Introduction

Since the discovery of superconductivity in the nonmagnetic uranium-based compound UTe2, the nature of its superconducting state has been extensively studied1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31. Recent studies have reported several anomalous superconducting properties, including extremely high upper critical field much beyond the Pauli limit1,2,3, reentrant superconductivity3,4, and little reduction of the Knight shift in the nuclear magnetic resonance (NMR)1,5. These results suggest odd-parity spin-triplet pairing in UTe2, as in the case of uranium-based ferromagnetic superconductors32. The symmetry of the superconducting order parameter is closely related to the superconducting gap structure, and the previous studies of low-energy quasiparticle excitations, such as specific heat, thermal transport, and magnetic penetration depth measurements6,7,8, support the presence of point nodes, consistent with the spin-triplet pairing states.

More intriguingly, recent scanning tunneling microscopy9, optical Kerr effect10, and microwave surface impedance measurements6 suggest time-reversal symmetry breaking (TRSB) in the superconducting state at ambient pressure. As the sign of imaginary part changes under the time-reversal transformation, the chiral TRSB state requires multiple order parameter components in complex form. We note that high-pressure studies reveal several superconducting phases11,12,13, suggesting the presence of multiple order parameters under pressure. Thus, UTe2 is a prime candidate of a topological chiral spin-triplet superconductor. However, the symmetry of the odd-parity vector order parameter d, whose magnitude is the gap size and whose direction is perpendicular to the spins of Cooper pairs, is still highly controversial. Especially, the nodal structure of order parameter and whether or not it is chiral in the ground state are important issues to understand the possible topological properties of UTe2.

The crystal structure of UTe2 (Fig. 1a) is classified into the point group D2h, whose irreducible representations (IRs) of odd-parity order parameters are listed in Table 1. In the cases of B1u, B2u, and B3u states, point nodes in the superconducting gap function exist on the kz-, ky-, and kx-axes (Fig. 1b), respectively, while the Au state is fully gapped. When the odd-parity order parameter d is represented by a single IR, the positions of nodes can be detected by the temperature dependence of the change in magnetic penetration depth, Δλ(T) ≡ λ(T) − λ(0). This is because the low-temperature superfluid density λ−2(T), which is determined by thermally-excited quasiparticles near the nodes, depends strongly on the directions of the shielding supercurrent density js and the point nodes (whose direction is defined as I). As a result, when the point nodes are directed along crystallographic α axis (I∥α), Δλα(T) follows T2 dependence, while for perpendicular axes β and γ, Δλβ(T) and Δλγ(T) should follow T4 dependence (Table 1)33. Here, the subscript i of λi represents the direction of supercurrent density js.

Fig. 1: Nodal positions of several pairing states in UTe2 and geometries of anisotropic penetration depth measurements.
figure 1

a Crystal structure of UTe2. Positions of the point nodes (red points) for B3u (b), B3u + iεB1u (c), B3u + iεAu (d), and B3u + iAu (e) order parameters, where ε is a sufficiently small real number. f Schematic relations between the directions of the ac magnetic field (big arrows) and λ components.

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Table 1 Basis functions, nodal types, and temperature dependence of the magnetic penetration depth for odd-parity order parameters in the point group D2h
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On the other hand, when two symmetries in different IRs accidentally admix to form a TRSB complex order parameter, point nodes are generally located away from the high symmetry axes, and various nodal structures become possible10,14. Considering the B3u + iεB1u (B3u + iεAu) state, for example, where ε is a sufficiently small real number, a point node of the B3u state splits into two (four) point nodes as depicted in Fig. 1c (Fig. 1d) (for more details, see Supplementary Information I). These split point nodes can be identified as topological Weyl nodes defined by a Chern number10,14, and corresponding Majorana arc surface states are expected34. Although an experimental determination of the exact positions of the nodes is quite challenging in these cases, we can summarize expected nodal positions for different relative sizes of two components in the complex order parameters in Table 2. Thus, by detecting the anisotropy of quasiparticle excitations through direction-dependent physical quantities, such as Δλi(T), we can pin down the superconducting symmetry among the non-chiral and chiral states listed in Tables 1 and 2, respectively.

Table 2 Expected positions of point nodes in chiral superconducting states with the order parameter d = d1 + id2, where d1 and d2 are classified into different IRs
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We use three independent measurements of resonant frequency of the tunnel-diode oscillator (see Methods) with weak ac magnetic field along the a-, b-, and c-axes, in which the shielding current flows perpendicular to the field as described in Fig. 1f. Thus the frequency shift Δf(T) consists of two penetration depth components perpendicular to the field direction. As a result, in the single component order parameter cases for point nodes along the α direction, Δf(T) for Hω∥α is the sum of Δλβ(T) and Δλγ(T) components and thus follows T4 dependence, while Δf(T) for Hω⊥α should follow T2 dependence at low T. We stress that in our measurements the sample is in the Meissner state. Therefore, our approach is an ideal way to investigate the superconducting symmetry in the ground state in the zero-field limit at ambient pressure.

Results

Specific heat

Figure 2a-c show Δf(T) in three single crystals of UTe2 denoted as #A1, #B1, and #C1 respectively (see Supplementary Information V). We observe a large change in Δf(T) at 2.1 K (#A1), 1.75 K (#B1), and 1.65 K (#C1) corresponding to the superconducting transition. We note that, while crystals #B1 and #C1 are grown by chemical vapor transport (CVT) method, crystal #A1 is grown by molten salt flux (MSF) method and the transition temperature is the highest value ever reported15. The clear superconducting transition at Tc = 2.1 K, 1.75 K, and 1.65 K for each sample, is also reproduced in the specific heat data (Fig. 2d). Here, we emphasize that a single jump clearly seen in crystals #A1 and #B1 do not necessarily contradict the multi-component order parameter discussed later, because in the Landau theory the jump heights have nontrivial dependence on the coefficients of the fourth power terms of the free energy for chiral superconducting order parameters11,35. However, a recent study16 has reported that the presence or absence of the double transitions depends strongly on the crystal growth conditions, and the origin of the double transitions is still highly controversial17,18. Figure 2e shows the low-temperature electronic specific heat Ce/T in crystal #A1 as a function of \({(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\). Here, the subtracted phonon contribution is estimated from the previous measurements19. We can find large quasiparticle excitations following \({C}_{{{{{{{{\rm{e}}}}}}}}}/T\propto {(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\) down to 0.12Tc, which is an expected behavior in a point-nodal superconductor. Furthermore, the residual electronic specific heat γ0 estimated from the extrapolation of the \({C}_{{{{{{{{\rm{e}}}}}}}}}/T\propto {(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\) relation is 4.6 mJK−2mol−1, leading to γ0/γn = 0.038, where γn is the Sommerfeld coefficient in the normal state. The small γ0/γn value confirms the high quality of crystal #A1. We note that the upturn in Ce/T observed below 0.12Tc is caused by the nuclear Schottky contribution15. While the observed \({C}_{{{{{{{{\rm{e}}}}}}}}}/T\propto {(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\) behavior and small γ0/γn value indicate the presence of point nodes in the gap structure, we cannot discuss the superconducting symmetry and position of point nodes from the specific heat. Therefore, we rather focus on the low-temperature penetration depth data.

Fig. 2: Superconducting transitions in UTe2.
figure 2

a-c, Overall temperature dependence of the frequency shift Δf in UTe2 for crystal #A1 (a), #B1 (b), and #C1 (c). The data of #A1 (#B1 and #C1) are normalized by the value at 2.2 K (2 K). d, Temperature dependence of the specific heat C divided by T in crystals #A1, #B1, and #C1 measured at zero field. e, Enlarged views of the electronic specific heat Ce/T in crystal #A1 at low temperatures as a function of \({(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\). The black broken line represents the linear fitting of the data. The gray solid line is the theoretical curve expected for a full gap structure with the gap size Δ = 2.2kBTc, which shows a similar size of the specific heat jump with the experiments.

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Magnetic penetration depth

The key results are the temperature dependence of Δf(T) at low T, which is shown for crystals #A1, #B1, and #C1 in Fig. 3a-c, respectively. The black solid lines represent fitting curves for the data below 0.3Tc using the power-law function, Δf(T) ∝ Tn. We note that the exponent values n = 2.11 (#A1), 2.07 (#B1), and 1.95 (#C1) for Hω∥c are consistent with the previous ab-plane penetration depth studies6,7. From the fittings, we find that the obtained exponent values for all field directions are nearly equal to 2 or less than 2 in all the samples. This feature can be more clearly seen by plotting Δf(T) as a function of \({(T/{T}_{c})}^{2}\) as shown in Fig. 3d-f, where all data show almost linear or convex downward curvatures at low T. Thus, our results contradict any cases of single-component odd-parity order parameters, in which Δf(T) should follow T4 dependence when the applied field is directed to the point node direction. Another feature of our data is that the exponent values obtained from Δf(T) for Hω∥a and Hω∥b are smaller than that for Hω∥c in crystals #B1 and #C1. Considering that Δf(T) consists of two Δλi(T) components perpendicular to the magnetic field (Fig. 1f), our exponent analysis on these samples indicates that the exponent value of Δλc(T) is smaller than those of Δλa(T) and Δλb(T), which will be discussed in more detail below.

Fig. 3: Anisotropic frequency shift in UTe2 for three field directions.
figure 3

Low-T behavior of Δf in UTe2 for crystal #A1 (a), #B1 (b), and #C1 (c) as a function of T normalized by Tc. Solid lines represent the fitting curves with the power-law function. Δf(T) normalized by the value at T = 0.3Tc as a function of \({(T/{T}_{c})}^{2}\) in UTe2 for crystals #A1 (d), #B1 (e), and #C1 (f). Dashed lines represent T2 dependence. The data with the field along the a- and b-axes are vertically shifted by 0.2 and 0.1, respectively.

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For further investigations of the gap structure, we extract Δλi(T) separately from the Δf(T) data for three different field orientations, by considering the geometry of the sample (see Supplementary Information VI). Such an analysis is valid when the magnetic penetration depth is much shorter than the sample dimensions, which holds at low temperatures. To compare the quasiparticle excitations along each crystallographic axis, we discuss the normalized superfluid density \({\rho }_{{{{{{{{\rm{s}}}}}}}},i}(T)={\lambda }_{i}^{2}(0)/{\lambda }_{i}^{2}(T)\) and normalized penetration depth Δλi(T)/λi(0) for three supercurrent directions i = a, b, and c, in which evaluations of λi(0) values are needed. The anisotropy of λ(0) can be estimated by the anisotropy of coherence length ξ which can be determined from the initial slope of the temperature dependence of upper critical field Hc2(T), when for simplicity we ignore the anisotropy of gap function (see Supplementary Information X). From the Hc2(T) data of an ultra-clean sample20, we estimate \({\lambda }_{a}(0):{\lambda }_{b}(0):{\lambda }_{c}(0)={\xi }_{a}^{-1}({T}_{c}):{\xi }_{b}^{-1}({T}_{c}):{\xi }_{c}^{-1}({T}_{c})=2.01:1:3.90\). By using the value \(\sqrt{{\lambda }_{a}(0){\lambda }_{b}(0)}\,\approx \,1\) μm estimated from the previous penetration depth studies6,7, we obtain λa(0) = 1420 nm, λb(0) = 710 nm, and λc(0) = 2750 nm.

The obtained normalized Δλi(T)/λi(0) as a function of T/Tc for three directions of crystals #A1, #B1, and #C1 are shown in Fig. 4a-c, respectively. First of all, the exponent values ni obtained from the power-law fitting in Δλi(T)/λi(0) data are all nearly equal to 2 or less than 2, which is again inconsistent with all the cases of the single component order parameter. Especially, as expected from Δf(T) data, nc = 1.60 in crystal #C1 and nc = 1.84 in crystal #B1 are smaller than na ≈ nb ≈ 2. The relatively small nc in crystals #B1 and #C1 can be more clearly seen by plotting the data as a function of \({(T/{T}_{c})}^{2}\) (Fig. 4b,c, inset). We note that the above discussions are independent from the fitting range of the power-law dependence of Δλ(T) ∝ Tn (see Supplementary Information VII). Another consequence of Δλ(T)/λ(0) results is that the quasiparticle excitations along the b- and c-axes are much larger than those along the a-axis, implying a highly anisotropic nodal structure. Figure 4d-f show the normalized superfluid density, \({\rho }_{{{{{{{{\rm{s,}}}}}}}}i}\equiv {\lambda }_{i}^{2}(T)/{\lambda }_{i}^{2}(0)\), along each crystallographic axis for crystals #A1, #B1, and #C1, respectively, plotted against T/Tc. Compared with theoretical curves for the single order parameter with the supercurrent density js parallel and perpendicular to the direction of point nodes I, the amount of the excitations along the a-axis is clearly smaller than the js∥I case, while those along the b- and c-axes are as large as the js∥I case.

Fig. 4: Anisotropic penetration depth in UTe2 for three supercurrent directions.
figure 4

Δλi(T) normalized by λi(0) for i = a, b, and c as a function of T/Tc for crystals #A1 (a), #B1 (b), and #C1 (c). Solid lines in a-c represent the fitting curves of the power-law function and the obtained exponent values are shown in the figures. The inset shows the same data plotted against \({(T/{T}_{c})}^{2}\), and the black dashed lines represent T2 dependence. Normalized superfluid density calculated from Δλi(T)/λi(0) data for crystals #A1 (d), #B1 (e), and #C1 (f). Solid and dashed lines represent the theoretical curves for the single order parameter case for js parallel and perpendicular to the nodal direction I, respectively.

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Discussion

Having established that our anisotropic superfluid density data exclude the single-component odd-parity order parameters, we now discuss the superconducting gap structure based on our experimental results. Adding two order parameters with preserving time-reversal symmetry will not split the point nodes and cannot account for our results (see Supplementary Information II), and thus we need to consider chiral superconducting states formed by two order parameters in different IRs (Table 2). First, we consider chiral superconducting states formed by two Bu IRs. As shown in Supplementary Information I, these chiral states have point nodes located on a symmetric plane of the momentum space. Therefore, small quasiparticle excitations and an exponent value ni > 2 of Δλi(T) are expected for the direction perpendicular to the plane. However, experiments show ni ≲ 2 for all directions, suggesting that these chiral states are unlikely to be realized in UTe2. Thus, we focus on the chiral superconducting states consisting of the Au and a Bu IRs. Considering the observed large quasiparticle excitations along the b- and c-axes, we conclude that the B3u + iAu pairing state is most consistent with our experiments (see Table 2). The reason is that for the B3u + iAu state with similar sizes of ∣dB3u∣ and ∣dAu∣ components, multiple point nodes can exist near the ky- and kz-axes, leading to larger excitations along the b- and c-axes than along the a-axis. Thus, the quicker decrease of our ρs,b and ρs,c data than the ρs,a data (Fig. 4d-f) is consistent with the B3u + iAu pairing state. This state can be supported by a recent theoretical study based on the periodic Anderson model, which suggests almost equally stable B3u and Au states at ambient pressure21. We note that the chiral B3u + iAu state is non-unitary with finite d × d* (see Supplementary Information IV), and for the system close to a ferromagnetic quantum critical point, theory shows that such a non-unitary complex order parameter may become stable22. Moreover, recent studies of NMR Knight shift 5 suggest finite \(\hat{y}\) and \(\hat{z}\) components of d, which is consistent with the B3u + iAu state. However, we note that the anisotropies of λi(0) and \({\xi }_{i}^{-1}\) are not completely the same in the case of anisotropic superconductors (Supplementary Information X). To confirm the chiral B3u + iAu state, more direct measurements of λi(0) are highly desired.

Next we discuss the sample dependence of the small exponent value nc. We found that the nc value systematically approach 2 as Tc gets higher (Fig. 4a-c), suggesting that impurity effect is related to the nc value. In the line node case, the exponent value larger than the clean limit n = 1 and smaller than n = 2 can be interpreted as a consequence of nonmagnetic impurity scatterings36, quantum criticality37, or non-local effects38. However, these possibilities can be excluded in the case of UTe2 (see Supplementary Information III). In the presence of point nodes in the gap structure, the impurity scattering can affect only the amplitude of the low-energy excitations, but does not symply change the exponent value of Δλ(T) ∝ T2 6,33. Here we propose an interference effect of the two point nodes located closely as the origin of the sample-dependent nc values. In the B3u + iAu state, the main contributions to Δλc(T)/λc(0) come from two pairs of two point nodes near the north and south poles along the kz-axis (Fig. 1e). When the distance of the two nodes near the pole gets sufficiently short, the low-energy excitations can no longer be treated as a sum of the contributions from two independent point nodes, and this interference effect can lead to an exponent value less than 2. We have confirmed that a simple model based on the B3u + iAu state can indeed lead to an exponent near the experimental nc value when the two point nodes are sufficiently close to each other (see Supplementary Figs. S12 and S13). Moreover, the observed trend that nc gets smaller for samples with lower Tc can be consistently explained if we consider that the anisotropic B3u component is more sensitive to impurity scattering than the isotropic Au component. The reduced B3u component in disordered samples would move the point nodes near the kz axis closer to each other (Supplementary Information IX), making the interference effect more prominent as observed. We note, however, that the detailed impurity effect on the chiral state requires more microscopic understanding, which deserves further studies.

We should mention that, while recent quantum oscillation measurements revealed quasi-two-dimensional Fermi surfaces (FSs) 23, their precise shape is still under debate. Some theoretical and ARPES studies suggest the presence of a heavy three-dimensional FS around the Z point, in addition to quasi-two-dimensional FSs mainly formed by U d orbitals and Te p orbitals14,21,24,25. The presence of the three-dimensional FS is also supported by the relatively isotropic transport properties26. These indicate that there are FSs near kx-, ky-, and kz-axes, which ensures the validity of our discussions on the nodal structure. Also, as for the spin fluctuations in UTe2, while NMR and μSR studies suggest the presence of ferromagnetic fluctuations27,28, recent neutron scattering measurements show only antiferromagnetic fluctuations and no ferromagnetic ones29,30,31. A possible origin of this descrepancy is the measurement time-scale because the NMR study suggests that the ferromagnetic fluctuations are as slow as the order of kHz, which is too slow to be detected by neutron scatterings. While the magnetic fluctuations are still highly controversial, the ferromagnetic fluctuations may play an important role to induce the B3u + iAu state, since the magnetic field along a-axis makes a degenerate state of the B3u and Au state. Thus, our experimental results promote further studies on the pairing mechanism in UTe2.

Finally, we note that our conclusion of the B3u + iAu pairing state is apparently different from the recent report of field angle-resolved specific heat measurements suggesting the point nodes only on the kx-axis8. However, our anisotropic measurements in the Meissner state probe the ground state in the zero-field limit, while the application of strong magnetic field can change the superconducting symmetry14,21,22,25. How the B3u + iAu state found here changes as a function of field also deserves further investigations.

Summary and perspectives

To sum up, we have found that the frequency shifts of TDO circuit follow Tn dependence with n nearly equal to 2 or less than 2 regardless of sample quality or magnetic field direction. This rules out any of single-component odd-parity states, and thus indicates a multi-component order parameter. The anisotropic penetration depth analysis reveals the presence of multiple point nodes near the ky- and kz-axes, which is most consistent with the chiral B3u + iAu superconducting state in UTe2. The presence of TRSB components splits the point node to multiple point nodes away from high-symmetry axes, and in analogy to topological Weyl points in Weyl semimetals, these nodes are expected to create surface arcs of zero-energy Majorana quasiparticles states34,39. Thus UTe2 is an ideal platform to investigate chiral superconductivity and its related topological physics. In particular, the positions of multiple point nodes (Weyl points) in the bulk studied here are fundamentally important to determine the topological properties of surface states.

Methods

Single crystals of UTe2 #B1, #C1, and #C2 were grown by the chemical vapor transport method with iodine as the transport agent. Crystal #A1 is grown by the molten salt flux method using a mixture of KCl-NaCl as flux15 and picked up from the same batch with the crystal showing the RRR about 1000. In both cases, a slightly uranium-rich composition was employed for the starting ratio of U to Te to avoid uranium deficiency17. The details of the characterizations of the crystals are shown in Supplementary Information V.

To obtain anisotropic components of penetration depth Δλa(T), Δλb(T), and Δλc(T) data separately, we have performed high-precision measurements of ac magnetic susceptibility shift Δχ(T) ≡ χ(T) − χ(0) using a tunnel diode oscillator technique operated at 13.8 MHz with weak ac magnetic field Hω along the three crystallographic axes40,41. The ac magnetic field Hω induced by the coil of the oscillator is the order of μT, which is much lower than the lower critical field of the order of mT in UTe220. In this technique, the frequency shift of the oscillator Δf(T) ≡ f(T) − f(0) is proportional to Δχ(T).

Specific heat capacity was measured by a long relaxation method in a 3He cryostat or a dilution refrigerator, where a Cernox resistor is used as a thermometer, a heater and a sample stage. The bare chip is suspended from the cold stage in order that it has weak thermal link to the cold stage, and electrical connection for the sensor reading. The samples are mounted on the bare chip using Apiezon N grease. The specific heat of the crystals is obtained by subtracting the heat capacity of bare chip and grease from the total data.