of UTe2, using group theory and topology. First, we consider an ordinary classification theory of the superconducting order parameter by the crystal point group Volovik and Gor’kov (1984, 1985); Anderson (1984); Sigrist and Ueda (1991). At zero magnetic field, UTe2 possesses $D_{2h}$ point group symmetry, in which an odd-parity order parameter is classified as one of four irreducible representations (IRs): $A_{u}$, $B_{1u}$, $B_{2u}$, and $B_{3u}$. Typical basis functions are shown in Table 4(a). We also consider magnetic fields along the $b$ axis in which UTe2 shows extremely high critical fields and metamagnetic transition Ran et al. (2019b, a); Knebel et al. (2019); Knafo et al. (2019); Miyake et al. (2019). Then, the symmetry is reduced to $C_{2h}^{y}$, which has two odd-parity IRs ($A_{u}$ and $B_{u}$). In this case, the $A_{u}$ and $B_{2u}$ ($B_{1u}$ and $B_{3u}$) states are not distinguished by symmetry since they are reduced to the $A_{u}$ ($B_{u}$) state. The correspondence is summarized in Table 4(b).

Although we can speculate gap structures from the order parameter, it is desirable to use the classification of gap structures in terms of symmetry and topology Yarzhemsky and Murav’ev (1992); Norman (1995); Micklitz and Norman (2009); Nomoto and Ikeda (2017); Micklitz and Norman (2017); Sumita et al. (2017); Sumita and Yanase (2018); Kobayashi et al. (2014, 2016, 2018); Sumita et al. (2019) because symmetry-protected gap nodes are precisely obtained. Detailed results of the topological classification are shown in the Supplemental Material sup . Using the results, gap structures of UTe2 are obtained for each pairing symmetry and FSs, as shown in Table 3.

Considering the FS topology in Figs. 2(b)-2(d), we find that UTe2 is a fully gapped superconductor at zero magnetic field when the order parameter belongs to $A_{u}$ IR with $U>1.0$ eV [regions (i)-(iii)], or $B_{1u}$ IR with $U>1.6$ eV [region (iii)]. Otherwise the superconducting state has some point nodes, whose positions depend on pairing symmetry and they can be distinguished by experiments. In Table 5, we show expected anisotropy of thermal conductivity Joynt and Taillefer (2002), which may determine the symmetry of superconductivity in UTe2.

When the magnetic field is applied along the $b$ axis, superconducting states are classified into $A_{u}$ or $B_{u}$ IR. The $A_{u}$ state of $C_{2h}^{y}$ can be regarded as a $A_{u}+B_{2u}$ state, a mixed representation in $D_{2h}$, while $B_{u}\uparrow D_{2h}=B_{1u}+B_{3u}$. According to the gap classification in Table S2(b) of Supplemental Materials sup , the $A_{u}$ state has symmetry-protected point nodes on the $k_{y}$ axes while the $B_{u}$ state has a line node on the $k_{y}=0$ plane. These results are consistent with speculation from classification of order parameter. Since the spin-triplet order parameter with $d$-vector parallel to the magnetic field does not cause the gap of Bogoliubov quasiparticles, we have only to consider the others. For $A_{u}$ IR, allowed bases are $k_{x}\hat{x}$, $k_{z}\hat{x}$, $k_{x}\hat{z}$, and $k_{z}\hat{z}$, which create point nodes. On the other hand, the $B_{u}$ order parameter of $k_{y}\hat{x}$ and $k_{y}\hat{z}$ results in line nodes on the $k_{y}=0$ plane.

Multiple phases under magnetic fields — Experimental data for UTe2 under magnetic fields along the $b$ axis reveal highly unusual behaviors Ran et al. (2019b, a); Knebel et al. (2019). The transition temperature shows a nonmonotonic behavior as a function of the magnetic field. It indicates the presence of two superconducting phases: The low-field and high-field phases may be distinguished by symmetry. Considering the gap structure discussed above, we propose the phase diagram in Fig. 3. Because point-nodal superconducting states gain more condensation energy than line-nodal one, the $A_{u}$ state may be stable at high magnetic fields, while the $B_{u}=B_{1u}+B_{3u}$ state may be favored by the spin-orbit coupling at low fields. In this case, the $B_{1u}$ or $B_{3u}$ state is realized at zero magnetic field. The order parameter of both states contains $d$-vector parallel to the $b$ axis, and therefore, the Knight shift decreases below $T_{\rm c}$. This is consistent with a recent NMR experiment Nakamine et al. (2019).

Anisotropy of pairing interaction Yanase et al. (2003) is also important. Strongly anisotropic ferromagnetic fluctuation may favor the $d$-vector perpendicular to the easy axis. Then, the easy axis along $a$ Ran et al. (2019a) implies the $B_{3u}$ state at $H=0$ while the fluctuation along the $b$ axis near the metamagnetic transition Knafo et al. (2019); Miyake et al. (2019) favors the $B_{2u}$ state. This is consistent with our proposal. Microscopic calculations are desirable for more precise discussions and left for a future study.

Magnetism — Figures 2(c) and (d) show nesting property of the FSs, and therefore, we expect a magnetic fluctuation with a finite-$q$ nesting vector coexisting with a widely believed ferromagnetic fluctuation. Thus, it is indicated that UTe2 is in the vicinity of the multiple magnetic orderings, and the long-range magnetic order is suppressed by magnetic frustration. This may explain why UTe2 does not undergo magnetic order in contrast to UCoGe and URhGe. A recent first-principles study proposes a similar scenario Xu et al. .

Summary and conclusion — In this Letter, we theoretically investigated the electronic state and superconductivity of UTe2. Using the GGA$+U$ calculation we clarified the insulator-metal transition due to Coulomb interaction. The metallic band structure for $U>1.0$ eV is compatible with the metallic conductance and superconducting instability, indicating a crucial role of electron correlation. For moderate $U$, all the odd-parity superconducting states expected in UTe2 were identified as time-reversal-invariant topological superconductivity. Superconducting gap structures under magnetic fields along the $b$ axis as well as at zero magnetic field were predicted. By this work, bulk and surface excitations characterizing the odd-parity superconductivity in UTe2 are elucidated. Our results pave a way to experimentally determine symmetry of superconductivity and identify intrinsic topological superconductivity which has been rare in nature.

After we wrote the first manuscript, we become aware of Ref. Shick and Pickett (2019), which reported DFT$+U$ calculations in both paramagnetic and ferromagnetic states. At their parameters $U=J=0.51$ eV, an insulating band structure was obtained consistent with our calculations, and insulator-metal transition was not shown. We are also aware of an experimental paper Metz et al. in which point nodes along the $a$ axis are concluded. This implies the $B_{3u}$ state at zero magnetic field consistent with our proposal in Fig. 3.

S1 Details of band calculation

We perform the DFT band structure calculation in the paramagnetic state of UTe2 using the wien2k code bla . The crystallographical parameters are shown in Table S1 Haneveld and Jellinek (1970). UTe2 crystallizes in an orthorhombic body-centered structure (space group No. 71, $Immm$) and the local site symmetry of uranium atoms is $C_{2v}$, lacking the inversion symmetry. The maximum reciprocal lattice vector $K_{\rm max}$ was given by $R_{\rm MT}K_{\rm max}=11.0$. The muffin-tin radii $R_{\rm MT}$ of 2.50 a.u. was chosen for U and Te atoms, and $12\times 12\times 12$ $k$-points sampling was used for the self-consistent calculation. For the GGA$+U$ calculation, we set Hund’s coupling $J=0$ eV. We use the around mean-field formula Czyżyk and Sawatzky (1994) for the double-counting correlation, and we crosschecked results with the self-interaction correction formula Anisimov et al. (1993); Liechtenstein et al. (1995).

Figure S1 shows the band structure by DFT and GGA$+U$ along high symmetry lines. Uranium $5f$ orbitals split into the $j=5/2$ and $j=7/2$ manifolds due to the spin-orbit coupling. We observe near the Fermi level doubly degenerated bands of the $j=5/2$ multiplet, two of which are valence bands, and other four bands are conduction bands. The $j=7/2$ states are located at around $1.0$ eV. The electron occupancy of $5f$ orbitals is around $n_{f}\sim 2.5$, which is larger than the nominal occupancy of U2+ ion $n^{0}_{f}=2$. Note that the dispersive bands at around $-0.5$ eV are composed of $5f$ orbitals mixed with U-$d_{3z^{2}-r^{2}}$ orbital and Te(2)-$p_{y}$ orbital. The DFT band structure is consistent with the previous result Aoki et al. (2019). Moreover, in Fig. S1, $j_{z}$ orbital characters are highlighted. The $j_{z}=\pm 1/2$, $\pm 3/2$, and $\pm 5/2$ states are quite entangled. The flat bands near the band gap are dominated by $j_{z}=\pm 1/2$ and $\pm 5/2$ states in the DFT calculation. The band structures by GGA$+U$ are shown for $U=1.0$, $1.1$, and $2.0$ eV. Compared to the DFT results, two $j=5/2$ bands are shifted downward in energy about $U/2$, leaving the band structure near the Fermi level dominated by the $5f$ $j_{z}=\pm 5/2$ and $\pm 3/2$ states. We can see an electron pocket at the $S$ and $X$ points and a hole pocket at the $R$ point. Figure S2 shows partial density of states for U-5$f$ orbitals $j=5/2$, $j_{z}=\pm 1/2$, $\pm 3/2$, and $\pm 5/2$ near the Fermi level. We can confirm the insulator-metal transition. The $j_{z}=\pm 5/2$ states are dominated at the Fermi level for $U=1.0$ and $1.1$ eV.

S2 Topological classification of superconducting gap structures

Table IV in the main text gives a classification of superconducting order parameter, which enables us to speculate gap structures of UTe2. However, it is desirable to use the classification of gap structures in terms of symmetry and topology Yarzhemsky and Murav’ev (1992); Norman (1995); Micklitz and Norman (2009); Nomoto and Ikeda (2017); Micklitz and Norman (2017); Sumita et al. (2017); Sumita and Yanase (2018); Kobayashi et al. (2014, 2016, 2018); Sumita et al. (2019) because symmetry-protected gap nodes are precisely obtained. This scheme focuses on high-symmetry points in the Brillouin zone in Fig. 2(a) of the main text: mirror planes ($k_{z}=0,\,2\pi/c$, $k_{y}=0$, and $k_{x}=0$) and rotational axes labeled by $\Lambda$, $\Delta$, $U$, $\Sigma$, and $F$. Crystal symmetry allowed on each high-symmetry $\bm{k}$ point is represented by a unitary little cogroup $\bar{\cal G}^{\bm{k}}$, whose irreducible representation is denoted by $\alpha$.

At zero magnetic field, we define time-reversal symmetry (TRS), particle-hole symmetry (PHS), and chiral symmetry (CS) operators preserving any $\bm{k}$ points by $\mathfrak{T}\equiv{\cal TI}$, $\mathfrak{C}\equiv{\cal CI}$, and $\Gamma\equiv{\cal TC}$, respectively. Thus the intrinsic symmetry is represented by the following group:

[TABLE]

Then, using the factor system $\{z_{g,h}^{\bm{k}}\}\in Z^{2}(\bar{\mathfrak{G}}^{\bm{k}},\mathrm{U}(1)_{\phi})$, we execute the Wigner criteria Wigner (1959); Herring (1937); Inui et al. (1990); Bradley and Cracknell (1972); Shiozaki et al. for $\mathfrak{T}$ and $\mathfrak{C}$, and the orthogonality test Inui et al. (1990); Shiozaki et al. for $\Gamma$:

[TABLE]

From Eqs. (S2)-(S4), we obtain the set of $(W_{\alpha}^{\mathfrak{T}},W_{\alpha}^{\mathfrak{C}},W_{\alpha}^{\Gamma})$, which indicates the effective Altland-Zirnbauer (EAZ) symmetry class of the Bogoliubov-de Gennes Hamiltonian on the high-symmetry point Altland and Zirnbauer (1997); Shiozaki et al. . The EAZ class gives us classification of gap structures, assuming that the high-symmetry planes or lines intersect normal-state Fermi surfaces (for details, see Ref. Sumita et al. (2019)). After some algebra, we obtain the gap classification shown in Table S2(a).

Under magnetic field along the $b$ axis, on the other hand, the intrinsic symmetry is changed. On the mirror plane ($k_{y}=0$), TRS $\mathfrak{T}$ and CS $\Gamma$ are ill-defined, while PHS $\mathfrak{C}$ is introduced by ${\cal CI}$. On the rotational axes ($\Delta,U$), we define TRS, PHS, and CS by $\mathfrak{T}\equiv{\cal T}C_{2z}$, $\mathfrak{C}\equiv{\cal CI}$, and $\Gamma\equiv{\cal T}C_{2z}{\cal CI}$, respectively. Eqs. (S2)-(S4) are also applied into the cases, which result in Table S2(b). Note that $W_{\alpha}^{\mathfrak{T}}=W_{\alpha}^{\Gamma}=0$ on the mirror plane since the little cogroup $\bar{\mathfrak{G}}^{\bm{k}}$ has no TRS and CS.