Evidence for odd-parity superconductivity underpinned by antiferromagnetism in heavy-fermion metal YbRh₂Si₂

Abstract

Topological superconductors, characterized by spin-triplet Cooper pairing, are important for exploring unconventional pairing mechanisms and protected quantum states. Yet experimentally established odd-parity, spin-triplet superconductors remain scarce. Here we demonstrate that the heavy-fermion compound YbRh₂Si₂ hosts distinct magnetic-field-tuned superconducting states, both Pauli limited and beyond this limit, revealed by high-resolution measurements of the complex electrical impedance. We also find that superconductivity is abruptly suppressed at the critical field associated with the primary antiferromagnetic transition of this compound. The onset of electro-nuclear spin density wave order enhances the superconductivity. We propose that this behaviour can be explained by the formation of a pair density wave that boosts a selected spin-triplet superconducting order parameter. Together, our findings indicate odd-parity superconductivity in YbRh₂Si₂ and point to one of the superconducting states being the topological helical phase.

Main

An unconventional superconductor is a macroscopic quantum state in which the wavefunction of the Cooper pairs varies over the Fermi surface, breaking the symmetry of the underlying material¹. If the pair condensate order parameter has odd parity, which in the simplest case of a spherical Fermi surface corresponding to pairs with orbital momentum L = 1, 3…, then the system may support topological superconductivity of different classes, in multiple superconducting states. However, candidates for odd-parity superconductivity are rare. Attention is currently focused on UTe₂ (refs. ^2,3) and CeRh₂As₂ (ref. ⁴). Other promising compounds include UGe₂, UGeRh, UCoGe (ref. ⁵), UBe₁₃ (ref. ⁶) and UPt₃ (refs. ^7,8). UPt₃ is noteworthy for having three odd-parity superconducting phases, one of which is a strong candidate for chiral superconductivity.

Odd-parity spin-triplet topological superfluid ³He exhibits both chiral and time-reversal-invariant phases with well-established order parameters⁹. The early identification of these superfluid states was possible because of the intrinsic purity and the lack of disorder in this unique quantum fluid, the simple spherical Fermi surface arising from the absence of crystal structure, weak spin–orbit coupling and the application of nuclear magnetic resonance to fingerprint the order parameters. Magnetic field⁹ and anisotropic confinement in regular geometries¹⁰ alter the relative stability of these phases. Moreover, new superfluid phases have been observed in ³He in aerogels^11,12,13,14. These results illustrate the impact of symmetry-breaking fields and disorder on odd-parity pairing states.

The present study focuses on the heavy-fermion metal YbRh₂Si₂ (ref. ¹⁵), with tetragonal D_4h symmetry¹⁶ and a complex Fermi surface^17,18. The electronic magnetism of YbRh₂Si₂ is highly anisotropic, reflected in a g-factor along the c axis, which is an order of magnitude smaller than that in the a–b plane¹⁹. The primary antiferromagnetic order AFM1, established at T_N = 70 mK, has small ~0.002μ_B staggered moments (here μ_B is the Bohr magneton)²⁰ and hitherto-unresolved magnetic structure. Superconductivity in YbRh₂Si₂ was revealed by the observation of diamagnetic screening, aligned with a magnetic transition at around 2 mK (ref. ²¹). The onset of superconductivity at around 8 mK has been subsequently observed in electrical transport, but this study resolved no feature near 2 mK (ref. ²²). Recent heat capacity measurements reveal an electro-nuclear spin density wave (SDW) order, which we refer to as AFM2, below T_A = 1.5 mK (ref. ²³), stabilized by the strong hyperfine interactions of ¹⁷¹Yb and ¹⁷³Yb. Figure 1a shows the magnetic phase diagram established for fields in the a–b plane²⁴.

Fig. 1: Magnetic and superconducting phase diagrams of YbRh₂Si₂ with an in-plane magnetic field.

a, Boundaries of the antiferromagnetic (AFM1 and AFM2) and paramagnetic (PM) phases inferred from calorimetry and magnetoresistance^23,24. The back-turn of the critical field H_N of AFM1 below 15 mK is well accounted for by a hyperfine field 〈b_hf〉 exerted on Yb electrons by Yb nuclear spins, averaged over Yb isotopes²⁴. b–e, Maps of resistance ${\rm{Re}}\,Z(T,H)$ for three samples (b–d) show sample-to-sample variation, in contrast to the reproducible transport signature of the Néel transition (e). For sample D, the measurements are limited by the critical field of the Al contacts. ${\rm{Re}}\,Z(T,H)$ is scaled by the normal-state resistance ${R}_{0}(H)={\rm{Re}}\,Z(11\,{\rm{mK}}, H)$. We identify the observed sharp contours in b–d with superconducting transitions in various parts of the heterogeneous samples (Extended Data Fig. 5). The magnetic phase boundaries (solid green and magenta lines reproduced from a) superimposed onto b–d highlight an abrupt suppression of superconductivity across the AFM1/PM phase boundary and markedly different superconducting behaviour inside the AFM1 and AFM2 phases. f, Superconducting signature of T_A in the kinetic inductance ${L}_{\rm{K}}={\rm{Im}}\,Z(T)/\omega$, shown for sample D at H = 0. This feature as a function of field is marked in b–d with open magenta circles. g–j, Contour classes observed in the AFM1 and AFM2 phases across samples B–D and their potential extrapolation beyond the magnetic phase boundaries.

Source data

Here we report the discovery of multiple superconducting states in YbRh₂Si₂, revealed by the high-resolution measurements of the complex electrical sample impedance as a function of temperature and magnetic field (both in the basal a–b plane and along the principal c axis of the tetragonal structure, referred to as the in-plane and out-of-plane directions, respectively). We establish that superconductivity is always underpinned by antiferromagnetism in this system. Weak non-uniformity in our single-crystal samples results in heterogeneous superconductivity, involving distinct superconducting states, both Pauli limited and exceeding the Pauli limiting magnetic field. This indicates spin-triplet pairing and allows us to identify one of the pairing states as the topological helical phase. Remarkably, this state is abruptly boosted on crossing from the electronic AFM1 phase into the electro-nuclear AFM2 phase. We propose a mechanism for this boost mediated by a spin-triplet pair density wave (PDW).

While the extreme conditions of a high magnetic field are important for UTe₂, superconductivity in YbRh₂Si₂ occurs at ultralow temperatures. This study required the development of precise and low-dissipation four-terminal superconducting quantum interference device (SQUID)-based transport measurement techniques with nano-ohm resolution, in situ sample Johnson–Nyquist noise thermometry and a mode to observe flux quantization (Methods, Extended Data Figs. 1–4 and Supplementary Note 1).

Signatures of superconductivity with in-plane magnetic field H⊥c

We study the superconductivity in YbRh₂Si₂ by measuring the complex electrical impedance Z (Methods). Figure 1b–d shows the key result—the temperature–magnetic field maps of the resistance (the real part of the impedance ${\rm{Re}}\,Z(T,H)$) of three single-crystal samples B–D in relation to the magnetic phase diagram (Fig. 1a) determined from samples A and B. The samples were selected for having similar residual resistivity ratio of RRR ≈ 50 and sharp reproducible signatures of the Neél transition at T_N = 70 mK (ref. ²⁵; Fig. 1e). The maps exhibit overall similarities: the onset of superconductivity at around 8 mK, re-entrant normal state below T_A at in-plane fields greater than about 10 mT and the abrupt suppression of superconductivity at critical field H_N of the AFM1 phase. Nevertheless, there are substantial sample-to-sample variations, which we understand in terms of heterogeneous superconductivity²⁶: ${\rm{Re}}\,Z$ drops below the residual normal-state value R₀ when superconducting regions appear and ${\rm{Re}}\,Z=0$ signifies the percolation of these regions. Each sample exhibits distinct contours of abrupt resistance change on the ${\rm{Re}}\,Z(T, H)$ maps (Extended Data Fig. 5) that we attribute to superconducting–normal phase boundaries in particular regions of the samples. The diverse field dependencies signify different superconducting order parameters stabilized in YbRh₂Si₂, and can discriminate between the many possible spin-triplet candidates.

A remarkable feature in ${\rm{Im}}\,Z(T)$ is the step-like drop at T_A (Fig. 1f), coincident with the sharp heat capacity peak that manifests the bulk transition between AFM1 and AFM2 phases²³. When fully imaginary, the impedance is proportional to the measurement frequency. Thus, we attribute ${\rm{Im}}\,Z$ to the kinetic inductance ${L}_{{\rm{K}}}\propto {n}_{{\rm{s}}}^{-1}$ (ref. ²⁷), reflecting the superfluid density n_s and geometry of the superconducting regions of the sample. This drop at T_A is a signature of a boost to superconductivity induced by the formation of the electro-nuclear SDW of the AFM2 phase²³. In samples C and D, the ${\rm{Im}}\,Z(T)$ signature of T_A is observed on the background of zero resistance (Fig. 1c,d). However, the resistance of sample B only vanishes at T_A (Fig. 1b), indicating that in this case, complete percolation requires the onset of superconductivity in previously normal regions of the sample, triggered by the boost in T_A.

Multiple superconducting order parameters and in-plane Pauli limit

The three classes of contours observed in the AFM1 phase on the ${\rm{Re}}\,Z(T, H)$ maps are illustrated in Fig. 1g–i: linear and parabolic suppression of the critical temperature with field, and non-monotonic field dependence, respectively (Extended Data Fig. 5). We focus on the second contour class—the parabolas—that we associate with the Pauli (paramagnetic) limit^28,29 with a large Maki parameter α ≫ 1 (ref. ³⁰). Figure 2 demonstrates a number of superconducting signatures with quadratic suppression of the critical temperature ${T}_{{\rm{c}}}(H)={T}_{{\rm{c}}0}[1-(H/H_{\rm{P}})^{2}]$ with in-plane field H, all with a reproducible ratio μ₀H_P/T_c0 = 0.76 ± 0.04 T K⁻¹ between the critical temperature T_c0 at zero field and critical field H_P at zero temperature.

Fig. 2: Signatures of Pauli limited superconductivity in resistance and flux quantization.

a, Quantization of magnetic flux in a loop comprising the YbRh₂Si₂ sample and conventional superconductors is detected by our SQUID-based transport measurement scheme (Methods). b–d, Parabolic contours with a reproducible H_P/T_c0 ratio are observed in all samples, despite strong sample-to-sample variations and arbitrary field orientations in the a–b plane. In samples B and C, the onsets of zero resistance and flux quantization are aligned. By contrast, sample D exhibits a zero-resistance state beyond the Pauli limit. Here the flux quantization is limited by the Pauli limited superconductivity in the contact regions.

Particularly clear parabolas are also exhibited in samples C and D above T_A by a complementary transport signature—the onset of flux quantization (Fig. 2 and Methods)—which demonstrates the macroscopic phase coherence of the superconducting state(s) in YbRh₂Si₂.

From the conventional expression for the Pauli limiting field ${\mu}_{0}H_{\rm{P}}=\varDelta\sqrt{2}/g{\mu}_{{\rm{B}}}$ (ref. ²⁸), we obtain a reasonable value Δ = 1.3k_BT_c0 of the energy gap at T = 0, considering the in-plane g-factor g_ab = 3.5 inferred from electron spin resonance³¹. In the presence of spin–orbit coupling, the pseudospin replaces the electron spin as a good quantum number^7,32. This results in a momentum-dependent g-factor of the quasiparticles involved in pairing⁷, which may affect the above gap estimate. Further corrections stem from the strong coupling effects³³, potential variations of the gap over the Fermi surface and finite susceptibility in the T = 0 limit in case of (pseudo-)spin-triplet pairing. Thus, the commonly used estimation of H_P from T_c is unreliable³³. By contrast, here we report the experimentally determined in-plane Pauli limiting field.

The two other contour classes we observe in AFM1 (Fig. 1g,i) exhibit significantly weaker suppression with an in-plane field than the parabolas. This demonstrates that the AFM1 phase of YbRh₂Si₂ also hosts superconducting state(s) that exceed the measured in-plane Pauli limiting field by an order of magnitude. This is a signature of spin-triplet pairing, as discussed below.

By contrast, in the AFM2 phase, all samples exhibit just one class of superconducting phase boundary (Fig. 1j), with a critical field of order 10 mT. This field is comparable to—but larger than—the Pauli limit observed above T_A; thus, we propose that below T_A, only one odd-parity order parameter is stable, the same as the state with the in-plane Pauli limit above T_A, but with a boosted gap and, hence, Pauli limiting field H_P. The abrupt boost to H_P at T_A, most clearly demonstrated by the flux quantization contours of samples C and D (Fig. 2c,d), is aligned with the kinetic inductance signature in ${\rm{Im}}\,Z(T)$ (Fig. 1f).

Phase diagram with out-of-plane magnetic field H∥c

When the magnetic field is applied along the c axis (Fig. 3), the superconductivity extends up to 0.6 T, close to the estimated critical field H_N of AFM1 for this field orientation²⁵ (Supplementary Note 2). The abrupt switching off of the superconductivity near H_N for both field orientations together with the clear superconducting transport signatures of T_A for H⊥c (Fig. 1b–d,f) demonstrates that the superconductivity in YbRh₂Si₂ is tuned by the magnetic orders and is only stable in their presence.

Fig. 3: Phase diagram of YbRh₂Si₂ with an out-of-plane magnetic field.

Superconductivity is observed up to the fields close to the critical field H_N of AFM1 (solid green line), estimated using the critical field anisotropy observed above 20 mK (ref. ²⁵; Supplementary Note 2). The kinetic inductance signature of T_A (step in ${\rm{Im}}\,Z(T)$) is observed all the way up to H_N and the re-entrant normal state below T_A is absent, in contrast to the in-plane fields; Fig. 1b–d. Our data indicate the PM/AFM1/AFM2 polycritical point, beyond which the PM/AFM2 phase boundary remains unchartered (shown schematically with a blue dotted line and indicated as ‘?’).

In the absence of calorimetry data for H∥c, we map the clear superconducting signature of T_A in ${\rm{Im}}\,Z(T)$ (Fig. 1f and Extended Data Fig. 3d). This phase boundary extends up to H_N, in contrast to H⊥c. Moreover, below T_A, there are no signs of a re-entrant normal state, exhibited by all samples with in-plane fields.

Considering the g-factor anisotropy g_ab/g_c = 20 (ref. ³¹), we estimate the Pauli limiting field along the c axis to be as high as 0.1 T above T_A and 0.3 T below T_A (Supplementary Note 3). These values are below H_N, suggesting that in both AFM1 and AFM2 phases, the superconductivity exceeds the Pauli limit for H∥c. According to this analysis, below T_A, the single superconducting state we identify has no Pauli limit for H∥c; above T_A, this property must be possessed by at least one of the possible superconducting states revealed by the H⊥c results.

Despite the strong anisotropy of magnetism and superconductivity in YbRh₂Si₂, the linearly suppressed features (Figs. 1b–d and 3) exhibit a similar initial slope dT_c/dH for H⊥c and H∥c. Attributing this behaviour to the conventional orbital suppression of superconductivity, we obtain a nearly isotropic coherence length of approximately 100 nm, consistent with a previous estimate²². This large value is reasonable for a heavy-fermion superconductor with low T_c observed in YbRh₂Si₂.

Identification of helical state

We now discuss the candidate superconducting order parameters in light of the observed selective anisotropic Pauli limit and the boost in superconductivity at the onset of electro-nuclear order SDW at T_A (Supplementary Note 4). In a crystalline superconductor, the possible order parameters are classified by the irreducible representations (IRs) of the crystalline symmetry group^32,34,35. For the D_4h group, which describes YbRh₂Si₂ above T_N (ref. ¹⁶), there are five odd-parity and five even-parity IRs. Even-parity (pseudo-)spin-singlet states would be Pauli limited for all field orientations. Thus, in view of the selective Pauli limit, we argue that odd-parity (pseudo-)spin-triplet pairing manifests in YbRh₂Si₂. These states are described by the d(k) vector that points in the direction of zero spin projection at a given position k on the Fermi surface. We now examine the five odd-parity IRs of the D_4h group: one-dimensional A_1u, B_1u, A_2u and B_2u, and two-dimensional E_u (refs. ^32,34,35). Strong spin–orbit coupling characteristic of heavy-fermion metals is predicted to further constrain d(k) of each IR, either to the easy axis d∥c or the easy plane d⊥c (refs. ^7,36). The D_4h symmetry may be lowered by the antiferromagnetism, strain and magnetic field. We assume the associated perturbation of the superconducting states we consider to be weak, consistent with the absence of in-plane anisotropy of the Pauli limit (Fig. 2).

Following these arguments, the superconducting state, which is Pauli limited for H⊥c, can be identified with the topological helical (planar) phase³⁷. Different orientations of the helical phase corresponding to the four one-dimensional IRs are ${\bf{d}}({\bf{k}})=\varDelta ({k}_{a}\widehat{{\bf{\kern.5pta\kern.5pt}}}+{k}_{b}\widehat{{\bf{b}}})$ (A_1u), ${\bf{d}}({\bf{k}})=\varDelta ({k}_{a}\widehat{{\bf{b}}}-{k}_{b}\widehat{{\bf{\kern.5pta\kern.5pt}}})$ (A_2u), ${\bf{d}}({\bf{k}})=\varDelta ({k}_{a}\widehat{{\bf{\kern.5pta\kern.5pt}}}-{k}_{b}\widehat{{\bf{b}}})$ (B_1u) and ${\bf{d}}({\bf{k}})=\varDelta ({k}_{a}\widehat{{\bf{b}}}+{k}_{b}\widehat{{\bf{\kern.5pta\kern.5pt}}})$ (B_2u), where the unit vectors $\widehat{{\bf{\kern.5pta\kern.5pt}}}$, $\widehat{{\bf{b}}}$ and $\widehat{{\bf{\kern.5pt c\kern.7pt}}}$ denote the principal crystallographic axes. All four have isotropic Pauli limit in the a–b plane and no Pauli limit for H∥c, consistent with observations. These states have an identical gap structure, with point nodes at k∥c or nodeless on a cylindrical Fermi surface. The A_2u order parameter is distinguished by the predicted Josephson coupling to an s-wave superconductor along the c axis³⁸. The observation of flux quantization requires a superconducting current between YbRh₂Si₂ and Al contacts (Fig. 2a), and may point towards A_2u (Supplementary Note 4).

Boost to helical state at T _A via PDW

We provide further evidence for the helical order parameter by demonstrating how this superconducting state can be boosted at T_A (Fig. 4 and Supplementary Note 5). At the heart of the boost mechanism is a spin-triplet PDW d_Q(k), which couples to the helical order parameter d_H(k) and the staggered magnetization M_Q of the AFM2 SDW via $F=\lambda {\langle {\rm{i}}{{\bf{d}}}_{\rm{H}}^{* }\times {{\bf{d}}}_{{\bf{Q}}}+{\rm{h}}.{\rm{c}}.\rangle }_{{\bf{k}}}\!\cdot {{\bf{M}}}_{{\bf{Q}}}$. This interaction describes the diffraction of the Cooper pairs with amplitude d_H(k) by the SDW, thereby forming a PDW d_Q(k) ∝ d_H(k) × M_Q of the same wavevector Q, and lowering the condensate energy by an amount $\propto {\langle | {{\bf{d}}}_{{\rm{H}}}({\bf{k}})\times {{\bf{M}}}_{{\bf{Q}}}{| }^{2}\rangle }_{{\bf{k}}}$. For instance, with H∥a, the A_2u state acquires a modulation ${\bf{d}}({\bf{k}},{\bf{r}})={\varDelta}_{\rm{H}}({k}_{a}{\widehat{\bf{b}}}-{k}_{b}{\widehat{\bf{\kern.5pta\kern.5pt}}})+i{\varDelta }_{{\bf{Q}}}{k}_{b}{\widehat{\bf{\kern.5ptc\kern.7pt}}}\cos ({\bf{Q}}\cdot {\bf{r}})$. The resulting spin-triplet state is non-unitary and possesses a staggered magnetization collinear with the SDW.

Fig. 4: Ginzburg–Landau model illustrating the boost of the helical superconducting order parameter on entering the AFM2 phase.

a–d, Below T_A, the magnetic order parameter M_Q of AFM2 (a) boosts the energy gap Δ_H of the helical superconducting state (b) via the formation of a Cooper PDW with gap Δ_Q (c). The Pauli limiting field, proportional to Δ_H is also boosted at T_A. This attractive coupling is absent for the easy-plane nematic superconducting state, and competition between the superconductivity and AFM2 suppresses its gap Δ_N below T_A (d). The heterogeneity of the superconductivity is represented in this model by spatial variations of T_c0, the local transition temperature in the absence of M_Q order. M_Q induces the helical superconducting state near T_A in the regions of the sample in which T_c0 is much smaller than T_A.

Source data

In the regions of the sample with a pre-existing helical state, its gap Δ_H and, hence, the Pauli limiting field and superfluid density abruptly increase at T_A. Moreover, for regions in which the tendency towards the helical order parameter is weak (T_c0 < T_A), the superconductivity switches on at T_A simultaneously with the PDW. These two outcomes of the model successfully describe the superconducting transport signatures of T_A observed in all samples (Fig. 1b–d,f). This boost mechanism relies on the vector nature of the order parameters involved, implying odd-parity spin-triplet pairing.

Other superconducting states

We now consider the regions of the sample in which in the AFM1 phase, the superconductivity exceeds the in-plane Pauli limit. If the easy-plane spin–orbit locking (d⊥c), required by the helical state, also applies here, we identify the nematic phase ${\bf{d}}({\bf{k}})={\varDelta}_{\rm{N}}{k}_{c}\,{\widehat{\bf{\kern.25ptd\kern.75pt}}}$ with arbitrary in-plane orientation ${\widehat{\bf{\kern.25ptd\kern.75pt}}}\perp {\rm{c}}$. This state, characterized by a line node at k⊥c, belongs to the E_u IR (Supplementary Note 4). Like the analogous polar phase of the superfluid ³He (refs. ^11,39), it may host half-quantum vortices. To avoid the in-plane Pauli limit, ${\widehat{\bf{\kern.25ptd\kern.75pt}}}$ must adjust to be perpendicular to the field. The in-plane field is understood to have the same effect on M_Q (ref. ²³). Since both vectors are restricted to the a–b plane ${{\bf{M}}}_{{\bf{Q}}}\parallel {\widehat{\bf{\kern.25pt d\kern.75pt}}}$, the nematic state receives no boost from the AFM2 SDW via the vector triple product interaction. To account for the re-entrant normal state below T_A, our model includes direct competition between the superconductivity and SDW in the free energy, which destabilizes the nematic state below T_A (Fig. 4d).

The superconductivity beyond the in-plane Pauli limit can also be accounted for by any easy-axis (d∥c) order parameter. The key characteristic of this scenario is the out-of-plane Pauli limit. However, for this field orientation, the moderate Maki parameter α ≈ 1 prevents us from unambiguously identifying Pauli limited phase boundaries (Supplementary Notes 3 and 4). A strong easy-axis candidate is the E_u chiral phase ${\bf{d}}({\bf{k}})={\varDelta }_{{\rm{C}}}({k}_{a}\pm {\rm{i}}{k}_{b})\widehat{{\bf{c}}}$, an analogue of the topological ³He-A (refs. ^9,40). Importantly, the chiral and helical states have the same gap structure ∣d(k)∣ and are found to be nearly degenerate on a quasi-two-dimensional Fermi surface^41,42. Such competition between the chiral and helical states has been discussed in the context of Sr₂RuO₄, a tetragonal unconventional superconductor previously considered to be spin-triplet⁴³.

Discussion and conclusions

Our results agree with the previously reported signatures of superconductivity in YbRh₂Si₂ (refs. ^21,22), if one allows for up to 1-mK discrepancy between the temperature scales. The key observation of the boost to superconductivity at T_A (Fig. 1f) aligns with the abrupt increase in diamagnetic screening at the onset of electro-nuclear magnetism²¹. Flux quantization in the AFM1 phase (Fig. 2c,d) demonstrates a long-range superconducting order in the regime of weak diamagnetism above T_A, attributed to superconducting fluctuations in ref. ²¹.

Conventional transport measurements on YbRh₂Si₂ and ¹⁷⁴YbRh₂Si₂ (ref. ²²) are consistent with our observations in the AFM1 phase, but show no sign of T_A or re-entrant normal state, suggestive of insufficient sample cooling to reach the AFM2 phase. Importantly, two contour classes (Fig. 1g,i) reported in ref. ²² indicate the multiplicity of superconducting order parameters in YbRh₂Si₂. Our discovery of another class (Fig. 1h) experimentally establishes the in-plane Pauli limit and provides key signatures of the spin-triplet superconductivity and helical state.

A major outstanding question is what makes the superconductivity in YbRh₂Si₂ heterogeneous? Candidates include crystalline defects with density varying on a scale of the coherence length, local strain and magnetic domains. Our study focuses on samples in which any strain is weak, as indicated by a sharp and reproducible T_N. However, in another sample, in which shifted and broadened T_N reveals stronger strain, the signatures of superconductivity are qualitatively unchanged (Extended Data Fig. 6).

In conjunction with our determination of the magnetic phase diagram²⁴, the present transport study demonstrates the abrupt destruction of superconductivity at the critical field of the AFM1 phase, discussed as a quantum critical point^21,22,25,44. This calls into question the proposed significance of quantum criticality for the pairing mechanism^21,22,44. The interplay of superconductivity with antiferromagnetic orders rather suggests spin-fluctuation-mediated pairing⁴⁵, in which the balance between ferromagnetic and antiferromagnetic spin fluctuations plays an important role. Both types of spin fluctuation are well established in YbRh₂Si₂ (refs. ^{19,25,46,47,48}).

The triplet PDW driven by the AFM2 SDW, proposed in this work, is related to the singlet and triplet PDWs stabilized by pre-existing charge density waves in NbSe₂ (ref. ⁴⁹) and UTe₂ (ref. ⁵⁰), respectively. A defining characteristic of YbRh₂Si₂ is that we have access to superconductivity both with and without the SDW order. We stress that although T_A is below the onset of superconductivity, the hyperfine energy scale of the AFM2 SDW is much higher than the superconducting pairing energy; therefore, the superconductivity is simply responding to this new magnetic order (Supplementary Note 5).

Two experimental observations, unusual for superconductors, remain beyond our model. In the AFM1 phase, the enhancement in T_c with magnetic field (Figs. 1b,c,i and 3) qualitatively resembles the phase diagram of UTe₂ (ref. ³) and invites explanation in terms of the field-dependent strength of the pairing interactions^22,51; under a high field, we cannot rule out superconducting order parameters different from the states we identify at low fields. In the AFM2 phase, the decrease in the Pauli limiting field by several milliteslas on cooling below T_A (Fig. 1b,c,d,j) may be connected to hyperfine effects, which are also responsible for the back-turn of H_N(T) (Fig. 1a)²⁴.

In conclusion, we have found signatures of odd-parity spin-triplet superconductivity with strong interplay with two AFM orders in YbRh₂Si₂. We distinguish multiple superconducting states and propose one of them to be the topological helical phase, for which the observed boost to superconductivity at the onset of the electro-nuclear SDW, would be mediated by a spin-triplet PDW.

The challenge for YbRh₂Si₂ is to stabilize uniform bulk superconductivity, for example, by improved sample quality, elimination of residual strain²⁶ and application of uniaxial strain^52,53. Future goals include phase-sensitive measurements to confirm and complete the identification of triplet order parameters and the investigation of emergent surface states in the candidate topological superconducting phases using scanning tunnelling microscopy⁵⁴.

In the wider context of heavy-fermion superconductivity, the rarity of superconductivity in Yb-based heavy-fermion compounds, 4f hole analogues of Ce-based 4f-electron heavy-fermion metals⁵⁵ and the low critical temperatures of the two discovered examples, namely, β-YbAlB₄ (ref. ⁵⁶) and YbRh₂Si₂, remains a mystery. Nevertheless, the low transition temperatures in YbRh₂Si₂ confer advantages through modest energy scales: the magnetism and superconductivity are effectively tuneable by magnetic field, uniaxial strain and Yb isotopic substitution. The relatively long coherence length makes YbRh₂Si₂ suitable for single-crystal superconducting quantum devices with nanobridge junctions. In light of the increased accessibility of ultralow-temperature platforms⁵⁷ and associated techniques, there is future promise in this flexibility, coupled with the new technological opportunities offered by topological superconductivity.

Methods

Single-crystal samples with RRR ≈ 50 were grown from In flux^16,58. Sample A was probed with calorimetry^23,24,59 and samples B–E were configured for transport measurements (Supplementary Fig. 1 shows all five samples for comparison).

SQUID-based electrical impedance measurements

The circuit diagram and high-resolution measurements of sample and contact impedances are illustrated in Extended Data Fig. 1a. Au wires spot welded to the sample thermally ground it to the refrigerator, also acting as a current sink I−. Ultrasonically bonded Al wires make other electrical connections. A SQUID current sensor^60,61 with input inductance L_i ≈ 1 μH is connected across the voltage probes V+ and V−, measuring the fraction ∂I_i/∂I₀ of the drive current I₀ diverted from the sample into the SQUID input coil. Together with the response ∂I_i/∂Φ_ext to the flux drive Φ_ext, it yields the sample impedance as Z = −iω(∂I_i/∂I₀)/(∂I_i/∂Φ_ext). Simultaneously, we obtain the contact impedance Z_c = − iω(1 − ∂I_i/∂I₀)/(∂I_i/∂Φ_ext) − iωL_i, which includes contributions from the regions of the sample adjacent to the V+ and V− contacts, and provides further evidence for heterogeneous superconductivity (Extended Data Fig. 1b–d). Supplementary Note 1 provides a comparison of this technique to a conventional four-terminal impedance measurement. The real and imaginary parts of Z and Z_c may be subject to small offsets due to unaccounted phase shifts and parasitic inductive or capacitive coupling in the circuit. This is the probable origin of the apparent small negative ${\rm{Im}}\,Z$ near T_A (Extended Data Fig. 1b).

Sample noise thermometry

When the sample and/or contacts are resistive, the circuit acts as a current-sensing noise thermometer⁶², and the spectrum of the I noise provides direct measure of the sample temperature T_sample. Extended Data Fig. 2 demonstrates good thermalization of sample D in the re-entrant normal state well below 1 mK. We expect similar behaviour in all samples, including when ${\rm{Re}}\,Z\ll {R}_{0}$, due to the heterogeneous nature of superconductivity. In the essentially zero-resistance regime, the I_i noise is dominated by the SQUID noise and does not reveal T_sample.

Thus, in all the measurements on samples B, C and E and in the studies of sample D in fields smaller than 100 mT, we infer T_sample from the precise noise and ³He melting curve thermometry of the refrigerator platform, ignoring the temperature gradient between the sample and refrigerator, which we estimate to be less than 0.1 mK. This figure includes the temporal lag, particularly significant in the vicinity of the heat capacity peak at T_A (ref. ²³), where the thermalization time constant exceeds 1 h.

The study of sample D in fields exceeding 100 mT required the use of a commercial sample magnet, and was performed on a refrigerator lacking local thermometry of the sample holder. Here we used noise thermometry in the normal Al contact wires attached to the sample.

The analysis of noise spectra can be also used to determine the combined resistance ${\rm{Re}}\,(Z+{Z}_{{\rm{c}}})$ (ref. ⁵²). The setup used to probe sample D with H⊥c lacked the Φ_ext drive, and ${\rm{Re}}\,(Z+{Z}_{{\rm{c}}})$ inferred from the noise was used together with ∂I_i/∂I₀ to determine Z. This technique was limited to fields in which Al contacts are superconducting (Supplementary Note 1).

Z(T, H) maps

Samples B, C and E were probed with step-wise field sweeps at stable refrigerator temperatures, whereas temperature sweeps at fixed fields were used to study sample D (Extended Data Fig. 3). After the measurements with H⊥c, sample D was re-mounted and re-contacted for the H∥c study. A somewhat different part of the crystal was probed with lower R₀ and slight variation in superconducting properties.

The observation of flux quantization is illustrated in Extended Data Fig. 4a–c. Different techniques were used in experiments with (Extended Data Fig. 4c) and without (Extended Data Fig. 4b) the Φ_ext drive.