Abstract
Chiral spintriplet superconductivity is a topologically nontrivial pairing state with broken timereversal symmetry, which can host Majorana quasiparticles. The heavyfermion superconductor UTe_{2} exhibits peculiar properties of spintriplet pairing, and the possible chiral state has been actively discussed. However, the symmetry and nodal structure of its order parameter in the bulk, which determine the Majorana surface states, remains controversial. Here we focus on the number and positions of superconducting gap nodes in the ground state of UTe_{2}. Our magnetic penetration depth measurements for three field orientations in three crystals all show the powerlaw temperature dependence with exponents close to 2, which excludes singlecomponent spintriplet states. The anisotropy of lowenergy quasiparticle excitations indicates multiple point nodes near the k_{y} and k_{z}axes in momentum space. These results can be consistently explained by a chiral B_{3u} + iA_{u} nonunitary state, providing fundamentals of the topological properties in UTe_{2}.
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Introduction
Since the discovery of superconductivity in the nonmagnetic uraniumbased compound UTe_{2}, the nature of its superconducting state has been extensively studied^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31}. Recent studies have reported several anomalous superconducting properties, including extremely high upper critical field much beyond the Pauli limit^{1,2,3}, reentrant superconductivity^{3,4}, and little reduction of the Knight shift in the nuclear magnetic resonance (NMR)^{1,5}. These results suggest oddparity spintriplet pairing in UTe_{2}, as in the case of uraniumbased ferromagnetic superconductors^{32}. The symmetry of the superconducting order parameter is closely related to the superconducting gap structure, and the previous studies of lowenergy quasiparticle excitations, such as specific heat, thermal transport, and magnetic penetration depth measurements^{6,7,8}, support the presence of point nodes, consistent with the spintriplet pairing states.
More intriguingly, recent scanning tunneling microscopy^{9}, optical Kerr effect^{10}, and microwave surface impedance measurements^{6} suggest timereversal symmetry breaking (TRSB) in the superconducting state at ambient pressure. As the sign of imaginary part changes under the timereversal transformation, the chiral TRSB state requires multiple order parameter components in complex form. We note that highpressure studies reveal several superconducting phases^{11,12,13}, suggesting the presence of multiple order parameters under pressure. Thus, UTe_{2} is a prime candidate of a topological chiral spintriplet superconductor. However, the symmetry of the oddparity 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 UTe_{2}.
The crystal structure of UTe_{2} (Fig. 1a) is classified into the point group D_{2h}, whose irreducible representations (IRs) of oddparity order parameters are listed in Table 1. In the cases of B_{1u}, B_{2u}, and B_{3u} states, point nodes in the superconducting gap function exist on the k_{z}, k_{y}, and k_{x}axes (Fig. 1b), respectively, while the A_{u} state is fully gapped. When the oddparity 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 lowtemperature superfluid density λ^{−2}(T), which is determined by thermallyexcited quasiparticles near the nodes, depends strongly on the directions of the shielding supercurrent density j_{s} 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), Δλ_{α}(T) follows T^{2} dependence, while for perpendicular axes β and γ, Δλ_{β}(T) and Δλ_{γ}(T) should follow T^{4} dependence (Table 1)^{33}. Here, the subscript i of λ_{i} represents the direction of supercurrent density j_{s}.
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 possible^{10,14}. Considering the B_{3u} + iεB_{1u} (B_{3u} + iεA_{u}) state, for example, where ε is a sufficiently small real number, a point node of the B_{3u} 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 number^{10,14}, and corresponding Majorana arc surface states are expected^{34}. 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 directiondependent physical quantities, such as Δλ_{i}(T), we can pin down the superconducting symmetry among the nonchiral and chiral states listed in Tables 1 and 2, respectively.
We use three independent measurements of resonant frequency of the tunneldiode oscillator (see Methods) with weak ac magnetic field along the a, b, and caxes, 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 H_{ω}∥α is the sum of Δλ_{β}(T) and Δλ_{γ}(T) components and thus follows T^{4} dependence, while Δf(T) for H_{ω}⊥α should follow T^{2} 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 zerofield limit at ambient pressure.
Results
Specific heat
Figure 2ac show Δf(T) in three single crystals of UTe_{2} 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 reported^{15}. The clear superconducting transition at T_{c} = 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 multicomponent 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 parameters^{11,35}. However, a recent study^{16} 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 controversial^{17,18}. Figure 2e shows the lowtemperature electronic specific heat C_{e}/T in crystal #A1 as a function of \({(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\). Here, the subtracted phonon contribution is estimated from the previous measurements^{19}. We can find large quasiparticle excitations following \({C}_{{{{{{{{\rm{e}}}}}}}}}/T\propto {(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\) down to 0.12T_{c}, which is an expected behavior in a pointnodal superconductor. Furthermore, the residual electronic specific heat γ_{0} estimated from the extrapolation of the \({C}_{{{{{{{{\rm{e}}}}}}}}}/T\propto {(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\) relation is 4.6 mJK^{−2}mol^{−1}, leading to γ_{0}/γ_{n} = 0.038, where γ_{n} is the Sommerfeld coefficient in the normal state. The small γ_{0}/γ_{n} value confirms the high quality of crystal #A1. We note that the upturn in C_{e}/T observed below 0.12T_{c} is caused by the nuclear Schottky contribution^{15}. While the observed \({C}_{{{{{{{{\rm{e}}}}}}}}}/T\propto {(T/{T}_{{{{{{{{\rm{c}}}}}}}}})}^{2}\) behavior and small γ_{0}/γ_{n} 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 lowtemperature penetration depth data.
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. 3ac, respectively. The black solid lines represent fitting curves for the data below 0.3T_{c} using the powerlaw function, Δf(T) ∝ T^{n}. We note that the exponent values n = 2.11 (#A1), 2.07 (#B1), and 1.95 (#C1) for H_{ω}∥c are consistent with the previous abplane penetration depth studies^{6,7}. 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. 3df, where all data show almost linear or convex downward curvatures at low T. Thus, our results contradict any cases of singlecomponent oddparity order parameters, in which Δf(T) should follow T^{4} 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 H_{ω}∥a and H_{ω}∥b are smaller than that for H_{ω}∥c in crystals #B1 and #C1. Considering that Δf(T) consists of two Δλ_{i}(T) components perpendicular to the magnetic field (Fig. 1f), our exponent analysis on these samples indicates that the exponent value of Δλ_{c}(T) is smaller than those of Δλ_{a}(T) and Δλ_{b}(T), which will be discussed in more detail below.
For further investigations of the gap structure, we extract Δλ_{i}(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 Δλ_{i}(T)/λ_{i}(0) for three supercurrent directions i = a, b, and c, in which evaluations of λ_{i}(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 H_{c2}(T), when for simplicity we ignore the anisotropy of gap function (see Supplementary Information X). From the H_{c2}(T) data of an ultraclean sample^{20}, 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 studies^{6,7}, we obtain λ_{a}(0) = 1420 nm, λ_{b}(0) = 710 nm, and λ_{c}(0) = 2750 nm.
The obtained normalized Δλ_{i}(T)/λ_{i}(0) as a function of T/T_{c} for three directions of crystals #A1, #B1, and #C1 are shown in Fig. 4ac, respectively. First of all, the exponent values n_{i} obtained from the powerlaw fitting in Δλ_{i}(T)/λ_{i}(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, n_{c} = 1.60 in crystal #C1 and n_{c} = 1.84 in crystal #B1 are smaller than n_{a} ≈ n_{b} ≈ 2. The relatively small n_{c} 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 powerlaw dependence of Δλ(T) ∝ T^{n} (see Supplementary Information VII). Another consequence of Δλ(T)/λ(0) results is that the quasiparticle excitations along the b and caxes are much larger than those along the aaxis, implying a highly anisotropic nodal structure. Figure 4df 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/T_{c}. Compared with theoretical curves for the single order parameter with the supercurrent density j_{s} parallel and perpendicular to the direction of point nodes I, the amount of the excitations along the aaxis is clearly smaller than the j_{s}∥I case, while those along the b and caxes are as large as the j_{s}∥I case.
Discussion
Having established that our anisotropic superfluid density data exclude the singlecomponent oddparity order parameters, we now discuss the superconducting gap structure based on our experimental results. Adding two order parameters with preserving timereversal 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 B_{u} 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 n_{i} > 2 of Δλ_{i}(T) are expected for the direction perpendicular to the plane. However, experiments show n_{i} ≲ 2 for all directions, suggesting that these chiral states are unlikely to be realized in UTe_{2}. Thus, we focus on the chiral superconducting states consisting of the A_{u} and a B_{u} IRs. Considering the observed large quasiparticle excitations along the b and caxes, we conclude that the B_{3u} + iA_{u} pairing state is most consistent with our experiments (see Table 2). The reason is that for the B_{3u} + iA_{u} state with similar sizes of ∣d_{B3u}∣ and ∣d_{Au}∣ components, multiple point nodes can exist near the k_{y} and k_{z}axes, leading to larger excitations along the b and caxes than along the aaxis. Thus, the quicker decrease of our ρ_{s,b} and ρ_{s,c} data than the ρ_{s,a} data (Fig. 4df) is consistent with the B_{3u} + iA_{u} pairing state. This state can be supported by a recent theoretical study based on the periodic Anderson model, which suggests almost equally stable B_{3u} and A_{u} states at ambient pressure^{21}. We note that the chiral B_{3u} + iA_{u} state is nonunitary with finite d × d^{*} (see Supplementary Information IV), and for the system close to a ferromagnetic quantum critical point, theory shows that such a nonunitary complex order parameter may become stable^{22}. Moreover, recent studies of NMR Knight shift ^{5} suggest finite \(\hat{y}\) and \(\hat{z}\) components of d, which is consistent with the B_{3u} + iA_{u} state. However, we note that the anisotropies of λ_{i}(0) and \({\xi }_{i}^{1}\) are not completely the same in the case of anisotropic superconductors (Supplementary Information X). To confirm the chiral B_{3u} + iA_{u} state, more direct measurements of λ_{i}(0) are highly desired.
Next we discuss the sample dependence of the small exponent value n_{c}. We found that the n_{c} value systematically approach 2 as T_{c} gets higher (Fig. 4ac), suggesting that impurity effect is related to the n_{c} 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 scatterings^{36}, quantum criticality^{37}, or nonlocal effects^{38}. However, these possibilities can be excluded in the case of UTe_{2} (see Supplementary Information III). In the presence of point nodes in the gap structure, the impurity scattering can affect only the amplitude of the lowenergy excitations, but does not symply change the exponent value of Δλ(T) ∝ T^{2} ^{6,33}. Here we propose an interference effect of the two point nodes located closely as the origin of the sampledependent n_{c} values. In the B_{3u} + iA_{u} state, the main contributions to Δλ_{c}(T)/λ_{c}(0) come from two pairs of two point nodes near the north and south poles along the k_{z}axis (Fig. 1e). When the distance of the two nodes near the pole gets sufficiently short, the lowenergy 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 B_{3u} + iA_{u} state can indeed lead to an exponent near the experimental n_{c} value when the two point nodes are sufficiently close to each other (see Supplementary Figs. S12 and S13). Moreover, the observed trend that n_{c} gets smaller for samples with lower T_{c} can be consistently explained if we consider that the anisotropic B_{3u} component is more sensitive to impurity scattering than the isotropic A_{u} component. The reduced B_{3u} component in disordered samples would move the point nodes near the k_{z} 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 quasitwodimensional Fermi surfaces (FSs) ^{23}, their precise shape is still under debate. Some theoretical and ARPES studies suggest the presence of a heavy threedimensional FS around the Z point, in addition to quasitwodimensional FSs mainly formed by U d orbitals and Te p orbitals^{14,21,24,25}. The presence of the threedimensional FS is also supported by the relatively isotropic transport properties^{26}. These indicate that there are FSs near k_{x}, k_{y}, and k_{z}axes, which ensures the validity of our discussions on the nodal structure. Also, as for the spin fluctuations in UTe_{2}, while NMR and μSR studies suggest the presence of ferromagnetic fluctuations^{27,28}, recent neutron scattering measurements show only antiferromagnetic fluctuations and no ferromagnetic ones^{29,30,31}. A possible origin of this descrepancy is the measurement timescale 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 B_{3u} + iA_{u} state, since the magnetic field along aaxis makes a degenerate state of the B_{3u} and A_{u} state. Thus, our experimental results promote further studies on the pairing mechanism in UTe_{2}.
Finally, we note that our conclusion of the B_{3u} + iA_{u} pairing state is apparently different from the recent report of field angleresolved specific heat measurements suggesting the point nodes only on the k_{x}axis^{8}. However, our anisotropic measurements in the Meissner state probe the ground state in the zerofield limit, while the application of strong magnetic field can change the superconducting symmetry^{14,21,22,25}. How the B_{3u} + iA_{u} 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 T^{n} dependence with n nearly equal to 2 or less than 2 regardless of sample quality or magnetic field direction. This rules out any of singlecomponent oddparity states, and thus indicates a multicomponent order parameter. The anisotropic penetration depth analysis reveals the presence of multiple point nodes near the k_{y} and k_{z}axes, which is most consistent with the chiral B_{3u} + iA_{u} superconducting state in UTe_{2}. The presence of TRSB components splits the point node to multiple point nodes away from highsymmetry axes, and in analogy to topological Weyl points in Weyl semimetals, these nodes are expected to create surface arcs of zeroenergy Majorana quasiparticles states^{34,39}. Thus UTe_{2} 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 UTe_{2} #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 KClNaCl as flux^{15} and picked up from the same batch with the crystal showing the RRR about 1000. In both cases, a slightly uraniumrich composition was employed for the starting ratio of U to Te to avoid uranium deficiency^{17}. The details of the characterizations of the crystals are shown in Supplementary Information V.
To obtain anisotropic components of penetration depth Δλ_{a}(T), Δλ_{b}(T), and Δλ_{c}(T) data separately, we have performed highprecision 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 H_{ω} along the three crystallographic axes^{40,41}. The ac magnetic field H_{ω} 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 UTe_{2}^{20}. 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 ^{3}He 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.
Data availability
The data that support the findings of this study are available within the paper and its Supplementary Information. Source data are provided with this paper.
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Acknowledgements
We thank S. Fujimoto, J. Ishizuka, T. Matsushita and Y. Yanase for fruitful discussions, and N. Abe, and T. Arima, M. Konczykowski, and Y. Tokunaga for technical supports. This work was supported by GrantsinAid for Scientific Research (KAKENHI) (Nos. JP22H00105, JP21J10737, JP21H01793, JP21KK0242, JP20H02600, JP19H00649, JP18H05227, JP16KK0106), GrantinAid for Scientific Research on innovative areas “Quantum Liquid Crystals” (No. JP19H05824), GrantinAid for Scientific Research for Transformative Research Areas (A) “Condensed Conjugation” (No. JP20H05869) from Japan Society for the Promotion of Science (JSPS), and CREST (No. JPMJCR19T5) from Japan Science and Technology (JST).
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K.H. and T.S. conceived the project. K. Ishihara, M.R., M.K., K.H., and T.S. performed magnetic penetration depth measurements and analyzes the results. K. Imamura and Y.M. carried out specific heat measurements. H.S., P.O., Y.T., and Y.H. synthesized and characterized single crystals of UTe_{2}. K. Ishihara, K.H., and T.S. prepared the manuscript with inputs from Y.H. All authors discussed the experimental results.
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Ishihara, K., Roppongi, M., Kobayashi, M. et al. Chiral superconductivity in UTe_{2} probed by anisotropic lowenergy excitations. Nat Commun 14, 2966 (2023). https://doi.org/10.1038/s4146702338688y
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DOI: https://doi.org/10.1038/s4146702338688y
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Journal of Low Temperature Physics (2024)
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