Unconventional superconductors can host topologically protected edge states provided that the right set of symmetries is broken at the superconducting transition temperature, Tc.

The superconducting state of UTe2 has attracted immense attention because several observations, including a temperature-independent NMR Knight shift ( 1 ), an anomalously large upper critical field (Hc2) ( 1, 2 ), re-entrant superconductivity at high fields ( 3 ), chiral behavior imaged by STM ( 4 ), and a point-node gap structure ( 5 ), all point to an odd-parity, spintriplet pairing state. However, the key question of whether time reversal symmetry is broken remains open. A prior attempt to measure time reversal symmetry breaking (TRSB) in UTe2 using muon spin relaxation was unsuccessful because of the presence of dynamic local magnetic fields ( 6 ). Furthermore, TRSB in UTe2 seems unlikely as the irreducible point group (D2h) representations of the orthorhombic crystal symmetry of UTe2 are all one dimensional ( 7 ). For a superconducting order parameter to break time-reversal symmetry, it needs to have two components with a relative phase. Owing to the presence of strong spin-orbit coupling in UTe2, which can be inferred from the strong anisotropy in its magnetic susceptibility ( 1 ), this means that the two components must belong to different irreducible representations of D2h. This would necessarily result in multiple superconducting transitions, a rare phenomenon only exhibited by three other systems: UPt3, Th-doped UBe13 and PrOs4Sb12 ( 8–10 ). In this study, we propose a multi-component order parameter that is experimentally supported by measurements of TRSB in UTe2, as well as two distinct phase transitions in specific heat measurements. Together, these experiments allow us to strongly constrain the symmetry classification of the order parameters to two unique candidates.

To test for possible time reversal symmetry breaking (TRSB) in the superconducting state of UTe2, we performed high resolution polar Kerr effects (PKE) measurements using a Zero-Area Sagnac Interferometer ( 11 ) (ZASI). In general, the generation of a Kerr effect arises from the unequal reflection (in both polarization and phase) of left and right circularly polarized light from a given material, resulting in reflected light relative to the incident light that is phase-shifted by a Kerr angle θK. The Kerr effect is not sensitive to Meissner effects, which normally prevent the measurement of global magnetic effects, and is therefore an optimal probe of TRSB in a superconducting system. At the same time, probing the system at frequencies ( ω ) much larger than the superconducting gap energy ( Δ ) will reduce a typical ferromagneticlike signal of order ~1 rad, by a factor of (Δ / ω)2 ~ 10−7 , where  is the reduced Planck constant, yielding a typical theoretically predicted signal of about 0.1-1 μrad ( 12–18 ).

However, owing to the high degree of common-mode rejection of the ZASI for any reciprocal effects (e.g., linear birefringence, optical activity, etc.), we are able to detect these small signals.

The design and operation of our ZASI interferometer is detailed in references ( 11, 19 ), and the basic operation is as follows: polar Kerr measurements are performed with 1550 nm wavelength light (20 uW incident power) that is polarized and then directed into a two-axes polarization maintaining (Page numbers not final at time of first release) 1 optical fiber that threads down into a He-3 cryostat until it reaches our UTe2 sample, which is mounted to a copper stage thermally anchored to the cold finger of the cryostat. There, a beam along each axis is reflected off the UTe2 crystal face (incident on the a-b plane of the crystal) and launched back up the opposite axis of the fiber. The two reflected beams which have both traveled along identical paths (passing through the same optical plates, lenses, and fiber axes) compose the arms of the Sagnac interferometer. Any relative phase shift between the two beams must arise from encountering the sample, and are revealed upon interference with one another to produce a signal from which the Kerr angle rotation can be extracted. The UTe2 single crystal used in this study and a basic schematic of the setup is shown in Fig. 1, A and B. Previously, this technique has been used to confirm TRSB in Sr2RuO4 ( 20 ) with a low-temperature saturation value of the Kerr effect of ~0.1 μrad. The heavy fermion uranium-based superconductors superconductors UPt3 ( 21 ) and URu2Si2 ( 22 ), and the filled-skutterdite PrOs4Sb12 ( 23 ) gave a larger signal of ~0.4 to 0.7 μrad, which is expected owing to their strong spin-orbit interaction. Crucially, testing the apparatus with reciprocal reflecting media such as simple conventional superconductors, gold mirrors, and the spin-singlet d-wave heavy-fermion compound CeCoIn5 ( 24 ), have yielded an expected null result.

To begin, we report the results of polar Kerr measurements performed at low temperatures on a single crystal of UTe2. The sample was first cooled below Tc of UTe2 (~1.6K) in ambient magnetic field (Hext <0.3Oe), and the Kerr angle was subsequently measured as the sample was warmed above Tc.

We find a small (~400 nrad at 300mK), field-trainable Kerr effect that onsets near Tc ~ 1.6K, which is consistent with a TRSB superconducting order parameter; Fig. 1C shows two runs performed identically in this manner. Whereas Run 1 shows a signal emerging around Tc, and saturating at ~500 nrad, Run 2 shows no discernible signal. This indicates that without an applied field, domains are formed in the sample that can orient in opposite directions, and give a finite signal or no signal at all, with an average signal of zero and a standard deviation dependent on the domain to beam size (10 um) ( 11 ). Similar results were found for a second UTe2 crystal, from a separate growth batch (fig. S1A). The detection of a finite positive Kerr signal indicates that a spontaneously large domain forms upon cooling the sample, due to TRSB in UTe2.

To orient all of the domains in one direction, the sample was cooled through Tc in a small applied field of +25 Gauss.

Experimentally, the magnitude of this training field has been found to be on the order of the lower superconducting critical field, Hc1 ( 21 ). Once the sample reaches base temperature (~300mK), the external field is removed, and the Kerr angle is measured as the sample is warmed slowly up past Tc. Figure 2 shows a positive finite Kerr value develop around Tc in this zero-field measurement. The sign of θK is reversed with a negative training field (-25 Gauss), indicating that the broken time-reversal symmetry shares the same symmetry as a magnetic moment. This is because trainability with field implies a linear coupling between the field and broken time-reversal symmetric order parameter.

The magnitude of the trained signal should indicate the maximum signal size when all of the domains are aligned in the same direction. Depending on the size of the domains as the sample is cooled through Tc, the signal can be very small, or close to the full signal that can be achieved with field-training field (see e.g., data on UPt3 in ref. 21). This may depend on the size of the sample, its purity and thus ability to pin domains, and the size of the probing beam. Additionally, small differences between signals in different runs may arise extrinsically owing to a change in background noise (from optical and electronic components and equipment) between runs. The measured Kerr signal was found to be independent of the magnitude of a small, applied training field (fig. S1B), indicating that the signal does not arise from magnetic vortices in the sample.

As discussed above, a TRSB order parameter in UTe2 can only be built out of two components belonging to two different irreducible representations ( 25 ). This makes UTe2 different from LaNiGa2, which also hosts TRSB superconductivity that develops out of a paramagnetic state ( 26 ). In that material, which also has a D2h point group, a multidimensional representation can be obtained by including the spin degree of freedom, an option that is precluded in UTe2 owing to spinorbit coupling. It is exceptionally rare for a system to support superconducting transitions in two symmetry channels, which might caution against our interpretation. However, we find direct evidence for the existence of two superconducting order parameters in the specific heat of UTe2. Normally, superconducting states are identified by resistivity or magnetization measurements, but neighboring superconducting states would both show zero resistance and diamagnetism.

For this reason, specific heat measurements have played a central role in identifying all previous examples of superconductors with multicomponent order parameters ( 8–10 ).

The specific heat at zero field was first measured using the small-pulse method with ΔT = 0.5 −1% ( 27 ). Four single crystals were measured from two growth batches ( 28 ). Figure 3 shows Cp/T near the superconducting transition for these four samples. In each case there is a shoulder-like feature at a temperature about 75 - 100mK above the peak in Cp/T. This feature is quite sharp and divides the jump in the specific heat into two local maxima in the derivative, d(Cp/T)/dT, representing two thermodynamic anomalies. The existence of two transitions seems to be stable to whatever perturbations are responsible for the notable difference in Tc between sample S4 and samples S1, S2, and S3. An additional three This picture can be checked by studying the two supersamples from a third growth batch also show two transitions conducting transitions as a function of magnetic field. The (fig. S4). The consistent splitting of Tc across seven samples symmetry considerations that suggest that TRSB can be despite differences in growth conditions and absolute value trained with a field applied along the c-axis imply that the of Tc provides firm support for our inference that this split- lower temperature transition should broaden and vanish ting is intrinsic to UTe2, reflecting the presence of a multi- with increasing field applied along that direction. This folcomponent order parameter, and is not an artifact of inclu- lows from the fact that the term ~iHc (ψ1ψ2* − ψ1*ψ2 ) leads to sions or intergrowths in these crystals. Furthermore, the inclusion scenario would not help explain the observed Kerr signal, because the order parameter has to have two components locally for a Kerr effect to exist. This is an essential point; prior observations of split superconducting transitions were only accepted slowly by the community because of the possibility that the two observed transitions were the result of inhomogeneity in the crystals. However, the fact that both the specific heat data and the polar Kerr data point to a twocomponent order parameter in UTe2 makes the conclusion compelling in the present case. This is especially true given that these two measurements probe the material on different scales: a 10 μm spot on the surface in the case of the Kerr effect, and the bulk of the crystal in the case of specific heat.

Finally, recent work has shown that there are two well-separated transitions under pressure ( 29, 30 ), which supports the idea that there are two nearly degenerate symmetry breaking possibilities for a superconducting state in UTe2. Both studies show that the splitting diminishes rapidly with decreasing pressure, with one study confirming the existence of splitting at ambient pressure ( 30 ), suggesting that the small splitting evident in our heat capacity experiments may be sensitive to sample quality and difficult to discern in other published work ( 2, 31 ).

Beyond demonstrating that the order parameter in UTe2 is two-component and TRSB, our data provide strong constraints on the particular irreducible representations to which the two components of the order parameter belong.

Our observation that the TRSB in UTe2 can be trained by a magnetic field along the crystallographic c-axis requires the a linear coupling between the two order parameters when a c-axis field is present. Symmetry considerations preclude the existence of terms like these for fields along the a- and b-axes.

Therefore, we expect the two transitions to remain distinct when a magnetic field is applied along those axes, but not when a field is applied along the c-axis.

The field dependence of the split Tc transition was measured on samples S1 and S2 by a large-pulse method ( 28 ) using oriented fields up to 5T in a vector magnet (Fig. 4). The crystal orientation was determined by measuring the field angle dependent Tc for a field of 2T. For fields oriented along the aand b-axes, there are two discernable features at all fields. For field along the c-axis, the two transitions broaden in field so that the splitting is no longer discernable above ~2T, consistent with what we expected based on the trainability of the TRSB with a c-axis field. The fact that both the trainability of the Kerr signal and the magnetic field dependence of the specific heat point to the same symmetry classification of the order parameters confirms that the two phenomena are connected and intrinsic to UTe2.

We thus arrive at a consistent picture of a superconducting state characterized by two order parameters that belong either to B3u and B2u, or B1u and Au, and that have a relative phase of π / 2 , leading to a TRSB state. Next, we turn to the question of the appearance of nodes. Thermal conductivity and penetration depth measurements suggest point node excitations ( 5 ). These have been interpreted as originating from the symmetry required points nodes of a time-reversal invariant odd-parity state ( 5 ). Naively, TRSB gaps these point nodes, leading to a fully gapped state which appears inconsistent with these experiments ( 5 ). However, topological arguments allow for the possibility of Weyl point nodes ( 32– 34 ), which we now consider in more detail. Weyl nodes are topologically protected by an integer Chern number, Z, and have associated surface Fermi arc states ( 32–34 ). For an oddparity superconducting state in a Kramer's doubly degenerate pseudo-spin band, the single-particle gaps in the quasiparticle spectrum for the two pseudo-spin-species are in general different and are given by ( 25 ) presence of a term ~iHc (ψ1ψ2* − ψ1*ψ2 ) in the free energy, where Hc is the c-axis component of the magnetic field and ψ1, ψ2 are the two components of the superconducting order parameter. Symmetry requires that this term exists for only four possibilities ( 28 ): 1 - ψ1 ∈ B3u and - ψ2 ∈ B2u 2 - ψ1 ∈ B1u and - ψ2 ∈ Au 3 - ψ1 ∈ B3g and - ψ2 ∈ B2g 4 - ψ1 ∈ B1g and - ψ2 ∈ A1g using the notation for irreducible representations adopted in Ref. ( 5 ). However, because UTe2 is known to be a spin-triplet superconductor, we can narrow the possibilities to the first two: B3u and B2u, or B1u and Au.

E+/− = ( (k ) − μ )2 + d (k ) 2 ± q (k ) 2

Here the gap function is Δ (k ) =d (k ) ⋅ σ  (iσ y ) (with Pauli matrices acting in pseudo-spin space), and q (k ) =d (k ) ×d * (k ) denotes the non-unitary part that naturally arises when time reversal symmetry is broken. In the two possibilities discussed above, the gap function is given by d =d1 + id2 , where d1 and d2 are both real. This gap function then gives rise to the single particle gaps Δ 2 ± = d (k ) |2 ± q (k ) |2 = d1 |2 + d2 |2 ±2 d1 × d2

=   d1 |2 + d2 |2 ±2 sin θ d1   d2 where θ is the angle between d1 and d2 . Nodes can only appear in Δ− and for this to occur two conditions must be met: i) d1 ⋅ d2 =0 ( sin θ = 1 ) and ii) | d1 =| d2 . In general, these two conditions will be satisfied on a line in momentum space. If this line intersects the Fermi surface, there will be a Weyl point. If it does not, the superconductor will be fully gapped.

Consequently, for the two gap structures discussed above, Weyl nodes can occur at arbitrary momenta on the Fermi surface.

Although the above argument reveals that Weyl points can generically occur, it does not guarantee that they exist in UTe2. Surprisingly, it can be shown that Weyl points are expected for the B2u + iB3u state. Given the current lack of detailed understanding of the electronic structure that gives rise to superconductivity, it is not possible to precisely identify the momentum dependence of the gap structure. However, symmetry places constraints on this. The symmetry dictated form of the of the corresponding gap functions are

and dB2u = fz,2 (k ) xˆ + fu,2 (k ) yˆ + fx,2 (k ) zˆ dB3u = fu,3 (k ) xˆ + fz,3 (k ) yˆ + f y,3 (k ) zˆ , where the unknown functions fx,i , f y,i , fz,i , fu,i share the same symmetry properties as kx , ky , kz , kxky kz . To find Weyl points for such a gap it is helpful to use insight for the Weyl semi-metal MoTe2 where it was found that Weyl points appear in mirror planes ( 35 ).

In particular, consider kx = 0 , then fx,i =fu,i =0 by symmetry. This immediately implies dB2u ⋅ dB3u

=0in this mirror plane. Furthermore, the nodal condition dB2u = dB3u implies that g z ≡ fz,22 − fz,32

=fy,32 . It is also possible to carry out a similar analysis for the mirror plane ky = 0 , for which dB2u ⋅ dB3u

=0is also satisfied. In this case, dB2u = dB3u implies −g z

=fx,22 . The relative minus sign in the two expressions g z = f y,32 ( kx = 0 ) and −g z =fx,22 ( ky = 0 ) ensures Weyl point will exist, provided that the cross section of the Fermi surface in both the kx and ky = 0 planes has a circular topology (to see, this note that g z = 0 for kz = 0 , fx,i = 0 , kx = 0 and f y,i = 0 for ky = 0 so that one of these two expressions must be satisfied somewhere on a closed Fermi surface encircling the origin in the kz − kx or kz − ky planes). Recent angle-resolved photoemission experiments suggest that such a Fermi surface exists ( 36 ), revealing a likely f-electron derived Fermi surface surrounding the Z point in the Brillouin zone. Consequently, this state will give rise to at least four Weyl points in either the kx = 0 or the ky = 0 planes. A similar analysis for the A + iB1u state ( 28 ) shows that, in this case,

u Weyl nodes are not guaranteed but are likely.

The above considerations, together with prior evidence for point nodes in UTe2 from thermal conductivity and penetration depth measurements ( 5 ), support the conclusion that UTe2 is a Weyl superconductor. The Weyl points in the superconducting gap would give rise to surface Fermi arc states, providing a natural explanation for the observation of chiral surface states ( 4 ). These findings suggest the possibility of using UTe2 for topological quantum computing, as well as for the exploration of superconducting analogs to phenomena in Weyl semimetals, including Fermi arcs and unusual Hall effects ( 37 ).

ACKNOWLEDGMENTS

SUPPLEMENTARY MATERIALS science.sciencemag.org/cgi/content/full/science.abb0272/DC1 Materials and Methods Supplementary Text Figs. S1 to S4 Table S1 References ( 39–42 ) 5 February 2020; resubmitted 30 June 2020 Accepted 30 June 2021 Published online 15 July 2021 10.1126/science.abb0272 (Page numbers not final at time of first release) 8