Orientation of point nodes and nonunitary triplet pairing tuned by the easy-axis magnetization in UTe 2

The gap structure of a novel uranium-based superconductor UTe 2 , situated in the vicinity of ferromagnetic quantum criticality, has been investigated via speciﬁc-heat C ( T , H ,(cid:2) ) measurements in various ﬁeld orientations. Its angular (cid:2) ( φ,θ ) variation shows a characteristic shoulder anomaly with a local minimum in H (cid:2) a at moderate ﬁelds rotated within the ab and ac planes. Based on theoretical calculations, these features can be attributed to the presence of point nodes in the superconducting gap along the a direction. Under the ﬁeld orientation along the easy-magnetization a axis, an unusual temperature dependence of the upper critical ﬁeld at low ﬁelds together with a convex downward curvature in C ( H ) were observed. These anomalous behaviors can be explained on the basis of a nonunitary triplet state model with equal-spin pairing whose T c is tuned by the magnetization along the a axis. From these results, the gap symmetry of UTe 2 is most likely described by a vector order parameter of d ( k ) = ( b + i c )( k b + ik c ).

Exotic superconductivity arising near ferromagnetic instability has been intensively studied for uranium-based superconductors [1], such as UGe 2 [2], URhGe [3], and UCoGe [4].These materials are itinerant ferromagnets but become superconducting even in the ferromagnetic phase.A remarkable feature is the upper critical field H c2 exceeding the Paulilimiting field.Furthermore, field re-entrant (reinforced) superconductivity occurs under high magnetic fields along the hard-magnetization axis in URhGe and UCoGe [1,5,6], in which spins of Cooper pairs would be polarized along the field orientation or the hard-magnetization axes.These facts demonstrate that these uranium-based superconductors are promising candidates of spin-triplet superconductors.The results of NMR measurements suggest that ferromagnetic spin fluctuations play a key role in mediating superconductivity [7,8].
Recently, a novel uranium-based superconductor UTe 2 has been discovered [9] and becomes a hot topic in the research field of superconductivity.Notably, it becomes superconducting at a relatively high T c of 1.6 K without showing a clear ferromagnetic transition.A first-order metamagnetic transition occurs under a magnetic field H at 35 T in H b with a critical end point at roughly 7 -11 K [10,11].NMR measurements revealed a moderate Ising anisotropy and suggest the presence of longitudinal magnetic fluctuations along the easy-magnetization a axis above 20 K [12].These facts imply that UTe 2 is close to ferromagnetic quantum criticality.Similar to the other three uranium-based ferromagnets, formation of spin-triplet Cooper pairing has been indicated by small decrease in the NMR Knight shift [13] and the large H c2 exceeding the Pauli-limiting field [14][15][16].Indeed, superconductivity survives up to an extreme high field of 35 T for H b, which is destroyed abruptly by the occurrence of a metamagnetic transition [14,15].Furthermore, re-entrant superconductivity arises under H beyond 40 T tilted away from the b axis toward c axis by roughly 30 degrees [15].These facts demonstrate that parallel spin pairing can be formed in UTe 2 .In other words, the vector order parameter is favorably aligned to the plane perpendicular to the a axis (i.e., d ⊥ a) at low fields.
One of remaining questions for UTe 2 is a large residual value of the Sommerfeld coefficient in the superconducting state, which is roughly half of the normal state value at T c .Whereas a non-unitary spin-triplet state was suggested to explain this feature in the early stage [9], magnetic contribution was recently proposed as a possible origin because the entropy balance between superconducting and normal states is not satisfied [17].Moreover, a primary question is the gap symmetry which is closely related to exotic pairing mechanisms.The presence of linear point nodes in the superconducting gap has been suggested from specific heat [9], nuclear relaxation rate 1/T 1 [13], penetration depth [17], and thermal conductivity [17] measurements.Although the results of recent STM experiments suggest a chiral order parameter [18], broken time-reversal symmetry in the superconducting state has not yet been detected from muon-spin-relaxation measurements [19].Furthermore, there is little information on the orientation of gap nodes.These issues need to be clarified from further careful experiments.
In this study, we have performed a field-angle-resolved measurement of the specific heat C(T, H, Ω) for UTe 2 , which is a powerful tool to identify the nodal structure [20][21][22][23][24]. Low-energy quasiparticle excitations detected by C(T, H, Ω) support that the superconducting gap possesses point nodes in the a direction alone.Furthermore, unexpected features, reminiscent of the Pauli-paramagnetic effect, were observed in H c2 (T ) and C(H) under H along the easy-magnetization a axis, although the Pauli-paramagnetic effect cannot destroy spin-triplet pairing with d ⊥ a when H a. To solve this puzzle, we propose a vector order parameter d(k) = (b + ic)(k b + ik c ), whose T c is tuned by the easy-axis magnetization.Single crystals of UTe 2 were grown by chemical vapor transport method [9].A single crystal with its mass of 5.9 mg wight was used in this study.The directions of the orthorhombic axes of the sample were confirmed by single crystal X-ray Laue photographs.The specific heat was measured using the quasi-adiabatic heat-pulse method in a dilution refrigerator.The addenda contribution was subtracted from the data shown below.The magnetic field was generated by using a vector magnet, up to 7 T (3 T) along the horizontal x (vertical z) direction.By rotating the refrigerator around the z axis using a stepper motor, the magnetic-field direction was controlled three-dimensionally.
Figure 1(a) plots C e /T in zero field and in the normal state (at 7 T for H a) as a function of temperature.Here, the phonon and nuclear contributions (C ph and C N , respectively) are subtracted, i.e., C e = C −C ph −C N ; the Debye temperature is set to 125 K and C N = 0.135H 2 /T 2 µJ/(mol K) is obtained by using a nuclear spin Hamiltonian for 123 Te and 125 Te nuclei (I = 1/2) with the natural abundances of 0.9 and 7%, respectively.In zero field, a superconducting transition is observed at T c = 1.56 K (onset).The jump size is as large as the previous results [16,17], ensuring high quality of the present sample.At low temperatures below 0.2 K, C e /T shows a rapid upturn on cooling, as already reported [17].To satisfy the entropy-balance law, it is expected that the normal-state C e /T is enhanced with decreasing temperature, as proposed in Ref. 17.However, in the normal state at 7 T for H a, C e /T does not show a substantial upturn at low temperatures.This result suggests that the normal-state C e /T varies with increasing H, as reported in the high-field measurements [25].
Figure 1(b) compares C e (T )/T at 7 T in several field orientations within the ab plane.Here, the field angle φ denotes the azimuthal angle measured from the a axis.Even with the same magnetic-field strength, the normal-state C e /T at 1.5 K becomes larger with tilting H away from the a axis.Furthermore, the entropy-balance law is not satisfied between the data at φ = 0 • and φ 0 • .These facts suggest that the normal-state C e /T of UTe 2 depends not only on the field strength but also on its orientation.
To examine the above possibility, the effects of H and its orientation on the normal-state C e /T have been investigated as shown in Fig. 2. Indeed, the normal-state C e /T changes with H at 1.8 K (> T c ) and shows a characteristic field-angle φ dependence under a rotating H within the ab plane.An anomalous peak-dip-peak feature in C e (φ) becomes evident around H b in the high-field region.This feature may be related to longitudinal spin fluctuations along the a axis because C e (φ) can be scaled approximately by the field component along the a axis, H a = H sin θ cos φ [26].The mechanism of this normal-state anomaly is beyond the scope of this paper and remains a future work.direction alone, which is consistent with a chiral p + ip type pairing concluded from STM experiments [18].With decreasing temperature as low as 0.15 K, a prominent peak appears around 0.2 T for H b and H c (see Supplementary Material [26]).This anomaly is probably related to the abnormal non-superconducting contribution in C(H)/T at low temperatures, which violates the entropy-balance law [17], and disturbs detection of low-energy quasiparticle excitations at this temperature.By contrast, C e /T remains to exhibit no clear anomaly at low fields for H a [Fig.3(d)].
In the high-field superconducting region for H a, C e (H) shows a convex downward curvature with increasing H at 0.15 K [Fig.3(d)].This high-field behavior is qualitatively different from the theoretical prediction for an axial state with two point nodes under H applied along the point-nodal direction [27] [open squares in Fig. 3(d)].Instead, this feature is apparently similar to the Pauli-paramagnetic effect which breaks Cooper pairs to make spins polarized along the field orientation [28,29].However, the Pauli-paramagnetic effect is not allowed for UTe 2 in H a because the a direction corresponds to the easy-magnetization axis and spins of triplet Cooper pairs (d ⊥ a) can be polarized in this direction.Therefore, unusual mechanism of spin-triplet superconductivity is required for UTe 2 .
The H-T phase diagram of the present sample is shown in Fig. 1(c the previous report from resistivity measurements [16].In this study, H c2 (T ) near T c is precisely determined from the thermodynamic measurements [Fig.1(d)].In sharp contrast to H c2 (T ) for H b and H c, H c2 (T ) for H a is clearly suppressed compared with the initial slope near H ∼ 0 [26].Again, the Pauli-limiting-like behavior is observed for UTe 2 in H a. A possible origin of these unusual phenomena will be discussed later.
To further investigate the gap anisotropy of UTe 2 , the fieldangle dependence of C e /T has been measured at 0.5 K in a rotating H within the ab, ac, and bc planes; the results are presented in Figs.4(a)-(c), respectively.As depicted in Figs. 2, θ (φ) denotes a polar (azimuthal) field angle measured from the c (a) axis.At a low field of 0.2 T, a local minimum exists in H a (φ = 0 • , θ = 90 • ).With increasing H, C e (Ω) around H a is gradually enhanced, and a shoulder structure appears in both C e (φ = 0 • , θ) and C e (φ, θ = 90 • ) at intermediate field angles tilted slightly away from the a axis, as indicated by arrows.For H bc, C e (φ = 90 • , θ) does not change drastically with increasing H [see Fig. 4(c)], suggesting that the effect of H c2 (θ) anisotropy is dominant even at low fields.This result supports that the superconducting gap has a rotational symmetry around the a axis.
It is noted that anomalous peaks are observed at φ = ±90 • (θ = 0 • and 180 • ) in Fig. 4(a) [4(b)], whose widths become narrower with increasing H [26].By contrast, such a sharp peak does not appear around H a. Plausibly, these anomalies are related to Ising-type spin fluctuations that are easily suppressed by H a , similar to the case at 1.8 K [Fig.2(b)], although detailed mechanisms remain unclear.
Let us discuss the gap symmetry of UTe 2 .On theoretical grounds, the low-temperature specific heat is proportional to the zero-energy quasiparticle density of states N(E = 0).The field and field-angle dependences of N(E = 0) calculated for a point-nodal superconductor were already reported in previous papers [27,30].The present observations in C(T, H, Ω), except for anomalous peaks in its angular dependence around H ⊥ a, are in good agreements with the calculated results based on a microscopic theory assuming the presence of linear point nodes in the gap along the a direction [30], as presented in Fig. 4(d).A slight deviation would mainly come from the effect of the H c2 anisotropy.The presence of linear point nodes has been indicated in the previous reports [9,13,17].Therefore, the present results, evidencing their orientation along the a direction [Fig.4(e)], lead to a conclusion that the orbital part of the order parameter for UTe 2 is a chiral state k b + ik c or a helical state k b c + k c b belonging to the B 3u representation classified for strong spin-orbit coupling [31][32][33][34].
Regarding a possible mechanism of anomalous behaviors in H c2 (T ) and C e (H) for H a, we here consider a phenomenological model based on Ginzburg-Landau framework in which the degeneracy of non-unitary order parameters d ∝ (b ± ic) with equal spin pairing (i.e., ∆ ↑↑ and ∆ ↓↓ ) is lifted by the easy-axis magnetization M a ; one of the order parameters (∆ ↑↑ ) arises at T c and the other (∆ ↓↓ ) appears at a lower temperature.In this model, T c of ∆ ↑↑ is written as T c (M) = T c0 + ηM a [31,35].Here, η is a positive constant coefficient.In general, M a has a non-linear component of H. Therefore, we can reasonably assume M a (H, T ) ∼ M 0 (T ) + α(T )H − β(T )H 2 at low fields by using positive coefficients α and β.The spontaneous magnetization or the root-mean-square average of longitudinal magnetization fluctuations, M 0 , breaks the degeneracy of ∆ ↑↑ and ∆ ↓↓ in zero field.Then, we obtain , where T * c0 = T c0 + ηM 0 (∼ 1.6 K).From this equation, it is suggested that the slope of H c2 (T ) is enhanced when H is sufficiently low [26], because of , but the slope is suppressed at higher fields due to the non-linear term in M a (H).Indeed, the slope of H c2 (T ) in H a for UTe 2 becomes small at low temperatures (in high fields) [see Fig. 1(d)].A similar behavior in H c2 (T ) was also reported for a re-entrant superconductor URhGe along the magnetic-easy axis direction in the lower-field superconducting phase [36].Furthermore, the convex downward curvature in the low-temperature C e (H) for H a [Fig.3(d)] can also be explained qualitatively by this model; if we assume C e (H)/C e (H = 0) ∼ H/H c2 (H) for the field direction parallel to the point nodes, . Under H along hard-magnetization axes, these unusual phenomena are not expected because M a does not change significantly with H. Thus, the present study may capture universal nature of non-unitary equal-spin triplet superconductivity.
On the basis of these results, the order parameter (b + ic)(k b + ik c ) is a leading candidate for the novel superconductivity in UTe 2 .In this case, a secondary superconducting transition is expected below T c , which was recently suggested by a sudden drop of 1/T 1 around T ∼ 0.15 K [13].In addition, a specific-heat anomaly was found within the superconducting state under hydrostatic pressure [37], suggesting a possible occurrence of multiple superconducting phases.In order to lift the degeneracy of the multiple order parameters in zero field, a spontaneous magnetization or very slow longitudinal spin fluctuations are needed.This requirement suggests a possibility that a short-range magnetic order develops in UTe 2 above T c , similar to the case of UPt 3 [38,39] which shows a double superconducting transition coupled with a short-range antiferromagnetic order [40,41].Although the proposed gap symmetry is not classified by group theory in D 2h , a chiral vector l pointing to the magnetic-easy axis may stabilize the proposed pairing via the energy of l•M .In the weak spin-orbit coupling case, SO(3) symmetry allows the b + ic state [31,32].
In summary, we have performed field-angle-resolved measurement of the specific heat on UTe 2 .Our results, in particular characteristic field evolution in C e (φ, θ), evidence that linear point nodes are located along the a direction in the superconducting gap.From this fact, the orbital part of the order parameter can be characterized by a chiral p-wave form k b + ik c or a helical state k b c + k c b. Furthermore, unusual H c2 (T ) and C e (H) behaviors have been found under H along the easy-magnetization a axis, which can be explained by a phenomenological model for a non-unitary equal-spin triplet pairing tuned by the easy-axis magnetization.On the basis of these findings, together with recent STM results [18], the vector order parameter Figures S1(a)-S1(b) show the field dependences of the specific-heat data C e /T of UTe 2 in three field orientations parallel to the a, b, and c axes at several temperatures.An anomaly at H c2 can be detected only for H a because of the limit of our measurement system.With decreasing temperature as low as 0.15 K, a prominent peak develops around 0.2 T for H b and H c. This anomaly may be related to the abnormal behavior in C e /T at low temperatures, which breaks the entropy-balance law.Unfortunately, this anomaly disturbs investigation of low-energy quasiparticle excitations at low temperatures.
Figures S2(a) and S2(b) show the field-angle φ dependence of C e /T at 0.15 and 0.5 K, respectively, in a rotating magnetic field within the ab plane.At high fields above 5 T, a specific-heat jump at H c2 is clearly detected in C e (φ).Furthermore, anomalous field-angle dependence is observed around H b and H c above 1 T; a peak develops and becomes sharper with increasing H.At 0.15 K, although the H c2 anomaly becomes small due to the suppression of the specific-heat jump at T c (H), the peak anomaly remains observed clearly.Such a peak anomaly does not arise around H a.
Figures S3(a)-S3(c) plot C e (φ)/T at 0.15, 0.5, and 1.8 K, respectively, as a function of the a-axis component of the magnetic field, i.e., H a = H sin θ cos φ.A sharp peak or dip exists in C e (H a ) centered at H a = 0 whose width is robust against the magnetic-field strength at any temperature.These results suggest that the specific heat in both normal and superconducting states is affected by longitudinal spin fluctuations along the a axis, which can be easily suppressed by H a .
II. Field orientation effect on T c at low fields Figure S4 compares the temperature dependences of C e /T at 0.2 T for H a and H c. Although the low-temperature H c2 is smaller in H a than in H c, T c (H) at 0.2 T is higher in H a. This result demonstrates that the initial slope of H c2 (T ) near T c is larger in H a. By contrast, at a slightly higher magnetic field of 0.75 T, T c (H) becomes lower in H a. These features clearly evidence that H c2 (T ) is suppressed on cooling compared with its initial slope at H ∼ 0 for H a.

III. Phenomenological model for non-unitary spin-triplet pairing
In general, spin-triplet pairing (S = 1) has spin degrees of freedom.Therefore, its multiple order parameters can be coupled with a magnetization.Here, we consider the case of non-unitary order parameters ∆ ↑↑ and ∆ ↓↓ , whose spins can be polarized along the easy-magnetization axis.If there exists a spontaneous magnetization or the root-mean-square average of longitudinal spin fluctuations along the easy-magnetization axis (M a ), the degeneracy of ∆ ↑↑ and ∆ ↓↓ is lifted and their transition temperatures are split.In this case, an onset T c can be described as with a positive constant coefficient η.In general, the low-temperature M a roughly behaves as Here, M * 0 is the temperature-independent part of M a .Because the temperature dependence of the upper critical field can be simply described as the slope of H c2 (T ) near H ∼ 0 can be expressed as The first term in the right-hand side of eq. ( 5) reinforces the initial slope of H c2 (T ).When we apply the magnetic field along the easy-magnetization axis, M a usually changes significantly with a non-linear term at low fields; e.g., M a (H) ∼ M 0 (T ) + α(T )H − β(T )H 2 . (6) Then, T c (M a ) changes as where T * c0 = T c0 + ηM 0 .The second term in the right-hand side of eq. ( 7) contributes to the increase of T c ; if it is dominantly large, the initial slope of H c2 (T ) near H ∼ 0 can even become positive.When the third term develops in the high-field region, the enhancement of T c is suppressed, and a relatively small slope can occur in H c2 (T ) at low temperatures in high fields.These features qualitatively match the experimental observations for UTe 2 .

FIG. 1 :
FIG. 1: (a) Temperature dependence of C e /T at 0 and 7 T for H a, where C e = C − C ph − C N .(b) Temperature dependence of C e /T at 7 T in various field orientations within the ab plane, where the field angle φ is measured from the a axis.(c) Field-temperature phase diagram in three field orientations parallel to the a, b, and c axes and (d) its enlarged view near T c .Dashed, dotted, and solid lines represent initial slopes of H c2 (T ) parallel to the a, b, and c axes, respectively.

FIG. 2 :
FIG. 2: (a) Field dependence of the normal-state C e /T at 1.8 K for H a and H b. (b) C e /T at 1.8 K as a function of the azimuthal field angle φ, taken under a rotating H within the ab plane, where the mirrored data with respect to φ = 90 • are also plotted (open symbols).Each dataset in (b) is vertically shifted by 0.01 J/mol K 2 .
FIG. 3: Field dependence of C e /T along the (a) a, (b) b, and (c) c axes at 0.25 and 0.5 K in the low-field region below 1.5 T. (d) C e (H)/T for H a at 0.15 K as a function of H/H c2 (circles), where µ 0 H c2 is 6 T. Open (closed) squares are the normalized zero-energy quasiparticle density of states, N(E = 0)/N 0 , obtained from theoretical calculations for an axial state with two point nodes under H parallel (perpendicular) to the nodal direction (taken from Ref. 27).

FIG. 4 :
FIG. 4: Field-angle dependences of C e /T at 0.5 K under several magnetic fields rotated within the (a) ab, (b) ac, and (c) bc planes, where θ is a polar angle between the magnetic field and c axis.Numbers labeling the curves represent the magnetic field in tesla.Solid lines in (a)-(c) show C e /T in zero field.Shoulder anomalies are indicated by arrows.In these figures, the mirrored data with respect to symmetric axes are also plotted (open symbols).(d) Calculated results of N(E = 0) normalized by N 0 for an axial state with two point nodes as a function of field angle, where θ = 0 • is the direction of point nodes (taken from Ref. 30).(e) The gap structure possessing point nodes along the a direction.
FIG. S1: Field dependence of C e /T at (a) 0.15, (b) 0.25, and (c) 0.5 K in three orientations along the a, b, and c axes.
FIG.S3: C e (φ)/T at (a) 0.15, (b) 0.25, and (c) 0.5 K in several magnetic fields rotated within the ab plane (θ = 90 • ) plotted as a function of the a-axis component of the magnetic field H a = H cos φ.