Unconventional hysteresis due to time-reversal symmetry breaking superconductivity in RbV_3Sb_5

Domain: physics
Published: 2026-01-17
Editorial outcome: major_revision

Abstract
The study of kagome materials has recently attracted much attention due to the presence of many electron-electron interaction-driven phases in a single material. In this work, we report the time-reversal symmetry-breaking superconductivity in the thin-flake kagome material RbV3Sb5. Firstly, when an in-plane magnetic field is swept in opposite directions, we observe an unconventional form of hysteresis in magnetoresistance, which is different from the hysteresis induced by extrinsic mechanisms. In contrast, no such hysteresis is observed in CsV3Sb5 samples below their superconducting transition temperature. Strikingly, at a fixed magnetic field, the finite-resistance state in RbV3Sb5 can be transitioned into the superconducting state by applying and subsequently removing a large current. Secondly, at temperatures around 400 mK, the re-entrance of superconductivity occurs during an in-plane field-sweeping process with a fixed sweeping direction. The observations of the unconventional hysteresis and re-entrance suggest the existence of time-reversal symmetry-breaking superconducting states in RbV3Sb5.

Introduction
The discovery of kagome superconductors1–4 AV3Sb5 (A=K, Rb, Cs) has garnered significant attention in recent years due to the intriguing interplay of superconductivity5–14, charge density waves5,15–25, nematic orders20,23, pair density waves6,12,13, all within a single material. Such intricate interplay of the intertwined orders holds promise for uncovering unconventional phenomena, notably unconventional superconductivity5–14. Indeed, the nodal superconducting states have been observed in the kagome superconductor RbV3Sb5 with μSR experiments9, while nodeless superconductivity was suggested in the CsV3Sb5 compound26–29. Up to now, the mechanism of the superconducting properties of the AV3Sb5 family remains an open question4, necessitating further investigation and exploration.

Here, we report the unconventional superconductivity in the kagome material RbV3Sb5. The observed superconducting states in RbV3Sb5 are characterized by the emergence of the in-plane magnetic field-driven unconventional hysteresis in the resistance measurements, followed by the zero-field superconducting diode effect observed in an irradiated RbV3Sb5 sample that breaks inversion symmetry. In addition, by heating the sample with a large current and subsequently removing it, all at a fixed magnetic field within the hysteresis loop, the sample can be brought from a finite-resistance state to a superconducting state. Moreover, an asymmetric re-entrance of superconductivity driven by the in-plane magnetic field is observed at around 400 mK, showing two superconducting domes in the temperature-magnetic field phase diagram. Through magnetotransport experiments, we observe a distinct anisotropic two-fold symmetry of the in-plane upper critical fields in this centrosymmetric material. These findings underscore the unconventional and time-reversal symmetry breaking nature of superconductivity in RbV3Sb5.

Results
Basicsuperconductingproperties
The kagome family AV3Sb5 represents a class of layered kagome superconductors, characterized by the alternating arrangement of V-Sb slabs, Sb slabs and A slabs. Specifically, RbV3Sb5 is a member of this family, which consists of V-Sb slabs interleaved with Rb layers. Each V-Sb slab exhibits a two-dimensional (2D) kagome net of V atoms, with Sb atoms occupying central positions of hexagons within the kagome lattice, as depicted in Fig. 1a. Evidently, this structural arrangement signifies a six-fold rotational symmetry. Figure 1b shows the calculated energy band of the kagome superconductor, highlighting an important feature near the Fermi energy: the parity mixing between the d-orbital of the V atom and the p-orbital of the Sb atom, introducing a spin-orbit-parity-coupling term30,31 in contrast to the conventional spin-orbit-coupling term. In the two-dimensional limit, this spin-orbit-parity-coupling term is responsible for the observation of the two-fold symmetric in-plane upper critical field in this centrosymmetric material30,31 which exhibits a nematic normal state20. In Fig. 1c, we present the current-voltage curves of the device at different temperatures, showcasing a characteristic superconducting behavior. Correspondingly, Fig. S1 shows the device image, in which the x-axis is roughly along the source-drain direction and the long axis of the sample. Figure 1d depicts the typical in-plane magnetic field dependence of the differential resistance dV/dI at a temperature T = 150 mK. Apparently, the critical current decreases with the magnetic field increasing, a hallmark of superconducting behavior. The in-plane superconducting coherence length is estimated to be around 137 nm according to the linearized Ginzburg–Landau (GL) expression32 Bc⊥ = Φ0 / 2πξ0^2(1 − T/Tc), where Φ0 is the magnetic flux quantum, ξ0 is the zero-temperature in-plane coherence length, and Tc is the critical temperature, as shown in Fig. S2.

Unconventional hysteretic superconducting behavior
We proceed by investigating the magnetic field dependence of the kagome superconductor RbV3Sb5. The configuration of the device is depicted in Fig. 2a. We employ a three-dimensional (3D) vector magnetic field to study the magnetic response of the superconducting states. In Fig. 2a, the direction of the magnetic field is defined by the θ and ϕ which are the polar angle and the azimuthal angle, respectively. Specifically, θ = ϕ = 0° denotes the direction of the x-axis. Remarkably, the magnetoresistance curve exhibits pronounced hysteretic behavior, as illustrated in Fig. 2b. When an in-plane magnetic field oriented along the x-direction is swept from −1 T to 1 T (red curve), the superconducting state is initially established at around −410 mT; then as the magnetic field turns positive, the resistance deviates from zero at about 210 mT, exhibiting pronounced asymmetry. Moreover, sweeping the field from 1 T back to −1 T (black curve), produces a mirrored response, thus uncovering hysteretic behavior. If we define the critical magnetic field by the magnitude of the magnetic field at which the resistance reaches 50% of the normal state resistance, then we get the critical values at the positive magnetic field during forward (denoted as B→) and backward (B←) sweeps as B→ = 380 mT and B← = 620 mT, respectively. Similar hysteresis is also observed when the magnetic field is applied along the y-axis as depicted in Fig. 2c. Figure 2e, f further show the hysteretic behavior observed during the field sweeping from 0 T to −1 T or 1 T, respectively. Notably, the critical field in the forward sweep is smaller than that in the backward sweep (B→ < B←), in sharp contrast to the trivial magnetic hysteresis (B→ > B←) typically observed in conventional superconductors33,34. This unusual behavior, as will be discussed later, points to broken time-reversal symmetry in the superconducting state. Additional evidence for such symmetry breaking is provided by the zero-field superconducting diode effect observed in the irradiated RbV3Sb5 sample that breaks inversion symmetry (Fig. S3).

An atural question is whether such hysteretic behavior is exclusively confined to the in-plane orientation. To answer this question, we conduct the out-of-plane magnetic field angle dependence of the hysteresis. As depicted in Fig. 2g, our findings demonstrate that at an out-of-plane angle θ = ±9°, the hysteresis is destroyed. Such out-of-plane magnetic field angle dependence suggests that the hysteresis is related to the in-plane magnetization of the superconducting state. Consequently, this rules out the possibility that the hysteresis is triggered by the out-of-plane component of the magnetic field.

The two-fold symmetric upper critical field
We further study the ϕ-dependence of the magnetoresistance under the in-plane magnetic field. The magnetoresistance as a function of ϕ are shown in Fig. 2d. A distinct two-fold symmetric behavior is evident. The extraction of the upper critical fields is further illustrated in Fig. S4a. As shown in the theoretical calculations (Fig. S4b) within the single-layer limit, the presence of the nematic order and the spin-orbit-parity-coupling (SOPC)30,31 can lead to a two-fold symmetric upper critical field (details can be found in the Supplementary Information) even when the orbital effects of the in-plane magnetic field are ignored. Due to the finite thickness of the sample, the orbital effects of the magnetic field can also induce this two-fold symmetry35,36, given the six-fold rotational symmetry is broken by the nematic order in the normal state20. The extra contribution of the orbital effect only changes the results quantitatively. The key conclusion of observed two-fold symmetric upper critical field is that the original D6h point group symmetry is reduced to D2h due to the presence of the nematic state.

The current heating and cooling experiment
In Fig. 2h, the result of the current heating and cooling experiment is depicted. Initially, as the magnetic field sweeps from −1 T to 270 mT (illustrated by the red curve) and the system enters into the state with non-zero resistance. At this point, we apply a large current to heat up the sample to suppress superconductivity completely. Subsequently, removing the current allows the system to cool down and enter into the superconducting state, as indicated by the red curve. Notably, with further increasing the magnetic field to 1 T, the resistance curve closely resembles the one observed during the magnetic field’s reversal from 1 T to 270 mT (shown by black curve). Furthermore, by repeating the current heating and cooling experiment while sweeping the magnetic field from 1 T to −270 mT (black curve), a similar phenomenon is observed. This finding clearly shows that the finite-resistance state within the hysteresis loop is a metastable state which has a higher energy than the ground state. As will be discussed below, this observation provides crucial evidence for the existence of superconducting domains in the sample.

Magnetic field driven re-entrance of superconductivity
Subsequently, an investigation into the temperature dependence of the magnetoresistance is studied. As temperature rises, superconductivity can be suppressed by thermal effects. Figure 3a, b show the temperature-magnetic field phase diagram and waterfall plot of the temperature dependence of the magnetoresistance curve, respectively, while the complete temperature-magnetic field phase diagram is available in the supplementary Fig. S6a. Remarkably, within Fig. 3a, two distinct superconducting domes are observed in the magnetoresistance measurements when the magnetic field changes from −1 T to 1 T. Initially, the superconductivity is destroyed at a finite magnetic field of 80 mT at 400 mK. However, a second dome of superconductivity is observed when the magnetic field becomes larger in the positive direction. Such a phenomenon of re-entrant superconductivity persists across a temperature span from 400 mK to 600 mK. When the sweep direction of the magnetic field is reversed, the corresponding mapping figure will be mirrored, as shown in Fig. S6b. Furthermore, the re-entrant superconducting states are also observed for magnetic fields in all in-plane directions, corroborated in the supplementary Fig. S7. We label these two distinct superconducting phases as “SC-I” (the superconducting dome obtained before the re-entrance of superconductivity) and “SC-II” (the superconducting dome obtained after the re-entrance of superconductivity) as shown in Fig. 3a. The temperature dependence of the re-entrant behavior is shown in Fig. 3b.

Figure 3c shows the θ-dependence of the magnetoresistance. It is shown that superconductivity in SC-II is eliminated when the out-of-plane magnetic field angle exceeds 6° while SC-I still exists. This observation underscores the greater fragility of SC-II to out-of-plane magnetic field components. To further display the two superconducting domes SC-I and SC-II of the material, we measured dV/dI as a function of the magnetic field and applied current at different temperatures37 as shown in Fig. 3d. Two superconducting domes SC-I and SC-II are clearly observed. Similar hysteresis and re-entrant behavior are reproduced and Figs. S8, 9 show the results for samples #2 and #3.

Ruling out trivial origins
Although hysteresis in magnetoresistance can be induced by superheating and supercooling effects33,34 or the flux-trapping effects38–40 as shown in previous works, the hysteresis loops observed in Fig. 2 are distinct from the hysteresis observed in previous experiments.

In the hysteresis induced by the superheating-supercooling effect, sweeping the magnetic field from 0 T to 1 T (forward sweep) and then gradually reducing the magnetic field from 1 T back to 0 T (backward sweep) must result in a higher upper critical field during the forward sweep compared to the backward sweep33. In contrast, our observations, as illustrated in Fig. 2f, reveal that during the forward sweep from 0 T to 1 T (indicated by the blue curve), the upper critical field B0 is smaller than the upper critical field B← in the backward sweep (indicated by the black curve). The same result is obtained when the magnetic field is swept from 0 T to −1 T and then reduced gradually, as shown in Fig. 2e by the blue and red curves. Such findings exclude the superheating and supercooling mechanism caused by the magnetic field induced local minima of the free energy of the superconductor41.

In addition, the hysteresis observed in our experiment exhibits behavior starkly distinct from that attributed to the flux-trapping mechanism. In the hysteresis induced by the flux-trapping, a magnetoresistance minimum at finite magnetic field can be observed in the backward-sweeping curve (returning curve)40,42,43. In contrast, as shown in Fig. 2b, c and Fig. S10 in Supplementary Information, in RbV3Sb5, no magnetoresistance minimum at finite field was observed in the backward-sweeping curve over a wider range of applied currents. Moreover, we conducted in-plane field-sweep experiments on over ten thin-flake kagome material CsV3Sb5. Although RbV3Sb5 and CsV3Sb5 belong to the same kagome material family and the thin flakes we used have similar thicknesses, no such hysteresis was observed in CsV3Sb5 below its superconducting transition temperature, as shown in Fig. S11. If the observed hysteresis were due to trivial mechanisms, such as flux trapping, rather than intrinsic properties of the superconducting states, it should also appear in CsV3Sb5 as well. This result suggests that the superconducting states of RbV3Sb5 and CsV3Sb5 are different, and the hysteresis originates from the intrinsic superconducting properties of RbV3Sb5. It is interesting to note that Knight shift experiments in CsV3Sb5 suggested spin singlet pairing44. However, since our control experiments suggest that the superconducting properties of RbV3Sb5 are quite different from CsV3Sb5, the pairing symmetry of RbV3Sb5 is possibly unconventional.

Furthermore, the re-entrant superconductivity observed in our experiments is different from the re-entrance observed in conventional superconductors45–47 which relies on vortex pinning induced by applied magnetic field. In contrast to some trivial re-entrant superconductivity observed only in specific directions, i.e. current parallel to magnetic field46, in RbV3Sb5, the re-entrant superconductivity appears in all in-plane magnetic field directions with respect to a fixed current direction (as shown in Fig. S7 of Supplementary Information). What’s more, the trivial reentrance induced by vortex pinning can only give rise to asymmetric B-T phase diagram45,47. This contradicts the asymmetric B-T phase diagram observed experimentally as shown in Fig. 3a. Therefore, the simultaneous observation of both in-plane magnetic field hysteresis and asymmetric re-entrant superconductivity rules out trivial origins, indicating the need for a novel explanation.

Possible pairing symmetries
Considering the key observations in experiments reported above, we discuss the possible pairing symmetries in the kagome superconductor RbV3Sb5. First of all, the observed novel hysteresis in magnetoresistance measurements suggests spontaneous breaking of time-reversal symmetry, which is further corroborated by the zero-field superconducting diode effect observed in an irradiated RbV3Sb5 sample that breaks inversion symmetry (see Fig. S3 in the Supplementary Information for details). It is noted that the time-reversal symmetry may be broken in the normal states of the kagome family AV3Sb5 due to loop currents in the charge density wave phase15,16,22,23. However, the loop currents would produce out-of-plane magnetization, in contrast to the in-plane magnetic moments observed in this work. Therefore, the in-plane hysteresis is likely caused by the superconducting state itself, rather than the normal state. Moreover, as shown in Fig. 2g, the magnetoresistance hysteresis is weakened by the out-of-plane component of the magnetic field. The reason is as follows: RbV3Sb5 is a quasi-two-dimensional superconductor48, where the out-of-plane resistivity is approximately 33 times greater than the in-plane resistivity at low temperatures. In such materials, an out-of-plane magnetic field rapidly suppresses superconductivity due to the orbital effects, which generates many vortices within the superconducting plane. When the applied magnetic field has a sizable out-of-plane component, many vortices will be created, and the vortices will dominate the transport properties of the superconducting films. As the transport is mostly dominated by vortices, the magnetoresistance behaves in a similar way for both the up-sweep and down-sweep fields, leading to the disappearance of hysteresis. A similar behavior is also observed in the heavy-fermion superconductor UCoGe49, where a slight tilt of the magnetic field toward the c-axis leads to a significant decrease in the upper critical field and the disappearance of hysteresis.

In addition to the pronounced hysteresis, the observation of re-entrant superconductivity as the magnetic field increases suggests that the SC-II phase can be stabilized by the magnetic field. After carefully ruling out trivial origins, the spin-singlet scenario cannot support the field-induced re-entrant superconductivity. This is because the Zeeman effect of the magnetic field tends to suppress the spin-singlet superconducting state by inducing a depairing effect on spin-singlet Cooper pairs, whereas it does not affect the equal-spin pairing states. Therefore, the reappearance of the superconducting state under a larger magnetic field is not consistent with the spin-singlet Cooper pairs scenario. Instead, the observed re-entrance is more consistent with equal-spin pairing, where the spin-magnetic moments of the Cooper pairs couple to the magnetic field, lowering the energy of the superconducting state. Moreover, within the hysteresis loop, the transition of the finite resistance state to the superconducting state upon applying and then removing the current indicates that the finite resistance state is a metastable state. This suggests that there can be different superconducting domains with opposite spin polarizations in the finite resistance state, similar to the case of a ferromagnet with magnetic domains. Furthermore, due to the centrosymmetric point group symmetry, the superconducting pairing should be odd parity due to the spin-triplet pairing50. Therefore, the orbital part of the pairing should be p-wave or f-wave in nature51–64. These large number of novel properties observed in the superconducting state place strong constraints on the pairing symmetry of RbV3Sb5.

Since the superconducting pairing is formed near the Fermi surface, we project the four-orbital (eight-band) model into the manifestly covariant pseudospin basis (MCPB)30,65,66 by choosing two degenerate bands (jk, αi, jk, αi) near the Fermi surface as the projecting targets. Here, k is the crystal momentum and α is the pseudo-spin index (The detail can be found in the Method Section and the Supplementary Information). It is important to note that the original point group symmetry of the material is D6h. Before the superconducting transition happens, the nematic order parameter comes in references20,23 and reduces the point group symmetry to D2h. Therefore, the superconducting order parameters can be constructed by using basis functions belonging to irreducible representations of the point group D2h. As mentioned above, this pairing state must possess net in-plane spin polarization. The possible superconducting order parameters represented by the d-vectors are listed in Table 1. Importantly, both of these are nodal topological p-wave, spin-triplet, time-reversal-symmetry-breaking states. As shown in the Supplementary Information, both of these states possess Majorana flat bands at the sample boundaries. In the following, we choose the B3u representation without loss of generality.

The matrix representation of the superconducting gap in the MCPB is Δ = (d·σ)iσ2, where the σi (i=1,2,3) are the Pauli matrices. The nodes of these two order parameters belonging to different representations are located on x-axis and y-axis, respectively. The parameter ξ controls the amplitude of the spin polarization: 1−η = 1−|cos(2ξ)2 + sin(2ξ)2|, which is determined by the specific form of the interactions in this material, while the parameter χ controls the spin-polarization direction of the Cooper pairs. The orientation of the spin polarization p of the Cooper pair is given by: p = id×d*/(sin(χ), −cos(χ), 0), which is coupled with the effective magnetic field gk in the MCPB in the form of −gk·p. Therefore, the direction of the spin polarization can be pinned by the applied magnetic field (See Supplementary Information for detail). When the magnetic field changes its sign, superconducting domains with opposite spin-polarization become energetically favorable and can be created.

A possible mechanism for the hysteresis, current heating and cooling experiments, and re-entrant superconductivity
Here, we provide a possible, schematic superconducting domain picture to explain the hysteresis in resistance, the current heating and cooling experiments as well as the reentrant superconductivity observed in RbV3Sb5. As discussed in last section, the system has spin-polarized triplet pairing components which can form the superconducting domains67,68 where Cooper pairs with net spin-polarization act like miniature magnets with different polarization directions, similar to ferromagnetic domains. At a low temperature for example (as denoted by T1 in Fig. 4), when the magnetic field sweeps from −1 T to 1 T, the system initially enters into a superconducting state dominated by negative domains. The negative domains are formed by spin-polarized Cooper pairs with magnetic moments pointing to the negative direction as indicated by blue arrows in Fig. 4. In other words, the spin-polarized Cooper pairs can form domains and the external magnetic field aligns the magnetization direction of the domains. When the magnetic field changes sign, superconducting domains with positive magnetization direction (positive domains) emerge. As the magnetic field further increases in the positive direction, positive domains become dominant, but the negative domains can still coexist. Therefore, at a positive magnetic field, we expect that the system enters into a metastable state characterized by the coexistence of the positive and negative domains as shown in Fig. 4. On the other hand, if the magnetic field is swept from 1 T to the same positive magnetic field, the domains are dominantly positive and the superconducting state will have a lower free energy. This provides a possible explanation for the hysteretic behavior of the resistance. Moreover, at a fixed positive magnetic field ~0.27 T (Fig. 2h), we can use a large current to heat the metastable finite-resistance state with both positive and negative domains and then remove the current under a fixed magnetic field, the system can transition to the superconducting ground state, where superconducting domains predominantly align with the applied magnetic field. This current heating and cooling experiment strongly supports the superconducting domain picture of our theory which suggests that the SC-II state is a metastable state containing superconducting domains with different spin polarizations. In addition, in Fig. S13, three curves of distinct colors correspond to different initial magnetic fields (1 T for the black curve, 0 T for the blue curve, and −1 T for the red curve) at the onset of the field sweeping. These disparate trajectories, as illustrated in Fig. S13, indicate that the magnetic response of the superconducting state in RbV3Sb5 is determined by the initial magnetic field, thereby further supporting the domain picture.

At a higher temperature (as denoted by T2 in Fig. 4), when the magnetic field sweeps from −1 T to 1 T, the system is initially in a superconducting state with negative domains. When the magnetic field changes sign from negative to positive, the negative domains shrink and the positive domains start to form as in the low temperature cases. However, due to the thermal effect, these domains are too small to be connected spatially to form a superconducting state. As the magnetic field increases, the positive domains become more energetically favorable to form. The enlargement of positive domains allows percolation of superconducting domains to give rise to superconducting state. This percolation picture gives a mechanism for the re-entrance of superconductivity at around 400 mK as observed in the experiment. Therefore, the nodal p-wave superconductivity belonging to the B2u or the B3u representation of the D2h point group, which enables Cooper pairs exhibiting net spin-polarization, can provide a reasonable explanation of all the observed novel phenomena in RbV3Sb5.

In summary, we report experimental evidence for time-reversal-symmetry-breaking superconductivity in RbV3Sb5. This conclusion is supported by several experimental observations. First, we observed a magnetoresistance hysteresis in the superconducting state. Importantly, the superconducting diode effect in inversion-symmetry-broken samples at zero field clearly demonstrated the time-reversal symmetry breaking properties of the superconducting state. Furthermore, the magnetic hysteresis is absent in our CsV3Sb5 samples, which suggests distinct superconducting mechanisms in the two systems, despite both belonging to the same kagome family. Our work suggests RbV3Sb5 is a new platform for realizing unconventional and time-reversal symmetry breaking superconductivity.

Methods
Sample fabrication
The thin films of RbV3Sb5 were exfoliated using polydimethylsiloxane (PDMS) from the bulk crystal onto silicon substrates with a 285 nm oxide layer. The fabrication of pre-patterned metal electrodes was achieved using the standard electron-beam lithography (EBL) method, followed by deposition of a Ti/Au (10 nm/50 nm) through electron-beam evaporation. Then, the device was encapsulated with hBN to prevent any degradation due to air atmosphere. All device fabrication procedures mentioned above were performed in a controlled environment within a nitrogen-filled glovebox with oxygen and water levels below 0.01 ppm to minimize any potential degradation of the samples.

Quantum transport measurements
The samples were measured in an Oxford dilution refrigerator Triton XL1000 equipped with a vector magnet. This setup allowed the samples to remain stationary while a real three-dimensional vector magnetic field was applied. Such a technique greatly improves the precision of the magnetic field angle-deviation measurements, effectively circumventing the issues of return differences and heat generation commonly associated with mechanical rotators.

The dV/dI measurements were conducted with a rather small alternative current (i.e., 100 nA or 10 nA) while applying a much larger direct current (i.e., 10 μA). To optimize the signal-to-noise ratio in these experiments, a standard lock-in technique was employed, along with the implementation of essential low-temperature filters.

Conducting the strict in-plane magnetic field measurements with the 3D vector magnet
1. Initially, we securely mounted the sample within the dilution refrigerator. Despite our meticulous efforts to align the sample with the magnetic field axis, a minor discrepancy persisted.
2. In response to this, our subsequent step involved the calibration of the vector magnetic field. What sets our approach apart from the traditional mechanical rotational system is that we did not need to physically rotate the sample plane, we could fine-tune the value of the vector magnetic field, which would greatly enhance the angle accuracy.
3. During the calibration process, we initially conducted angle-dependent measurements of the critical magnetic field of the sample in both the XZ and YZ planes. Given that our sample plane was mounted in the XY plane, these measurements allowed us to determine the angle deviation from the sample plane. With the calibration of the XZ-plane and YZ-plane, then the new coordinates are achieved, with strict alignment of the vector magnetic field within the XY plane. It’s worth noting that our device substrate consisted of a rigid silicon wafer with a 285 nm SiO2 layer. Through this dual-axis calibration process, we can confidently assert that the calibrated XY plane remains strictly within the sample plane, with an error margin of less than 0.1°.

Ruling out trivial origins of two-fold symmetric superconductivity
It has been suggested that the anisotropy may arise from current-induced vortex motion. Such a scenario was ruled out by different configurations in our previous paper69. The two-fold symmetric axis is independent on the current direction. Specifically speaking, in both cases that the current is applied in-plane with ϕ=90° and ϕ=0°, the in-plane azimuthal angular dependence of the magnetoresistance exhibited the same twofold symmetry pattern, irrespective of the current direction. Thus, the two-fold symmetry is independent of the current direction, ruling out the possible vortex origin.

Manifestly covariant pseudospin basis (MCPB)
Due to the existence of the inversion symmetry, though there is SOPC30,31, one can always define a pseudospin basis, in which the orbital and pseudospin degrees of freedom are decoupled. On the other hand, since the superconducting pairing happens near the Fermi surface, we can project the four-orbital (eight-band) model into a two degenerate band basis with different pseudospins, which is named as the manifestly covariant pseudospin basis (MCPB)30,65,66. In this basis, the magnetic field couples to the spin in this form:
Hz = μB Σk,α,β,i,j c†kα aij(k)σij βckβ, (1)
where μB means the Bohr magnetron, α, β label the pseudospin of the fermion in band basis, σi (i=1,2,3) are Pauli metrices. Here, aij(k) (i,j=1,2,3) come from the projection to MCPB (See Supplementary Information for detail), which change the original magnetic field Bi into a k-dependent effective magnetic field gk = μB a(k)B. Including the pairing order parameter, the total Hamiltonian of the superconducting state can be written as:
Htotal = Σk,α εk c†kα ckα + Σk,αβ [ (d·σ)iσ2 ] αβ c†kα c†−kβ + h.c. + Hz, (2)
where the d-vector is the possible order parameter of the odd-parity superconductivity which is show in Table 1. With the total Hamiltonian Htotal, one can calculate the topological properties and the magnetic responses conveniently. Importantly, this Hamiltonian describes a nodal topological p-wave superconductor which possesses the Majorana flat bands70–77 on the edges of the samples78–82 (See Fig. S13 in Supplementary Information for the result of the open boundary calculations).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In their manuscript entitled \"Spin-polarized p-wave superconductivity in the kagome material RbV3Sb5\", Shuo Wang et al. measure magnetotransport in devices of RbV3Sb5. They have three main observations: 1) hysteresis of the resistance in field; 2) field-reentrant superconductivity that is asymmetric in field direction; and 3) two-fold anisotropy in the in-plane critical field. They use these observations to motivate the presence of a chiral p-wave superconducting state.\nPossible chiral p-wave superconductors have a long and messy history, and thus these strong claims require equally strong evidence. Unfortunately, Wang et al. do not appear to provide it. My main concerns are:\n1) Hysteresis in superconductors usually arises due to vortex physics. Their argument against this is based on their quenching scheme, whereby they apply a large current to drive the sample normal, then cool it in a finite field. This resets the flux trapping state based on whatever field they cool in. They claim flux will be trapped at weak pinning sites that should be mobile enough be dissipative. I don’t know how they can make that claim. This depends entirely on the microscopics of the the pinning site strength, the amount of current, the energy barrier at the edge of the sample for vortices, etc, none of which are known. Also, the authors state that \"In the positive magnetic field region, we obtain a newly formed superconducting state with zero resistance\", but see point 3 below where I point out that zero resistance is never seen.\n-1b) Did the authors check how the hysteresis depends on the applied current? Do the authors get same hysteretic behavior with more or less current?\n2) The authors claim that the superconducting state breaks the C6 rotation symmetry of the lattice down to C2. However, this symmetry is already broken by the CDW at higher temperatures! See e.g. https://iopscience.iop.org/article/10.1088/1674-1056/acd8b1/pdf Once a symmetry is reduced, all measured quantities in general have that lower symmetry, e.g. the critical field is expected to be C2 symmetric in the CDW state even if it is a trivial s-wave state.\n-2b) There may be domains of the CDW that produce a higher effective symmetry at long length scales. However, this is a mesoscopic device with a very high aspect ratio that will naturally favor one domain of the others. This has the prediction that the anisotropy in the critical field could be device dependent, and indeed a subset of the present authors published exactly this data in an earlier paper https://www.nature.com/articles/s42005-023-01496-3#Sec10 that shows a MUCH higher anisotropy than the data in the current paper. Why should the anisotropy be device dependent if it was intrinsic to the superconducting state?\n3) The authors claim re-entrant superconductivity. However, the authors never measure a zero-resistance state at ANY field or temperature - there is always a substantial offset. Why is this the case? The reentrance only goes close to the zero-field resistance (which again, is not zero resistance) in a very, very narrow temperature range around 400 mK, otherwise it is just a dip in resistance.\n-3b) Very similar measurements were performed on the same material in an earlier paper by a subset of the current authors https://www.nature.com/articles/s42005-023-01496-3#Sec10 why was no reentrant superconductivity observed in this device?\n4) The absolute highest the upper critical field could be is 0.75 tesla for field in plane (taken as the field where the resistance becomes flat in Fig 2b). This is very close to the Pauli limit of ~1 tesla (calculated using a g-factor of 2). If RbV3Sb5 is spin triplet, why is the upper critical field Pauli limited for field in the plane?", "reviewer_2": "The authors present intriguing magnetotransport findings in kagome superconductor RbV3Sb5 thin films, including hysteresis in magnetoresistance, reentrance of superconductivity, and a two-fold symmetric in-plane upper critical field. By conducting additional measurements, they ruled out trivial explanations for these phenomena. The authors propose that spin-polarized p-wave pairing could account for the observed results. Currently, the pairing symmetry of kagome superconductors is still under debate from both experimental and theoretical perspective. The present results are interesting and offer more insights into the pairing state of thin-film kagome superconductors. However, it is important to note that this evidence presented is not definitive proof of p-wave pairing, particularly given knight shift measurements suggesting spin-singlet pairing in the bulk (Chin. Phys. Lett. 38 077402 (2021)).\nI suggest that the authors should temper their assertion of p-wave pairing in the manuscript and explicitly acknowledge the indirect nature of the evidence provided. Nevertheless, the paper offers valuable experimental insights and features a well-executed theoretical analysis. The authors should properly address the following concerns before my recommendation of publication:\n1. The authors discuss the band dispersion with the spin-orbit-parity-coupling term and its relevance to the two-fold in-plane upper critical field. However, according to fig.4a in PRL 125, 247002 (2020), the SOC gap between p- and d-orbital band along Gamma-M (located at 0.5 eV) is far away from the Fermi level. Why is such term relevant for the band dispersion and properties of kagome superconductor AV3Sb5?\n2. Clarification is needed regarding the measuring temperature in Fig.2h. Additionally, if a quenching process is involved in the measurement shown in Fig.3b, it is essential to determine if a similar zero resistance can be achieved for a positive magnetic field. If this is the case, can this be called re-entrant superconductivity by sweeping magnetic field (as superconductivity is not destroyed)? Conventionally, re-entrant superconductivity is characterized by the reappearance of superconductivity as the magnetic field increases after being suppressed by a magnetic field.\n3. How might the temperature-magnetic field phase diagram in Fig.2 be altered by conducting measurements with a fixed magnetic field and varying temperature?\n4. The bulk superconductivity (CsV3Sb5) is spin-singlet according to Chin. Phys. Lett. 38 077402 (2021). In contrast, the authors claim a spin-triplet pairing in RbV3Sb5 thin films, with a transition temperature close to that of the bulk material. It is difficult to understand why there will be a transition from spin-singlet to spin-triplet pairing between bulk and thin films. The author should provide more discussions about the difference between bulk and thin films. The relatively low upper critical field observed in the thin films challenges the expectation for a spin-triplet pairing. Will a time-reversal symmetry breaking spin-singlet pairing explain some of experimental phenomena? If not, possible mechanism of spin-triplet pairing in thin films should be further discussed.", "reviewer_3": "In this manuscript, the authors present magnetoresistance measurements of the kagome material RbV3Sb5 in order to investigate the nature of its superconducting state. One of their most notable findings is the hysteresis in magnetoresistance —namely in the boundaries of the superconducting state—for magnetic fields swept in opposite directions. The authors have measured this hysteresis for a variety of field directions and temperatures.\nThe authors have demonstrated that this hysteresis is present for all directions of in-plane fields, yet disappears with a small tilt of field outside the ab plane. They also show that there is a range of temperatures, roughly 400-600 mK, for which the superconductivity is split into two domes as a function of field, with a higher-resistance state appearing at roughly 80 mT for forwards field sweeps and -80 mT for backwards field sweeps between the two domes.\nIn addition, with the sample in a finite field, the authors have used applied current to heat their sample into the normal state and then removed the current and allowed it to cool. They have showed that this process can bring the sample from a finite-resistance state to a zero-resistance state. This supports the authors’ hypothesis that the finite-resistance state in the hysteresis loop is related to domain formation, rather than being due to flux trapping. They posit that this is due to time-reversal symmetry-breaking superconducting order that can couple to applied magnetic fields.\nThe hysteretic behavior presented here is novel and interesting. The authors have studied this behavior thoroughly in terms of field direction and temperature. Their use of current heating/cooling to further understand the finite-resistance state is excellent. The authors also put forward a sensible model by which to interpret their data, which indicates that the material is a topological superconductor, certainly of interest to the field.\nHowever, there are a few points that I would like to see addressed:\n•The authors note that “the AV3Sb5 family lacks magnetic orders in the normal state”, and thus attribute the time-reversal symmetry-breaking to the superconducting phase itself. However, as pointed out in the authors’ Ref. 4, the charge density wave orders of this family of materials do appear to break time-reversal symmetry. The authors should discuss whether time-reversal symmetry-breaking from the CDW of RbV3Sb5 could be the cause of the observed hysteresis.\n•The authors indicate that the superconductivity is spin-triplet based on the observed time-reversal symmetry-breaking. However, it is possible for spin-singlet superconductors to break time-reversal symmetry; see, e.g., Phys. Rev. B 105, 155101 (2022). If the authors have experimental or theoretical reasons to support this superconducting phase being spin-triplet, they should make those arguments explicitly.\n•In the explanation of their proposed mechanism, the authors do not address the effect of out-of-plane magnetic fields as seen in their measurements. In the model involving superconducting domains, why should an out-of-plane field component destroy hysteresis?\n•It would be useful to see data showing if there are any memory effects. For example, if the sample goes from -1 T to 1 T, back to -1 T and again to 1 T, does the initial forwards sweep look the same as the second forward sweep?\n•Given the difference in critical fields for different field directions, it is interesting to note that the split between SC-I and SC-II seems to occur at the same magnetic field value regardless of the magnetic field direction, if I am interpreting Fig. S6 correctly. I am curious if the authors have any comment on this behavior.\n•The similarity between the hysteresis seen in this compound and that observed in UCoGe is interesting; see Appl. Phys. Lett. 98, 132507 (2011). The hysteresis in UCoGe has the same qualitative behavior and also disappears for fields tilted slightly out of plane. I believe this similarity merits at least a mention, if not discussion.\nIn addition, certain points in the text could benefit from clarification:\n•In the abstract:\n“Strikingly, at a fixed magnetic field, the finite-resistance state can be quenched to the zero-resistance state by applying a large current.”\nIf I understand correctly, the statement should be: “…by applying and subsequently removing a large current.” It also might be worth noting here that the purpose of the current is sample heating.\n•Lines 13-15:\n“In addition, at a fixed magnetic field, a large current can suppress superconductivity completely at the finite-resistance state by heating. Incredibly, upon the removal of the current and cooling, the sample can be quenched to a zero-resistance state.”\nI understand why the authors refer to superconductivity in the finite-resistance state, having read the full article. For the sake of introducing the subject to new readers, I suggest something like the following:\n“In addition, by heating the sample with a large current and subsequently cooling it, all at a fixed magnetic field within the hysteresis loop, the sample can be brought from a finite-resistance state to a zero-resistance state.”\n•Lines 35-36:\n“Each V-Sb slab exhibits a two-dimensional (2D) kagome net of V atoms, forming a hexagonal structure with Sb atoms occupying central positions…”\nI believe it would be more accurate to say,\n“Each V-Sb slab exhibits a two-dimensional (2D) kagome net of V atoms, with Sb atoms occupying central positions of the hexagons within the kagome lattice…”\n•Throughout the text, “quench” is used to refer to the heating of the sample above the superconducting temperature as well as the cooling of the sample. This terminology is somewhat confusing, as “quench” can be used to refer to superconducting magnets unintentionally becoming normal, or to a quick-cooling process. It may be more clear to refer simply to heating and cooling via current.\n•Were all sweeps starting at a finite magnetic field actually cooled in that field, as opposed to zero-field-cooled? If so, was in-field cooling always done from room temperature, or from some other temperature?\n•In the section “A proposed mechanism…”, when discussing the current heating/cooling experiment, the authors could be more explicit that in their model, cooling under an applied field results in superconducting domains that are predominantly aligned with the field, thus leading to a zero-resistance state.\n•In Fig. S3a, “lower critical field” is used. I believe the authors are referring to the critical field that is at a lower value due to hysteresis. However, given that “lower critical field” already has a defined meaning for superconductors, a different term should be used (perhaps “greater” and “lesser” critical fields?) and the caption should clarify what is meant by these.\n•The caption of Fig. S6 indicates that it is showing data for a variety of magnetic field directions, presumably with the current direction fixed. However, the text (lines 166-168) says, “In RbV3Sb5, the re-entrant superconductivity appears in all current directions with respect to a fixed in-plane magnetic field direction (as shown in Fig. S6 of Supplementary Information).” Which of these is accurate?\n•Additionally, at what temperature were the data for Fig. S6 taken? This should be indicated in the figure caption.\n•Similarly, at what temperature were the data for Fig. 3c taken? This should be indicated in the figure caption.\n•Is the normal-state resistance of the sample actually 1 Ohm, as shown in Fig. 1d, or is this a normalization?\n•The authors cite Ref. 22 on line 143 when discussing the superheating-supercooling effect. However, I do not believe that Ref. 22 touches on superheating/supercooling. Is this the correct reference?\nThe following changes are also suggested to improve figure legibility:\n•Adding tick marks along the axes for all figures that don’t have them:\n2d, 3a, 3c, S4, S5a&b, S6a\n•Possibly adding arrows and/or labels to Fig. 2h along the “quenching” segments, to indicate the heating and subsequent cooling\n•Changing the color of font used for “T1” and “T2” in Fig. 4 to make them more visible against the background color", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "In their manuscript entitled \"Spin-polarized p-wave superconductivity in the kagome material RbV Sb \", Shuo Wang et al. measure magnetotransport in devices of RbV Sb . 3 5 3 5 They have three main observations: 1) hysteresis of the resistance in field; 2) field- reentrant superconductivity that is asymmetric in field direction; and 3) two-fold anisotropy in the in-plane critical field. They use these observations to motivate the presence of a chiral p-wave superconducting state. Possible chiral p-wave superconductors have a long and messy history, and thus these strong claims require equally strong evidence. Unfortunately, Wang et al. do not appear to provide it. Response: We sincerely thank the referee for the insightful and critical comments, which have provided us with an opportunity to clarify and elaborate on our viewpoints more effectively. We fully agree that the history of chiral p-wave superconductors is complex and contentious, and as such, robust claims demand equally robust evidence. This principle has guided our approach, as we have strived to present a comprehensive chain of evidence rather than relying on a single observation. In this work, we argue that the collective phenomena (hysteresis, re-entrant superconductivity, quench experiments, and two-fold symmetric superconductivity) put strong constraints on the possible pairing symmetries, with the p-wave scenario emerging as the most plausible explanation. Inspired by the referee’s comments, we conducted additional control experiments on ten thin flakes of CsV₃Sb₅ (with thicknesses ranging from 20 to 40 nm and the superconducting transition temperature ranging from 2.7 to 3.5 K). No hysteresis was observed in all ten thin-flake CsV₃Sb₅ samples under the superconducting transition temperature. This result strongly suggests that the superconducting states of RbV₃Sb₅ and CsV₃Sb₅ are different and the hysteresis originates from the intrinsic superconducting properties of RbV₃Sb₅ rather than the flux trapping mechanism. Below, we address the reviewer’s specific concerns in detail. My main concerns are: Comments: 1) Hysteresis in superconductors usually arises due to vortex physics. Their argument against this is based on their quenching scheme, whereby they apply a large current to drive the sample normal, then cool it in a finite field. This resets the flux trapping state based on whatever field they cool in. They claim flux will be trapped at weak pinning sites that should be mobile enough be dissipative. I don’t know how they can make that claim. This depends entirely on the microscopics of the pinning site strength, the amount of current, the energy barrier at the edge of the sample for vortices, etc, none of which are known. Also, the authors state that \"In the positive magnetic field region, we obtain a newly formed superconducting state with zero resistance\", but see point 3 below where I point out that zero resistance is never seen. Response: Thank you very much for your insightful comment. We fully agree with you that the flux trapping mechanism can lead to hysteresis, particularly in nonmagnetic granular superconductors. However, the hysteresis behavior, which is induced by the flux trapping mechanism, differs fundamentally from that observed in RbV Sb . 3 5 Importantly, to address the question of the referee, we performed control experiments on kagome superconductor CsV₃Sb₅, in which the hysteresis in magnetoresistance was not observed in all ten samples. The following are the details of the discussion: A. No observation of resistance minimum at finite field in the returning curve over a wide range of currents. We thank the referee for pointing out that the hysteresis in the magneto-resistance measurements can be induced by flux trapping mechanism. We fully agree with this comment of the referee. However, we would like to point out that the hysteresis behavior induced by flux trapping is different from the one observed in RbV Sb Fig. 3 5. R1a shows the hysteresis loop in nonmagnetic superconductor YBCO [Phys. Solid State 63, 1069 (2021)]. As the magnetic field increases from zero (the virgin curve is denoted by the black dots), the resistance starts to deviate quickly from zero at small magnetic field. At the returning curve starting from 200 Oe, a clear hysteresis loop was observed, as expected by the referee. However, there is a magnetoresistance minimum at around 20 Oe for the returning curve, as shown by the red cycle in Fig. R1a. The appearance of the magnetoresistance minimum at a finite field is a rather common feature for flux trapping induced hysteresis. Fig. R1b shows these features in another experiment [Physica C: Superconductivity 390, 363 (2003)]. Fig. R1c shows the numerical simulation that reproduced the resistance minimum for the returning curve over a wide range of parameters, when the hysteresis is induced by the flux trapping mechanism [Phys. stat. sol. (a), 204, 805-813 (2007)]. In Fig. R1d, we present the hysteresis curves of RbV Sb No magnetoresistance 3 5. minimum at a finite field was observed in the returning curves over a wide range of currents. Instead, we observed a (nearly) zero resistance plateau for the returning curve. [The nearly zero residue resistance will be discussed later to answer the comment 3 from the referee]. Fig. R1 a. Typical hysteretic dependences of the magnetoresistance for the Y–Ba– Cu–O compounds (which is reproduced from Phys. Solid State 63, 1069 (2021)). Arrows show the direction of the change in external field H, and the solid cycles mean the virgin curve which starts from zero field. At the returning curve starting from 200 Oe, there is a magnetoresistance minimum at around 20 Oe for the returning curve which is emphasized by the red circle. b. Magnetoresistance R(B ) curves taken on the a sample YBa 2 Cu 3 O 7-𝛿-10 wt.% Ag for two different values of excitation current: I ex = 55 mA (filled squares) and I = 75 mA (blank circle) [Physica C: Superconductivity ex 390, 363 (2003)]. The magnetoresistance minimum in the returning curve is observed under both cases. c. Theoretical virgin and returning curves of R(B ) dependence from a 10 and 20 mT [Phys. stat. sol. (a), 204, 805-813 (2007)]. d. The current-dependent hysteresis behavior of the thin-flake RbV Sb at a temperature of T = 150 mK. The red 3 5 and black curves represent the up-sweep and down-sweep processes, respectively. No magnetoresistance minimum at a finite field was observed in the returning curves over a wide range of currents. This figure is now presented in Supplementary Information (Fig. S9). B. No hysteresis is observed in the thin-flake kagome material CsV₃Sb₅ which belongs to the same family with RbV₃Sb₅ If the magneto-resistance hysteresis is induced by flux trapping, the hysteresis should appear generally in similar materials. Recently, we conducted the same in-plane magnetic field sweeping experiments on ten thin-flake samples of kagome material CsV₃Sb₅ (with thicknesses ranging from 20 to 40 nm and the superconducting transition temperature ranging from 2.7 to 3.5K). No hysteresis was observed in all ten samples. A typical magneto-resistance curve for CsV₃Sb₅ is shown in Fig. R2a. For comparison, we also show the field-sweeping experiment result of thin-flake RbV₃Sb₅ (whose thickness is 27.6 nm) in Fig. R2b (Fig. 2b in our main text). Although RbV₃Sb₅ and CsV₃Sb₅ belong to the same kagome material family and the thin flakes we used have similar thicknesses, their field-sweeping experiment results differ significantly. Unlike in RbV₃Sb₅, no hysteresis was observed in CsV₃Sb₅ below its superconducting transition temperature. If the observed hysteresis were due to trivial mechanisms, such as flux trapping, rather than intrinsic properties of the superconducting states, it should also appear in CsV₃Sb₅ as well. This result strongly suggests that the superconducting states of RbV₃Sb₅ and CsV₃Sb₅ are different and that the hysteresis originates from the intrinsic superconducting properties of RbV₃Sb₅. We thank the referee for the comments, and we have now added the data of the control experiments of CsV₃Sb₅ in the Supplementary Information (Fig. S10). Fig.R2 a. A typical magneto-resistance curve of thinflake kagome material CsV Sb 3 5 below its superconducting transition temperature under an in-plane magnetic field sweep. There is no hysteresis during the whole field-sweeping process. b. The hysteresis observed in thin-flake RbV Sb , which is reproduced from the Fig. 2b in our 3 5 main text. The thickness of this thin-flake RbV Sb is 27.6 nm. 3 5 In summary, the hysteresis behavior observed in RbV 3 Sb 5 is different from that induced by the flux trapping mechanism in nonmagnetic granular superconductors. The superconducting domain picture proposed in our manuscript provides a unified framework to explain both the hysteresis and reentrant behavior in this material. This superconducting domain picture is also consistent with the results obtained from the current-quenching experiment (even though we agree with the referee that the current- quenching experiment itself cannot rule out the flux trapping mechanism). Moreover, the distinctly different magnetoresistance behavior of RbV Sb and CsV Sb during the 3 5 3 5 field-sweeping process suggests that the hysteresis originates from the intrinsic unconventional superconducting properties of RbV Sb . 3 5 For clarity, we modified the related paragraphs in the main text as: “In addition, the hysteresis observed in our experiment exhibits behavior starkly distinct from that attributed to the flux-trapping mechanism. In the hysteresis induced by the flux-trapping, a magnetoresistance minimum at finite magnetic field can be observed in the backward-sweeping curve (returning curve)28,72,73. In contrast, as shown in Fig. 2 b&c and Fig. S9 in Supplementary Information, in RbV Sb , no 3 5 magnetoresistance minimum at finite field was observed in the backward-sweeping curve over a wide range of applied currents. Moreover, we conducted in-plane field-sweep experiments on over ten thin-flake kagome material CsV Sb . Although RbV₃Sb₅ and CsV₃Sb₅ belong to the same kagome 3 5 material family and the thin flakes we used have similar thicknesses, no such hysteresis was observed in CsV Sb below its superconducting transition temperature, as shown 3 5 in Fig. S10. If the observed hysteresis were due to trivial mechanisms, such as flux trapping, rather than intrinsic properties of the superconducting states, it should also appear in CsV Sb as well. This result suggests that the superconducting states of 3 5 RbV Sb and CsV Sb are different, and the hysteresis originates from the intrinsic 3 5 3 5 superconducting properties of RbV Sb . It is interesting to note that Knight shift 3 5 experiments in CsV Sb suggested spin singlet pairing74. However, since our control 3 5 experiments suggest that the superconducting properties of RbV Sb are quite different 3 5 from CsV Sb , the pairing symmetry of RbV Sb is possibly unconventional. ” 3 5 3 5 The answer to the question about “zero-resistance” can be found in our reply to comment 3. Comments: -1b) Did the authors check how the hysteresis depends on the applied current? Do the authors get same hysteretic behavior with more or less current? Fig. R3: The current-dependent filed-sweeping experiments of the thin-flake sample at a temperature of T = 150 mK. Response: Fig. R3 presents current-dependent magnetic resistances at a fixed 1 Ω temperature of T = 150 mK. The hysteresis persists consistently across all applied currents below the critical current. For clarity, in Fig. R3, the curves have been vertically offset. Comments: 2) The authors claim that the superconducting state breaks the C rotation 6 symmetry of the lattice down to C . However, this symmetry is already broken by the 2 CDW at higher temperatures! See e.g. https://iopscience.iop.org/article/10.1088/1674- 1056/acd8b1/pdf Once a symmetry is reduced, all measured quantities in general have that lower symmetry, e.g. the critical field is expected to be C symmetric in the CDW 2 state even if it is a trivial s-wave state. Response: We appreciate the referee’s comment and we fully agree with the referee that the C rotational symmetry of the lattice is indeed reduced to C symmetry by the 6 2 CDW-induced nematic state at temperatures above the superconducting transition temperature. Our classification of the pairing symmetry based on the D point group 2h is indeed a consequence of this symmetry breaking in the normal state. We did not claim the superconducting state breaks the C rotation symmetry spontaneously. Our 6 statement in the main text (original main text 199-203 lines) was: “It is important to note that the original point group symmetry of the material is D . 6h Before the superconducting transition happens, the nematic order parameter comes in10,15 and reduces the point group symmetry to D . Therefore, the superconducting 2h order parameters can be constructed by using basis functions belonging to irreducible representations of the point group D .” 2h Therefore, we fully agree with the referee that the symmetry breaking from D to D 6h 2h is a result of the normal-state nematicity. The superconducting state itself is not a reason for the symmetry breaking. Comments: -2b) There may be domains of the CDW that produce a higher effective symmetry at long length scales. However, this is a mesoscopic device with a very high aspect ratio that will naturally favor one domain of the others. This has the prediction that the anisotropy in the critical field could be device dependent, and indeed a subset of the present authors published exactly this data in an earlier paper https://www.nature.com/articles/s42005-023-01496-3#Sec10 that shows a MUCH higher anisotropy than the data in the current paper. Why should the anisotropy be device dependent if it was intrinsic to the superconducting state? Response: We thank the referee for these comments. We fully agree with the referee that the anisotropy of the upper critical filed can be device-dependent due to the properties of the normal states that vary between devices. Indeed, we did not claim that the anisotropic upper critical field was induced by the unconventional pairing. As shown by some of the authors in previous theoretical [Physical Review Letters 125, 107001 (2020)] and experimental [Nature Physics 19, 106 (2023)] works, the upper critical field can be anisotropic for centrosymmetric superconductors (such as in some 2M-WS and in our current case of RbV₃Sb₅) even if the pairing is a conventional s- 2 wave pairing. Therefore, the anisotropic upper critical field is indeed a piece of evidence that there is nematicity in the normal state and that there is spin-orbit-parity coupling at the Fermi energy for this centrosymmetric superconductor. From our four orbital model calculations in the Supplementary Information, we did show that there is spin-orbit-parity coupling in this centrosymmetric superconductor. [The concept of spin-orbit-parity coupling was first introduced by Law in Physical Review Letters 125, 107001 (2020) to explain the anisotropic upper critical fields and other novel properties in some two-dimensional superconductors]. Therefore, we concluded that the anisotropic upper critical field provides evidence that the normal state is nematic. Given the nematicity in the normal state, we use the D point group, instead of the 2h original D point group to classify the pairing symmetry. In this work, we did not claim 6h that the anisotropy of the upper critical field was caused by the unconventional p-wave pairing of the superconducting state. This is in full agreement with the expectation from the referee. Comments: 3) The authors claim re-entrant superconductivity. However, the authors never measure a zero-resistance state at ANY field or temperature - there is always a substantial offset. Why is this the case? The reentrance only goes close to the zero-field resistance (which again, is not zero resistance) in a very, very narrow temperature range around 400 mK, otherwise it is just a dip in resistance. Response: We thank the referee for raising these two critical questions. A. Concerning finite residual resistance: First of all, zero resistance states are observed in bulk RbV Sb samples [Chinese 3 5 Phys. Lett. 38, 037403 (2021)]. While the finite residual resistance in our thin-film samples is observed, we emphasize that this does not negate the presence of the superconducting state. Below, we address the origin of residual resistance in thin-film superconductors. i. The ratio of the normal-state resistance to residual resistance of superconducting states in our samples ranges from 10 to 100, depending on the micro-fabrication quality. This variability is an inherent challenge in thin-film fabrication and measurements. Unlike bulk superconductors, thin films are quasi-2D systems that are particularly susceptible to quasiparticles, thermal fluctuations, and fabrication defects. Specifically, RbV Sb has a T of only ~ 900 mK. Even at 3 5 c temperatures as low as 100 mK (well below T ), thermal fluctuations can induce c phase decoherence between different superconducting regions which can cause dissipations. In comparison, thin-flake CsV Sb , with a higher T 2.5 - 3.8 K, 3 5 c exhibits residual resistance more than an order of magnitude smaller than that of RbV Sb , as confirmed by our experiments. 3 5 ii. During sample fabrication, contact resistance between the electrodes and the thin-film samples can arise. While bulk AV Sb samples are relatively stable, 3 5 their exfoliated thin flakes are highly sensitive to environmental factors, such as oxygen exposure. Our fabrication is conducted in a noble-gas or high-vacuum environment to minimize oxidation. However, minor contact resistance persists due to inevitable interface contamination between the thin flakes and electrodes, as well as chemical potential mismatches between the sample and gold electrodes. Although four-probe measurements eliminate most contact resistance, minor inductive or capacitive coupling can still be picked up by lock- in amplifiers at finite frequencies (e.g. 10-100 Hz), mimicking a small resistive behavior. Therefore, the presence of minor residual resistance does not invalidate the claim of the superconducting state. Additionally, the resistance-temperature curves, the dV/dI measurements, and the magnetoresistance data all consistently demonstrate superconducting behavior. B. Concerning Re-entrant Superconductivity: We thank the referee’s question concerning re-entrance of the superconducting state. The referee’s observation that the magnetic field driven re-entrance of the superconducting state appears only in a small range of temperature is correct. Indeed, our observation in RbV Sb is consistent with the magnetic field driven re-entrance in 3 5 magic angle twisted bilayer graphene (MATBG) (Nature 595, 526–531 (2021), which also appears only in a small range of temperatures ranging from 300 mK to 400 mK. From 400 mK to 600 mK, the re-entrance is manifested as a resistance dip, as expected by the referee. The magnetoresistance under different currents and temperatures of MATBG is shown in Fig. R4. In Fig. R4a, the differential resistance (dV /dI) plot as a function of magnetic field xx and current at 250 mK for MATBG is shown. Two superconducting domes, depicted in blue, were observed. This is similar to our RbV Sb data at 400 mK (Fig. R4b) in 3 5 which two superconducting domes were observed (in black). In MATBG, below 250 mK, it is believed that the two domes should be connected at small currents and there is no magnetic field driven re-entrance. Similarly, for RbV Sb , below 400 mK, there 3 5 is no magnetic field driven re-entrance as shown in Fig. R4d and Fig. R5 below (Fig. 3a of the main text). Fig. R4 a. the differential resistance plot as a function of magnetic field and current for MATBG at 250 mK. b. the differential resistance plot as a function of magnetic field and current for RbV Sb at 400 mK. c. The magnetoresistance curves under different 3 5 temperatures for MATBG. d. The magnetoresistance curves under different temperatures for RbV Sb . 3 5 Fig. R5. The temperature dependence of the magnetoresistance when the magnetic field sweeps from -1 T to 1 T, adapted from Fig. 3a in the manuscript. With temperature increasing, the critical magnetic field gradually decreases. No re-entrant superconductivity is observed below 400 mK. In MATBG, when the temperature is increased above 400 mK, magnetoresistance will have a dip at higher fields as shown in Fig. R4c. We observed a similar behavior in our RbV Sb samples as shown in Fig. R4d. So, the referee is correct that the re- 3 5 entrance of superconductivity only appears in a narrow range of temperature. At higher temperatures, the magnetoresistance acquires a dip driven by the magnetic field. Our observation in RbV Sb is similar to the re-entrance behavior observed in MATBG. 3 5 However, we have to emphasize that even though the magnetic field driven re-entrant superconductivity is observed in both MATBG and RbV Sb , the microscopic origin 3 5 of the re-entrance can be very different for the two systems. Here, we want to emphasize that for the re-entrant superconductivity in MATBG, it appears only in a very small temperature range. At higher temperatures, the re-entrance is manifested as a resistance dip, as pointed out by the referee. Concerning the explanation of the re-entrant superconductivity, the MIT group suggested that the magnetic field induced re-entrance can be induced by superconducting state with equal spin pairing. However, there was a lack of detailed theoretical analysis (Nature 595, 526–531 (2021) about the pairing symmetry in MATBG. In this work, using the point group symmetry analysis, we suggest that the pairing probably belongs to the B or B representation of the D point group in 2u 3u 2h RbV Sb . Even though the observed re-entrance behavior in MATBG and RbV Sb 3 5 3 5 look similar, we emphasize again that the exact microscopic pairing mechanism of MATBG and RbV Sb can still be rather different. 3 5 Comments: -3b) Very similar measurements were performed on the same material in an earlier paper by a subset of the current authors F#Sec10 why was no reentrant superconductivity observed in this device? Response: We thank the referee for pointing out this important issue. Indeed, re-entrant superconductivity is not observed in all samples. Statistically, over 50% of our samples exhibit such re-entrant superconductivity. This is because the emergence of the second superconducting dome requires percolation of superconducting domains. When sweeping the magnetic field from negative to positive values, the magnetic field initially suppresses superconductivity after the first dome. Subsequently, the superconducting domains must grow and reconnect to form the second superconducting dome. In an ideal case, re-entrant superconductivity would be observed at an intermediate temperature. However, in practice, the reconnection of superconducting domains is highly sensitive to disorder and defects, which vary between devices, particularly in thin-flake samples. Thus, there is a possibility that the domains will fail to reconnect again and form the second superconducting dome. During the preparation of our previously published work, we also observed re-entrant superconductivity in several samples. However, at the time, we did not fully understand the underlying mechanism, and thus, the focus of that manuscript was primarily on the two-fold symmetric superconductivity. The current study represents significant progress in our understanding. Comments: 4) The absolute highest the upper critical field could be is 0.75 tesla for field in plane (taken as the field where the resistance becomes flat in Fig 2b). This is very close to the Pauli limit of ~1 tesla (calculated using a g-factor of 2). If RbV Sb is 3 5 spin triplet, why is the upper critical field Pauli limited for field in the plane? Response: We thank the referee for this insightful question. You are correct that in the single-layer limit, the upper critical field for spin-triplet superconducting states should not be Pauli-limited. Actually, in our experiments, the upper critical field is strongly suppressed rather than being Pauli-limited. The key factor contributing to the observed decrease in the upper critical field is the material thickness: In the main text, we presented the data for Sample 1 (Fig. 2b) which has a thickness of 𝑑 = 27.6 nm. 0 Given the layer separation of about 1 nm, our sample is far from the atomically thin limit. On the other hand, the resistivity along the out-of-plane direction is approximately 33 times greater than the in-plane resistivity at low temperatures, as reported in Chinese Phys. Lett. 38, 037403 (2021). Therefore, our system consists of multiple superconducting kagome layers. The interlayer coupling between these superconducting layers can contribute to an orbital effect, reducing the in-plane upper critical field compared to the single-layer limit (RA Klemm. Layered Superconductors. Oxford University Press, 2012). Consequently, the low upper critical field which does not exceed the Pauli limit is observed in our experiments. Reviewer #2 (Remarks to the Author): The authors present intriguing magnetotransport findings in kagome superconductor RbV Sb thin films, including hysteresis in magnetoresistance, reentrance of 3 5 superconductivity, and a two-fold symmetric in-plane upper critical field. By conducting additional measurements, they ruled out trivial explanations for these phenomena. The authors propose that spin-polarized p-wave pairing could account for the observed results. Currently, the pairing symmetry of kagome superconductors is still under debate from both experimental and theoretical perspective. The present results are interesting and offer more insights into the pairing state of thin-film kagome superconductors. However, it is important to note that this evidence presented is not definitive proof of p-wave pairing, particularly given knight shift measurements suggesting spin-singlet pairing in the bulk (Chin. Phys. Lett. 38 077402 (2021)). I suggest that the authors should temper their assertion of p-wave pairing in the manuscript and explicitly acknowledge the indirect nature of the evidence provided. Nevertheless, the paper offers valuable experimental insights and features a well-executed theoretical analysis. The authors should properly address the following concerns before my recommendation of publication: Response: We sincerely thank the referee for the thoughtful comments and valuable suggestions. We greatly appreciate the reviewer’s positive assessments and the feedback has inspired us to further refine and improve the manuscript. We acknowledge that the assertion of p-wave pairing is a strong claim and have taken the referee’s suggestion to “temper their assertion of p-wave pairing” to heart. Accordingly, we have revised the title to “Signatures of spin-polarized p-wave superconductivity in the kagome material RbV Sb ” and softened the language in the 3 5 abstract to state: “We point out that the pairing symmetry, which is consistent with the crystal symmetry and all the observed novel properties, is possibly a time-reversal symmetry-breaking, p-wave pairing with net spin polarization”. We also thank the referee for the comments concerning the Knight shift experiments. We would like to emphasize that the Knight shift experiments [Chin. Phys. Lett. 38 077402 (2021)] were performed with CsV Sb . As we show in our control experiments 3 5 with ten thin-flake CsV Sb samples, no magnetoresistance hysteresis was observed in 3 5 all ten samples [details are given in the response to Comment 4 of the referee]. Therefore, we believe that the pairing symmetry of RbV Sb can be different from that 3 5 of CsV Sb 3 5. In the main text, we now acknowledge that: “It is interesting to note that Knight shift experiments in CsV Sb suggested spin singlet pairing74. However, since our control 3 5 experiments suggest that the superconducting properties of RbV Sb are quite different 3 5 from CsV Sb , the pairing symmetry of RbV Sb is possibly unconventional.” 3 5 3 5 Comments: 1. The authors discuss the band dispersion with the spin-orbit-parity- coupling term and its relevance to the two-fold in-plane upper critical field. However, according to fig.4a in PRL 125, 247002 (2020), the SOC gap between p- and d-orbital band along Gamma-M (located at 0.5 eV) is far away from the Fermi level. Why is such term relevant for the band dispersion and properties of kagome superconductor AV3Sb5? Response: We thank the referee for this insightful question. The referee is correct that the spin-orbit-parity coupling (SOPC) gap is far from the Fermi level. However, while the orbital mixing near the Fermi level is weaker compared to that near the SOPC gap, the parity-mixing effects still exist near the Fermi energy (which is about 0.5 eV below the parity mixing gap as pointed out by the referee) as shown in the right panel of Fig. R6, in which the p-orbital (blue) and d-orbital (red) contribute jointly to the band near the Fermi surface (colors are mixed). For example, in Fig. R6, the band inside the green circle has an orbital composition of 82.6% p-orbital and 17.4% d-orbital. This parity mixing effect can still influence the superconducting properties, including the amplitudes and anisotropic behavior of the upper critical field. The details of the model were given in the Supplementary Information. Fig.R6. The band structure without (left) and with (right) spin-orbit-parity coupling (SOPC) of the four-orbital model, including d-orbital (red) from V atoms and p-orbital (blue) from in-plane Sb atoms. Comments: 2. Clarification is needed regarding the measuring temperature in Fig.2h. Additionally, if a quenching process is involved in the measurement shown in Fig.3b, it is essential to determine if a similar zero resistance can be achieved for a positive magnetic field. If this is the case, can this be called re-entrant superconductivity by sweeping magnetic field (as superconductivity is not destroyed)? Conventionally, re- entrant superconductivity is characterized by the reappearance of superconductivity as the magnetic field increases after being suppressed by a magnetic field. Response: We thank the referee for this helpful comment. The measuring temperature in Fig. 2h has been clarified. No quenching process is involved in the measurements of re-entrant superconductivity. Fig. R7 (Fig. 3b in the main text) presents the temperature dependence of the magnetoresistance, obtained by sweeping the magnetic field from - 1 T to 1T and back from 1 T to -1 T, without any additional processes such as quenching. When the magnetic field sweeps from -1 T to 1 T (or 1 T to -1 T), the sample first enters into the superconducting states; after the magnetic field reverses and increases further, the superconducting state would be first suppressed and then appears again after a critical value. This behavior shows the unconventional nature of the superconducting state in the kagome material RbV Sb and provides further support for our proposed 3 5 pairing mechanism and domain picture. Fig. R7. The waterfall plot of the resistance at different temperatures, adapted from Fig. 3b in the manuscript. The re-entrant superconductivity starts at 400 mK. The red curves denote a forward sweep, while the black curves represent a backward sweep. Comments: 3. How might the temperature-magnetic field phase diagram in Fig.2 be altered by conducting measurements with a fixed magnetic field and varying temperature? Response: We thank the referee for this insightful question. The temperature-magnetic field phase diagram is presented in Fig. R8 (Fig. 3a in the main text). If we vary the temperature at a fixed magnetic field, the resulting phase diagram would not exhibit two distinct superconducting domes, SC-I and SC-II. This is because re-entrant superconductivity relies on the disconnection and percolation of superconducting domains during the field-sweeping process, which requires the reversal of the magnetic field's direction. When the magnetic field is fixed and the temperature is varied, the fixed magnetic field inhibits the formation of two distinct superconducting domains with opposite magnetic moments. As a result, the reentrant behavior does not appear. Fig. R8. The temperature dependence of the magnetoresistance when the magnetic field sweeps from -1 T to 1 T, adapted from Fig. 3a in the manuscript. With temperature increasing, the critical magnetic field gradually decreases. However, a re-entrant superconductivity occurs at around 400 mK. Here we label the superconducting dome obtained before the re-entrance of superconductivity as SC-I and the superconducting dome obtained after the reentrance of superconductivity as SC-II. Comments: 4. The bulk superconductivity (CsV Sb ) is spin-singlet according to Chin. 3 5 Phys. Lett. 38 077402 (2021). In contrast, the authors claim a spin-triplet pairing in RbV Sb thin films, with a transition temperature close to that of the bulk material. It 3 5 is difficult to understand why there will be a transition from spin-singlet to spin-triplet pairing between bulk and thin films. The author should provide more discussions about the difference between bulk and thin films. Response: We thank the referee for this helpful comment, which we believe is crucial for advancing the understanding of kagome superconductors AV Sb in the broader 3 5 scientific community. We would like to address this comment in three key aspects: i. No hysteresis observed in thin-film CsV Sb . Recently, we conducted the same 3 5 in-plane magnetic field sweeping experiments on ten thin-film samples of kagome material CsV₃Sb₅ (with thicknesses ranging from 20 to 40 nm and the superconducting transition temperature ranging from 2.7 to 3.5 K). No hysteresis was observed in any of the samples. A typical magneto-resistance curve for CsV₃Sb₅ is shown in Fig. R9a. For comparison, we also show the field-sweeping experiment result of thin-film RbV₃Sb₅ (whose thickness is 27.6 nm) in Fig. R9b (Fig. 2b in our main text). Although RbV₃Sb₅ and CsV₃Sb₅ belong to the same kagome material family and the thin films we used have similar thicknesses, their field-sweeping experiment results differ significantly. This result indicates that the superconducting states in RbV Sb and CsV Sb are intrinsically different. 3 5 3 5 The data for CsV₃Sb₅ is now presented in the Supplementary Information. Fig.R9 a. A typical magneto-resistance curve of thin-film kagome material CsV Sb 3 5 below its superconducting transition temperature under an in-plane magnetic field sweep. There is no hysteresis during the whole field-sweeping process. b. The hysteresis observed in thin-film RbV Sb , which is reproduced from the Fig. 2b in our 3 5 main text. The thickness of this thin-film RbV Sb is 27.6 nm. 3 5 ii. Difference between bulk RbV Sb and CsV Sb . At first glance, RbV Sb and 3 5 3 5 3 5 CsV Sb , as members of the same kagome superconductor family, might be 3 5 expected to exhibit similar properties. However, experimental evidence reveals significant differences in both their normal and superconducting states. For instance, ARPES measurements indicate distinct charge density wave (CDW) structures in the two materials (Nature Materials 22, 186-193(2023)), which could influence their superconducting properties, given the competitive relationship between CDW and superconductivity. Additionally, μ-SR experiments suggest that RbV Sb is more likely to be a nodal superconductor 3 5 while CsV Sb appears to be nodeless (Nature Communications 14, 153(2023)). 3 5 The presence of residual in-gap states in RbV Sb further hints at unconventional 3 5 pairing mechanisms. Consequently, current research largely supports the view that superconductivity in CsV Sb exhibits different behavior from that in 3 5 RbV Sb . 3 5 iii. Theoretical perspective on CsV Sb : While some experiments suggest that 3 5 CsV Sb may exhibit spin-singlet or even s-wave superconductivity, theoretical 3 5 studies caution against oversimplifying its nature. For example, recent works (e.g. Physical Review B, 110, 144516 (2024), Physical Review B 108, 144508 (2023)) highlight that “unconventional superconductivity on the kagome lattice is deceptive in the sense that its properties may appear similar to conventional non- sign-changing superconductivity”. Thus, the community has not yet reached a definitive conclusion that CsV Sb must be pure spin-singlet or s-wave. 3 5 Specifically, while Knight shift suppression and the observation of a Hebel- Slichter peak below T in the spin-lattice relaxation rate measured by NMR c experiments were interpreted as evidence for s-wave spin-singlet superconductivity (Chin. Phys. Lett. 38 077402 (2021)), Brain Andersen et al. have demonstrated that such features can also arise in unconventional kagome superconductors (Phys. Rev. B 110, 144516 (2024)). Given these considerations, we do not interpret the observed phenomena as evidence of a spin-singlet to spin-triplet transition from bulk to thin films. Instead, we propose that RbV Sb itself is possibly a nonunitary p-wave superconductor. This interpretation 3 5 aligns with the experimental observations and theoretical insights discussed above, providing a coherent framework for understanding its unique superconducting properties. We hope this detailed response clarifies our perspective and contributes to the ongoing discussion about the intriguing physics of kagome superconductors. We thank the referee again for this thought-provoking question. Comments 5: The relatively low upper critical field observed in the thin films challenges the expectation for a spin-triplet pairing. Will a time-reversal symmetry breaking spin-singlet pairing explain some of experimental phenomena? If not, possible mechanism of spin-triplet pairing in thin films should be further discussed. Response: We thank the referee for this insightful question. We would like to clarify two points below: A. the low upper critical field observed in the thin films does not contradict the spin-triplet pairing scenario. B. the time-reversal symmetry breaking spin-singlet pairing cannot explain the observed magnetic field-driven reentrance in superconducting states. A. Decreasing in the upper critical field of the spin-triplet superconductor: The low upper critical field does not exclude spin-triplet pairing. Even if the superconducting state in our sample is spin-triplet, the upper critical field can still be significantly decreased and exhibit a value lower than the Pauli limit. The key factor contributing to the observed decrease in the upper critical field is the material thickness: In the main text, we presented the data for Sample 1 (Fig. 2b), which has a thickness of 𝑑 = 27.6 nm. Given the layer separation of about 1 nm, our sample is 0 far from the atomically thin limit. On the other hand, the resistivity along the out-of- plane direction is approximately 33 times greater than the in-plane resistivity at low temperatures, as reported in Chinese Phys. Lett. 38, 037403 (2021). Therefore, our system consists of multiple superconducting kagome layers. The interlayer coupling between these superconducting layers can contribute to an orbital effect, reducing the in-plane upper critical field compared to the single-layer limit (RA Klemm. Layered Superconductors. Oxford University Press, 2012). Consequently, the low upper critical field observed in our experiments does not exclude the spin-triplet superconducting state. B. Considering time-reversal symmetry breaking spin-singlet scenario. After meticulously ruling out trivial origins, especially the vortex physics, the reentrance in superconducting states cannot be explained by time-reversal symmetry breaking spin-singlet pairing. The reason is that the Zeeman effect of the magnetic field tends to suppress the spin-singlet superconducting state by inducing a depairing effect on spin-singlet Cooper pairs, whereas it does not affect the equal-spin pairing states. Therefore, the reappearance of the superconducting state under a larger magnetic field is not consistent with the spin-singlet Cooper pairs scenario. Instead, the observed re-entrance is more consistent with equal-spin pairing, where the spin magnetic moments of the Cooper pairs couple to the magnetic field, lowering the energy of the superconducting state. Overall, to address the referee’s comments, we conducted control experiments on CsV Sb thin films and did not observe magnetoresistance hysteresis in all ten samples. 3 5 The data are presented in Fig. S10 of Supplementary Information. We thank the referee again for the excellent questions. Reviewer #3 (Remarks to the Author): In this manuscript, the authors present magnetoresistance measurements of the kagome material RbV Sb in order to investigate the nature of its superconducting state. One of 3 5 their most notable findings is the hysteresis in magnetoresistance—namely in the boundaries of the superconducting state—for magnetic fields swept in opposite directions. The authors have measured this hysteresis for a variety of field directions and temperatures. The authors have demonstrated that this hysteresis is present for all directions of in- plane fields, yet disappears with a small tilt of field outside the ab plane. They also show that there is a range of temperatures, roughly 400-600 mK, for which the superconductivity is split into two domes as a function of field, with a higher-resistance state appearing at roughly 80 mT for forwards field sweeps and -80 mT for backwards field sweeps between the two domes. In addition, with the sample in a finite field, the authors have used applied current to heat their sample into the normal state and then removed the current and allowed it to cool. They have showed that this process can bring the sample from a finite-resistance state to a zero-resistance state. This supports the authors’ hypothesis that the finite-resistance state in the hysteresis loop is related to domain formation, rather than being due to flux trapping. They posit that this is due to time-reversal symmetry-breaking superconducting order that can couple to applied magnetic fields. The hysteretic behavior presented here is novel and interesting. The authors have studied this behavior thoroughly in terms of field direction and temperature. Their use of current heating/cooling to further understand the finite-resistance state is excellent. The authors also put forward a sensible model by which to interpret their data, which indicates that the material is a topological superconductor, certainly of interest to the field. Response: We sincerely thank the referee for the supportive and constructive feedback, which has been invaluable in guiding us to refine and enhance the manuscript. Your thoughtful comments have not only encouraged us but also provided clear directions for improving the clarity and rigor of our work. We deeply appreciate the time and effort the referee has made to review our paper, and we are truly thankful for the referee’s contribution to this process. However, there are a few points that I would like to see addressed: Comments 1: The authors note that “the AV3Sb5 family lacks magnetic orders in the normal state”, and thus attribute the time-reversal symmetry-breaking to the superconducting phase itself. However, as pointed out in the authors’ Ref. 4, the charge density wave orders of this family of materials do appear to break time-reversal symmetry. The authors should discuss whether time-reversal symmetry-breaking from the CDW of RbV Sb could be the cause of the observed hysteresis. 3 5 Response: We thank the referee for this insightful question. We fully agree with the referee that “the charge density wave orders of this family of materials do appear to break time-reversal symmetry”. Indeed, the time-reversal breaking order in the charge density wave (CDW) phase, possibly due to orbital loop currents, was proposed by one of us (Xilin Feng) and collaborators in Science Bulletin 66, 1384-1388 (2021), which is Ref.181 of the review paper by Yaojia Wang et al (Ref. 4 of our main text). However, in the CDW phase, the loop currents have local orbital magnetic moments which are pointing out-of-plane. At the same time, the overall orbital magnetic moment of the sample is zero (as the loop currents with opposite chirality cancel each other). The existence of this time-reversal symmetry breaking (TRSB) CDW order in this family of materials cannot explain the observed hysteresis. This is because the magnetization of the TRSB CDW lies along the out-of-plane direction. However, the hysteresis in our work is observed under an in-plane magnetic field, which cannot be explained by the out-of-plane orbital magnetization. Therefore, we suggest that the in-plane hysteresis is due to the superconducting state itself instead of the normal state. Moreover, the signature of TRSB CDW phase was also observed in CsV Sb [Nature 3 5 physics, 18, 1470 (2022)]. However, as shown in our control experiments (Fig. R10a), we measured the magnetoresistance curves of ten CsV Sb thin films. 3 5 Magnetoresistance hysteresis was not observed in all ten samples of CsV Sb . The data 3 5 for CsV Sb are presented in the Supplementary Information. Therefore, the loop 3 5 current is not related to the magneto-resistance hysteresis observed, and the hysteresis is a property of the superconducting state of RbV Sb . 3 5 Fig.R10 a. A typical magneto-resistance curve of thin-film kagome material CsV Sb 3 5 below its superconducting transition temperature under an in-plane magnetic field sweep. There is no hysteresis during the whole field-sweeping process. b. The hysteresis observed in thin-film RbV Sb , which is reproduced from the Fig. 2b in our 3 5 main text. Concerning the statement “the AV Sb family lacks magnetic orders in the normal 3 5 state”, it was for emphasizing that “the AV Sb family lacks spin magnetic orders in 3 5 the normal state”. We sincerely thank the referee for this important comment. To avoid confusion, we have revised the main text as follows: “It is noted that the time-reversal symmetry may be broken in the normal states of the kagome family AV Sb due to loop currents in the charge density wave phase5,6,13,14. 3 5 However, the loop currents would produce out-of-plane magnetization, in contrast to the in-plane magnetic moments observed in this work. Therefore, the in-plane hysteresis is likely caused by the superconducting state itself, rather than the normal state.” Comments 2: The authors indicate that the superconductivity is spin-triplet based on the observed time-reversal symmetry-breaking. However, it is possible for spin-singlet superconductors to break time-reversal symmetry; see, e.g., Phys. Rev. B 105, 155101 (2022). If the authors have experimental or theoretical reasons to support this superconducting phase being spin-triplet, they should make those arguments explicitly. Response: We thank the referee for these helpful comments. We fully agree that it is possible for spin-singlet superconductors to break time-reversal symmetry (such as 𝑑+𝑖𝑑 superconductor). However, we would like to clarify that the time-reversal- symmetry-breaking spin-singlet superconducting states cannot explain the observed re- entrance in superconductivity under an applied magnetic field. Excluding the possibility for spin-singlet superconducting states: After meticulously ruling out trivial origins, especially the vortex physics, the reentrance in superconducting states cannot be explained by time-reversal symmetry breaking spin-singlet pairing. The reason is that the Zeeman effect of the magnetic field tends to suppress the spin-singlet superconducting state by inducing a depairing effect on spin-singlet Cooper pairs, whereas it does not affect the equal-spin pairing states. Therefore, the reappearance of the superconducting state under a larger magnetic field is not consistent with the spin-singlet Cooper pairs scenario. Instead, the observed re-entrance is more consistent with equal-spin pairing, where the spin magnetic moments of the Cooper pairs couple to the magnetic field, lowering the energy of the superconducting state. To clarify this point, we have revised the main text as follows: “In addition to the pronounced hysteresis, the observation of re-entrant superconductivity as the magnetic field increases suggests that the SC-II phase can be stabilized by the magnetic field. After carefully ruling out trivial origins, the spin- singlet scenario cannot support the field-induced re-entrant superconductivity. This is because the Zeeman effect of the magnetic field tends to suppress the spin-singlet superconducting state by inducing a depairing effect on spin-singlet Cooper pairs, whereas it does not affect the equal-spin pairing states. Therefore, the reappearance of the superconducting state under a larger magnetic field is not consistent to the spin-singlet Cooper pairs scenario. Instead, the observed re-entrance is more consistent with equal-spin pairing, where the spin-magnetic moments of the Cooper pairs couple to the magnetic field, lowering the energy of the superconducting state.” Comments 3: In the explanation of their proposed mechanism, the authors do not address the effect of out-of-plane magnetic fields as seen in their measurements. In the model involving superconducting domains, why should an out-of-plane field component destroy hysteresis? Response: We thank the referee for this very insightful question. As noted in Chinese Phys. Lett. 38, 037403 (2021), RbV₃Sb₅ is a quasi-two-dimensional superconductor, where the out-of-plane resistivity is approximately 33 times greater than the in-plane resistivity at low temperatures. In such materials, an out-of-plane magnetic field rapidly suppresses superconductivity due to the orbital effects, which generates many vortices within the superconducting plane. When the applied magnetic field has a sizable out-of-plane component, many vortices will be created, and the vortices will dominate the transport properties of the superconducting films. As the transport is primarily dominated by vortices, the magnetoresistance behaves in a similar way for both the up-sweep and down-sweep fields. Therefore, the magnetoresistance hysteresis is weakened by the out-of-plane component of the magnetic field. As shown in Fig. R11 (Fig. 2g in the main text), the upper critical field decreases rapidly as the out-of-plane tilt angle of the magnetic field increases, which is consistent with our expectation above. Fig.R11. The out-of-plane angle dependence of the magnetoresistance, adapted from Fig. 2g in the manuscript. To address this point, we have revised the main text as follows: “ Moreover, as shown in Fig. 2g, the magnetoresistance hysteresis is weakened by the out-of-plane component of the magnetic field. The reason is as follows: RbV₃Sb₅ is a quasi-two-dimensional superconductor75, where the out-of-plane resistivity is approximately 33 times greater than the in-plane resistivity at low temperatures. In such materials, an out-of-plane magnetic field rapidly suppresses superconductivity due to the orbital effects, which generates many vortices within the superconducting plane. When the applied magnetic field has a sizable out-of-plane component, many vortices will be created, and the vortices will dominate the transport properties of the superconducting films. As the transport is mostly dominated by vortices, the magnetoresistance behaves in a similar way for both the up-sweep and down-sweep fields, leading to the disappearance of hysteresis. A similar behavior is also observed in the heavy-fermion superconductor UCoGe76, where a slight tilt of the magnetic field toward the c-axis leads to a significant decrease in the upper critical field and the disappearance of hysteresis.” Comments 4: It would be useful to see data showing if there are any memory effects. For example, if the sample goes from -1 T to 1 T, back to -1 T and again to 1 T, does the initial forwards sweep look the same as the second forward sweep? Fig. R12. The magnetoresistance curves show no memory effects since three consecutive magnetic field sweeps exhibit no discernible differences. Response: We thank the referee for this helpful comment. As demonstrated in Fig. R12, no memory effects are observed during three consecutive sweeps of the magnetic field between±1 T. This lack of memory effect suggests its intrinsic reversibility under magnetic fields, driven by spin-triplet pairing. We have added this result to our new Supplementary Information (Fig. S11). Comments 5: Given the difference in critical fields for different field directions, it is interesting to note that the split between SC-I and SC-II seems to occur at the same 0.2 Ω 3 2 1 r n s d d t magnetic field value regardless of the magnetic field direction, if I am interpreting Fig. S6 correctly. I am curious if the authors have any comment on this behavior. Response: We thank the referee for this very insightful question. The upper critical field of our sample shows a clear two-fold symmetry, as shown in Fig. R13. This two-fold symmetry provides evidence that there is nematicity in the normal state. For a sample with finite thickness, the nematicity effect dominates the upper critical field and gives rise to the two-fold symmetry. However, the magnetic field-driven re-entrance is due to the Zeeman coupling between the magnetic field and the spin-magnetic moments of the Cooper pairs, which is nearly independent of the magnetic field direction. Therefore, we expect that the critical magnetic field of the reentrance, which is controlled by the Zeeman effect, is insensitive to the magnetic field direction. Fig.R13. The re-entrant superconductivity under different in-plane magnetic fields with angles ranging from -180° to 180°. Comments 6: The similarity between the hysteresis seen in this compound and that observed in UCoGe is interesting; see Appl. Phys. Lett. 98, 132507 (2011). The hysteresis in UCoGe has the same qualitative behavior and also disappears for fields tilted slightly out of plane. I believe this similarity merits at least a mention, if not discussion. Response: We thank the referee for providing this useful information. We agree that this interesting work should be mentioned. The in-plane hysteresis observed in our work is indeed similar to the behavior seen in heavy fermion superconductors UCoGe. In both cases, a slight tilt of the magnetic field toward the c-axis leads to a significant decrease in the upper critical field and the disappearance of hysteresis. We have incorporated the relevant citation into the manuscript to highlight this connection: “A similar behavior is also observed in the heavy-fermion superconductor UCoGe76, where a slight tilt of the magnetic field toward the c-axis leads to a significant decrease in the upper critical field and the disappearance of hysteresis.” In addition, certain points in the text could benefit from clarification: •In the abstract: “Strikingly, at a fixed magnetic field, the finite-resistance state can be quenched to the zero-resistance state by applying a large current.” If I understand correctly, the statement should be: “…by applying and subsequently removing a large current.” It also might be worth noting here that the purpose of the current is sample heating. •Lines 13-15: “In addition, at a fixed magnetic field, a large current can suppress superconductivity completely at the finite-resistance state by heating. Incredibly, upon the removal of the current and cooling, the sample can be quenched to a zero-resistance state.” I understand why the authors refer to superconductivity in the finite-resistance state, having read the full article. For the sake of introducing the subject to new readers, I suggest something like the following: “In addition, by heating the sample with a large current and subsequently cooling it, all at a fixed magnetic field within the hysteresis loop, the sample can be brought from a finite-resistance state to a zero-resistance state.” •Lines 35-36: “Each V-Sb slab exhibits a two-dimensional (2D) kagome net of V atoms, forming a hexagonal structure with Sb atoms occupying central positions…” I believe it would be more accurate to say, “Each V-Sb slab exhibits a two-dimensional (2D) kagome net of V atoms, with Sb atoms occupying central positions of the hexagons within the kagome lattice…” Response: We are deeply grateful for the referee’s exceptionally thorough and thoughtful review. We fully agree with your suggestions and have carefully incorporated the advice into the revised manuscript. Comments: Throughout the text, “quench” is used to refer to the heating of the sample above the superconducting temperature as well as the cooling of the sample. This terminology is somewhat confusing, as “quench” can be used to refer to superconducting magnets unintentionally becoming normal, or to a quick-cooling process. It may be more clear to refer simply to heating and cooling via current. Response: We thank the referee for this very helpful suggestion. We agree that the terminology “quench” may lead to confusion, so we replaced it with “current heating and cooling,” as you suggested. Comments: Were all sweeps starting at a finite magnetic field actually cooled in that field, as opposed to zero-field-cooled? If so, was in-field cooling always done from room temperature, or from some other temperature? Response: To clarify, the measurement process begins with the sample being cooled in zero field, after which the magnetic field is swept. During the initial sweep from 0 to 1 T, the superconducting domains are aligned. Then, the subsequent sweeps (1 T to -1 T and -1 T to 1 T) are performed, as shown in our earlier response regarding the absence of memory effects. Additionally, we performed a field-cooling process, where the sample was cooled under a 1 T field and then sweeping the magnetic field from 1 T to -1 T and -1 T to 1 T. The results from the field-cooling process and zero-field-cooling process (sweeping between ±1 T after zero-field cooling) are identical. Comments: In the section “A proposed mechanism…”, when discussing the current heating/cooling experiment, the authors could be more explicit that in their model, cooling under an applied field results in superconducting domains that are predominantly aligned with the field, thus leading to a zero-resistance state. Response: We thank the referee for this suggestion. We modified the text as: “Moreover, at a fixed positive magnetic field ~ 0.27 T (Fig. 2h), we can use a large current to heat the metastable finite-resistance state with both positive and negative domains and then remove the current under a fixed magnetic field, the system can transition to the superconducting ground state, where superconducting domains predominantly align with the applied magnetic field. This current heating and cooling experiment strongly supports the superconducting domain picture of our theory which suggests that the SC-II state is a metastable state containing superconducting domains with different spin polarizations.” Comments: In Fig. S3a, “lower critical field” is used. I believe the authors are referring to the critical field that is at a lower value due to hysteresis. However, given that “lower critical field” already has a defined meaning for superconductors, a different term should be used (perhaps “greater” and “lesser” critical fields?) and the caption should clarify what is meant by these. Response: We thank the referee for this excellent suggestion. To avoid any potential ambiguity or misunderstanding, we have revised the text to use the terms ‘greater’ and ‘lesser’ instead. Comments: The caption of Fig. S6 indicates that it is showing data for a variety of magnetic field directions, presumably with the current direction fixed. However, the text (lines 166-168) says, “In RbV Sb , the re-entrant superconductivity appears in all 3 5 current directions with respect to a fixed in-plane magnetic field direction (as shown in Fig. S6 of Supplementary Information).” Which of these is accurate? Response: We apologize for the earlier misinformation and appreciate your attention to this detail. To clarify, we used a 3D vector magnet to rotate the in-plane magnetic field while keeping the current direction fixed. We have now corrected this in the manuscript. Comments: Additionally, at what temperature were the data for Fig. S6 taken? This should be indicated in the figure caption. •Similarly, at what temperature were the data for Fig. 3c taken? This should be indicated in the figure caption. Response: We thank the referee for these comments. The temperature information has been modified. Comments: Is the normal-state resistance of the sample actually 1 Ohm, as shown in Fig. 1d, or is this a normalization? Response: Yes, the actual measured resistance of the sample is around 1 Ohm, not a normalization. Comments: The authors cite Ref. 22 on line 143 when discussing the superheating- supercooling effect. However, I do not believe that Ref. 22 touches on superheating/supercooling. Is this the correct reference? The following changes are also suggested to improve figure legibility: •Adding tick marks along the axes for all figures that don’t have them: 2d, 3a, 3c, S4, S5a&b, S6a Response: We thank the referee for these comments. The citation has been modified, and the tick marks have been added. Comments: Possibly adding arrows and/or labels to Fig. 2h along the “quenching” segments, to indicate the heating and subsequent cooling •Changing the color of font used for “T1” and “T2” in Fig. 4 to make them more visible against the background color Response: We added arrows in Fig. 2h and changed the color of font to white. Reviewer #4 (Remarks to the Author): I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, {"round": 2, "reviewer_comments": {"reviewer_1": "First I'd like to thank the authors for putting together such a detailed response, including further measurements. I appreciate that this takes a lot of time and effort.\nUnfortunately, I do not think that the case for p-wave superconductivity has been strengthened. While I appreciate that the authors have added \"Signatures of\" to the title, I don't think this qualifier is sufficient. I think claims of p-wave need to be robust, and I do not see robust evidence here.\n1) Regarding hysteresis. The authors claim to have measured 10 CsV3Sb5 samples and seen no hysteresis. They show data for only one sample, however, and this sample DOES show hysteresis. This is seen as a shift in the \"forward sweeping\" curve compared to the \"backward sweeping\" curve. This is particularly clear in the vector version of the image show in the SI (as opposed to the bitmap in the rebuttal), where one can zoom in. The hysteresis is approximately 40 mT. This is not particularly different than the hysteresis the authors see for RbV3Sb5, which is around 250 mT at most. Crucially, the size of the hysteresis depends sensitively on angle, and without an angle dependence for the CsV3Sb5 sample, it is unclear that the authors are actually at 0 degrees (if they were off by even 5 degrees then they would see a substantial reduction in hysteresis, as per figure 2 in the main text). While the hysteresis seen in the CsV3Sb5 sample is fractionally smaller than the value of Bc itself, I have no reason to believe that it SHOULD be scaling with Bc.\nIt is worth noting that there is now another paper on the arxiv https://arxiv.org/pdf/2406.13164v1 that sees hysteresis in CsV3Sb5 devices and claims that this supports chiral superconductivity in THAT material as well. I do not find their claims to be convincing, either.\n2) Regarding arguments about resistance minima in the hysteresis curves: the theory that the authors cite, as well as the two measurements they show, are for granular superconductors. It is clear that granular superconductors will behave differently in a magnetic field than single crystals do because the nature of the pinning sites is completely different. I don't think this comparison is valid for the present study, which uses single crystals.\n3) Regarding \"nematicity\". The authors state in their rebuttal that \"Therefore, we fully agree with the referee that the symmetry breaking from D6h to D2h is a result of the normal-state nematicity. The superconducting state itself is not a reason for the symmetry breaking.\" If that is the case, then I do not understand why the authors are including the anisotropy in upper critical field at all. In the SI, they use a triplet state to explain the anisotropy in the upper critical field, and make connections to calculations that show \"The upper critical field exhibit anisotropic behavior in single-layer limit when considering both the spin orbital-parity-coupling effect and nematic normal state.\"\nI don't understand why \"spin orbital parity coupling\" is needed at all, in fact. Once the symmetry is D2h, an \"s-wave\" superconductor, with exactly zero spin orbit coupling, can have an anisotropic Hc2. For example, both the phonons and the density of states will have D2h symmetry, so there is no reason that the critical field should not as well. So I don't understand this argument about a \"spin orbital-parity-coupling effect\".\n4) Regarding reentrance and zero resistance: the authors essentially attribute the lack of zero resistance and the lack of reproducibility on the reentrant behaviour to device inhomogeneity, invoking percolation and disorder even at temperatures as low as 100 mK, allegedly. It is extremely hard to understand how one could rule out flux trapping as a source of hysteresis if the devices are this inhomogeneous. This also directly contradicts the earlier statement \"If the observed hysteresis were due to trivial mechanisms, such as flux trapping, rather than intrinsic properties of the superconducting states, it should also appear in CsV₃Sb₅ as well.\" If it is indeed an intrinsic property of the superconducting state, then why does it only occur in ~50% of the samples studied?\nEven more concerning, in response to point 4 (the Pauli limit), the authors write \"In the main text, we presented the data for Sample 1 (Fig. 2b) which has a thickness of (cid:1)0 = 27.6 nm. Given the layer separation of about 1 nm, our sample is far from the atomically thin limit.\" This directly contradicts what the authors wrote in response to point 3, \"Unlike bulk superconductors, thin films are quasi-2D systems that are particularly susceptible to quasiparticles, thermal fluctuations, and fabrication defects\". 27.6 nm is not anywhere near the \"thin film\" limit where thermal fluctuations could play such a role.\n5) Regarding orbital suppression. If the resistivity ratio is approximately 33, then, if the critical field is orbital limited, the ratio of Hc2_ab to Hc2_c should be equal to (rho_c/rho_ab)^.5 = 6 (see e.g. https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.66.1125) While I cannot find data on these devices for B||c, studies I can find (e.g. https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.043018) suggest that the ratio of Hc2_ab to Hc2_c is only half this value (i.e. it is more strongly suppressed for field in plane than if it was orbitally limited based on the resistance ratio). This suggests it is instead pauli limited for field in plane.", "reviewer_2": "The authors have properly addressed most of my questions, and the manuscript has been improved after the revision.\nHowever, I still have a few minor suggestions and comments. Once there are addressed, the paper should be suitable for publication in Nature communications. My comments are as follows:\n1. It seems that pairing states of CsV3Sb5 and RbV3Sb5 are different, at least according to experimental evidence. I think it deserves to add some discussions of this distinction in the supplementary materials, as it may be helpful for readers in the field.\n2. In page 9, the authors argue that spin-singlet scenario cannot support the field-re-entrant superconductivity, by stating that “This is because the Zeeman effect of the magnetic field tends to suppress the spin-singlet superconducting state by inducing a depairing effect on spin singlet Cooper pairs,…”. I agree with this statement for the case of large magnetic field. However, in Fig.3b, the critical magnetic field B is only about 0.1 T, which may not induce strong depairing effect for spin-singlet pairing. Can d+id and d-id pairings account for observed two domains, given that they are time-reversal breaking and can couple with magnetic field?\n3. The hysteresis and re-entrant superconductivity is explained by the formation of superconducting domains for varying magnetic field. The current heating and coupling measurements support this scenario for hysteresis (Fig.2h at 230 mK). I am curious whether similar measurements can be done at the temperature where re-entrant superconductivity appears, such as at 450 mK. This measurement will further strength the proposed scenario and conclusions of the paper.", "reviewer_3": "The authors have addressed the majority of my concerns and questions with their revisions, which have clarified their arguments for their interpretation of the data. While the results presented are not proof of p-wave superconductivity, they are certainly suggestive; the new title of the article is more appropriate to this level of evidence.\nOne minor correction from the first review seems to have been missed.\nOriginal comment: The caption of Fig. S6 indicates that it is showing data for a variety of\nmagnetic field directions, presumably with the current direction fixed. However, the\ntext (lines 166-168) says, “In RbV3Sb5, the re-entrant superconductivity appears in all\ncurrent directions with respect to a fixed in-plane magnetic field direction (as shown in\nFig. S6 of Supplementary Information).” Which of these is accurate?\nResponse from authors: We apologize for the earlier misinformation and appreciate your attention to this detail. To clarify, we used a 3D vector magnet to rotate the in-plane magnetic field while keeping the current direction fixed. We have now corrected this in the manuscript.\nLines 188-190 in the new version still read, “in RbV3Sb5, the re-entrant superconductivity appears in all current directions with respect to a fixed in-plane magnetic field direction.”\nPlease correct this to indicate that the current direction is fixed and the magnetic field direction is changing.\nIn addition, please note the temperatures of measurements shown in Figs. S3, S4, S6, S8, S11, and S12 (either in figures themselves or in captions).\nWith the above changes, I would recommend the manuscript for publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "The referee has misunderstood the nature of the hysteresis observed in our measurements. Here we will explain it in details.\n1. First, the minor hysteresis observed in CVS, as cited by the referee, differs fundamentally from what we report in RVS. In CVS, a careful inspection reveals that the critical field in the forward sweep (positive field direction) is slightly higher than in the backward sweep, i.e. H > H . This is a characteristic of trivial hysteresis, F B commonly attributed to sweep-rate-dependent artifacts or weak vortex pinning. In contrast, our data exhibit the opposite trend, with H < H , clearly distinguishing it F B from such conventional behavior.\n2. Moreover, the hysteresis in CVS is extremely subtle and strongly dependent on the magnetic field sweep rate. As shown in the attached figure of another sample, increasing the sweep rate leads to a more pronounced offset, while reducing the sweep rate to a sufficiently slow value eliminates the hysteresis entirely. This reinforces our conclusion that the hysteresis in CVS is trivial and unrelated to the unconventional effects we observed.\nFig. R1: The hysteresis of CVS as a function of magnetic field sweep rate. At a sweep rate of 0.2 T/min, a small difference on order of ~40 mT under ±9 T sweeping is observed with the trivial case: H F > H B. This tiny difference gradually vanishes as the sweep rate is reduced.\n3. Finally, while we refrain from commenting on the details of the other experiment cited by the referee (arxiv preprint), we must emphasize two critical differences:\na) In that study, the forward sweep again yields H > H , opposite to our F B observation.\nb) The measurements there are performed with the magnetic field applied out- of-plane, whereas our experiments use an in-plane field configuration. This fundamental difference in geometry significantly affects the underlying physics and cannot be overlooked.\n4. In light of these distinctions, the hysteresis in our experiment is qualitatively different from the cases referenced by the referee and should not be conflated.\nFurthermore, the time-reversal symmetry breaking in the superconducting states is further evidenced by our new added experiments of zero-field superconducting diode effect, as seen in the new supplementary Fig. S3. In general, observing a zero-field superconducting diode effect (SDE) requires the breaking of both inversion symmetry and time-reversal symmetry. Since AV Sb possesses the inversion 3 5 symmetry, an ideal device would not, in principle, exhibit an SDE unless inversion symmetry is locally broken by inhomogeneities such as oxidation, disorder, or defects. This is why no SDE is observed in our previous samples as shown in Fig. 1. In our supplementary experiment, the sample was plasma-irradiated to break inversion symmetry, and thus the SDE could be observed if the time reversal symmetry is broken in the superconducting states. In the following Fig. R2, the differential resistance curve shows different critical currents for forward and backward sweeps, demonstrating the existence of superconducting diode effect. The observation of zero-field SDE provides evidence of time-reversal symmetry breaking in the superconducting state.\nFig. R2 The superconducting diode effect observed in irradiated RbV 3 Sb 5 sample.\n2) Regarding arguments about resistance minima in the hysteresis curves: the theory that the authors cite, as well as the two measurements they show, are for granular superconductors. It is clear that granular superconductors will behave differently in a magnetic field than single crystals do because the nature of the pinning sites is completely different. I don't think this comparison is valid for the present study, which uses single crystals.\nResponse: The referee has misunderstood the logic of our argument regarding the origin of the magnetic hysteresis.\nTo clarify, our conclusion does not rely on a direct analogy to granular superconductors. In fact, we explicitly state that our sample is high-quality single crystal, and therefore the hysteresis we observed cannot be attributed to grain boundary effects or intergranular Josephson coupling, which are typical in granular systems.\nThe reasoning is as follows:\nWe observe a distinct and reproducible magnetic hysteresis that cannot be explained by any known trivial mechanism in single-crystalline superconductors. To further explore possible origins, we surveyed the literature and found that some superficially similar hysteresis has been reported in granular superconductors. However, we emphasize that our case is fundamentally different:\n1. Our sample is single crystal, not granular, which is also what the referee pointed out, ruling out intergranular effects.\n2. The resistance minima scenario discussed in granular systems is also not applicable here, as it does not fit the nature of the hysteresis we report.\nThus, our reference to granular superconductor is not a claim of similarity, but rather a demonstration that even the most superficially comparable phenomena cannot account for our observations.\nIn conclusion, we stand by our statement that the hysteresis we report represents a novel and previously unreported magnetic response in a single-crystalline superconductor, distinct from any known behaviors in either granular or conventional systems.\n3) Regarding \"nematicity\". The authors state in their rebuttal that \"Therefore, we fully agree with the referee that the symmetry breaking from D6h to D2h is a result of the normal-state nematicity. The superconducting state itself is not a reason for the symmetry breaking.\" If that is the case, then I do not understand why the authors are including the anisotropy in upper critical field at all. In the SI, they use a triplet state to explain the anisotropy in the upper critical field, and make connections to calculations that show \"The upper critical field exhibit anisotropic behavior in single-layer limit when considering both the spin orbital-parity-coupling effect and nematic normal state.\"\nI don't understand why \"spin orbital parity coupling\" is needed at all, in fact. Once the symmetry is D2h, an \"s-wave\" superconductor, with exactly zero spin orbit coupling, can have an anisotropic Hc2. For example, both the phonons and the density of states will have D2h symmetry, so there is no reason that the critical field should not as well. So I don't understand this argument about a \"spin orbital-parity-coupling effect\".\nResponse: The referee has misunderstood our intention in referencing spin-orbit-parity coupling. We are not attempting to invoke it arbitrarily or forcefully. Rather, we emphasize that this coupling is an intrinsic feature of the system, rooted in its symmetry and band structure—whether or not one chooses to highlight it. As noted in our previous response, spin-orbit-parity coupling is fundamentally present in the material and should be considered in interpreting the full physical behavior.\nTo clarify the logic:\nThe two-fold symmetric superconductivity observed in our experiment arises from two intertwined contributions:\n1. Nematicity and orbital effect, which have been widely discussed in prior studies.\n2. Spin-orbit-parity coupling, which naturally emerges from the symmetry constraints of the crystal and the nature of the electronic states.\nWe do not claim that spin-orbit-parity coupling is the only origin of the observed anisotropy, but it is a necessary and unavoidable component of the overall physical picture. Ignoring its role would oversimplify the interpretation and overlook an aspect of underlying mechanisms of twofold superconducting symmetry.\n4) Regarding reentrance and zero resistance: the authors essentially attribute the lack of zero resistance and the lack of reproducibility on the reentrant behaviour to device inhomogeneity, invoking percolation and disorder even at temperatures as low as 100 mK, allegedly. It is extremely hard to understand how one could rule out flux trapping as a source of hysteresis if the devices are this inhomogeneous. This also directly contradicts the earlier statement \"If the observed hysteresis were due to trivial mechanisms, such as flux trapping, rather than intrinsic properties of the superconducting states, it should also appear in CsV₃Sb₅ as well.\" If it is indeed an intrinsic property of the superconducting state, then why does it only occur in ~50% of the samples studied?\nEven more concerning, in response to point 4 (the Pauli limit), the authors write \"In the main text, we presented the data for Sample 1 (Fig. 2b) which has a thickness of 𝑑0 = 27.6 nm. Given the layer separation of about 1 nm, our sample is far from the atomically thin limit.\" This directly contradicts what the authors wrote in response to point 3, \"Unlike bulk superconductors, thin films are quasi-2D systems that are particularly susceptible to quasiparticles, thermal fluctuations, and fabrication defects\". 27.6 nm is not anywhere near the \"thin film\" limit where thermal fluctuations could play such a role.\nResponse: The referee appears to misunderstand the experimental context and interpretation of our data. In experimental condense matter physics, it is well understood that not all devices yield identical results, even when targeting the same physical phenomenon. For instance, in the case of twisted graphene—a system now widely accepted to host superconductivity—not all fabricated sample exhibit superconductivity, and among those that do, only a subset reveal more delicate or emergent features. This variability is not due to the absence of intrinsic physics, but rather reflects the technical difficulty and sensitivity of device fabrication, as well as the phase rigidity of some phenomena.\nThis distinction is critical: the fact that a phenomenon is observed in only a portion of devices—in our cases, approximately 50%—does not imply that it is extrinsic. Instead, it reflects the fragility of the re-entrant superconductivity, which is clearly more sensitive than the primary superconducting state. Such fragility is expected for emergent phases that sit near instability boundaries in the phase diagram and require precise tuning of the experimental conditions.\nWe note that the discussion of thermal fluctuations in shaping this behavior has already been discussed in our previous response. The “thin films” are “quasi-2D systems” and “far from the atomically thin limit”. These claims are not mutually exclusive.\nFinally, we emphasize that developing the ability to reliably detect such subtle and fragile phenomena is precisely what separates technically advanced experimental programs from routine measurements. As experimentalists, refining our technique and improving device yield is a lifelong pursuit—not an indication of uncertainty, but a testament to the complexity of the physics involved.\n5) Regarding orbital suppression. If the resistivity ratio is approximately 33, then, if the critical field is orbital limited, the ratio of Hc2_ab to Hc2_c should be equal to (rho_c/rho_ab)^.5 = 6 (see e.g. https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.66.1125) While I cannot find data on these devices for B||c, studies I can find (e.g. https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.043018) suggest that the ratio of Hc2_ab to Hc2_c is only half this value (i.e. it is more strongly suppressed for field in plane than if it was orbitally limited based on the resistance ratio). This suggests it is instead pauli limited for field in plane.\nResponse: The referee has an inappropriate assumption, which leads to an opposite conclusion. The out-of-plane magnetic field critical value is about 12.5 mT (See Fig. S1), the in-plane critical field is about 600 mT (See Fig. 2b). So, the ratio is about 40, not the referee’s assumption ~3.\nThe referee’s objection regarding orbital versus Pauli limitation is based on an incorrect assumption. If we follow the established relation Hc /Hc ≈√𝜌 /𝜌 , the expected ratio 2ab 2c 𝑐 𝑎𝑏 would be ~6. However, in our system, we observe a much larger anisotropy (~40), then we can get the opposite conclusion of the referee’s.\nReferee #2\nThe authors have properly addressed most of my questions, and the manuscript has been improved after the revision. However, I still have a few minor suggestions and comments. Once there are addressed, the paper should be suitable for publication in Nature communications. My comments are as follows:\n1. It seems that pairing states of CsV Sb and RbV Sb are different, at least according 3 5 3 5 to experimental evidence. I think it deserves to add some discussions of this distinction in the supplementary materials, as it may be helpful for readers in the field.\nResponse: We thank the referee for the suggestion and fully agree with it. In response, we have added a one-page discussion on the differences between CsV Sb and RbV Sb 3 5 3 5 in the supplementary materials.\nComment 2. In page 9, the authors argue that spin-singlet scenario cannot support the field-re-entrant superconductivity, by stating that “This is because the Zeeman effect of the magnetic field tends to suppress the spin-singlet superconducting state by inducing a depairing effect on spin singlet Cooper pairs,…”. I agree with this statement for the case of large magnetic field. However, in Fig.3b, the critical magnetic field B is only about 0.1 T, which may not induce strong depairing effect for spin-singlet pairing. Can d+id and d-id pairings account for observed two domains, given that they are time-reversal breaking and can couple with magnetic field?\nResponse: We thank the referee for this thoughtful question. The referee raises a valid point regarding the scale of the magnetic field in Fig. 3b.\nWe wish to emphasize two key clarifications:\n1. Field-Reentrant Superconductivity Is Observed at 0.1 T — a Signature Incompatible with Spin-Singlet Pairing\nWhile 0.1 T is indeed moderate in absolute terms, it is significant in our system because superconductivity is first suppressed and then re-emerges as the magnetic field increases — a hallmark of field-reentrant behavior. In conventional spin-singlet superconductors, the magnetic field, whether via Zeeman effect or orbital suppression, usually acts monotonically to destroy superconductivity.\nIn fact, this kind of non-monotonic behavior is widely considered a strong indicator of unconventional pairing, often pointing toward spin-triplet states or other field-enhanced mechanisms. The revival of the superconducting state with increasing field, even at ~0.1 T, rules out the monotonic suppression expected from singlet states.\n2. On the d + id and d – id Scenario: It Cannot Account for the Observed Behavior\nWhile it is true that d+id and d−id pairings are time-reversal symmetry-breaking spin-singlet states, and that they can form domains with opposite chirality, this scenario cannot explain the field-reentrant superconductivity we observe. Here’s why:\nThis is because that although the d+id and d−id states can couple to the magnetic field, the energy difference between the d+id and d−id states, arising from the coupling between the magnetic field and orbital magnetization, is very small under 0.1 T compared to the energy scale of the superconducting gap. This small energy difference is insufficient to account for the pronounced reentrance we observe.Therefore, while the domain formation in d+id and d-id superconductivity is an interesting possibility in other contexts, it does not provide a mechanism for the field-induced re-entrance of superconductivity, as observed in our experiment.\nConclusion\nIn summary, although the field scale is modest, the non-monotonic superconducting response to the field is the key evidence here. The behavior is incompatible with spin-singlet scenario, including chiral d+id or d-id, and is better understood in terms of field-stabilized spin-triplet pairing. We hope this clarifies our reasoning and appreciate the referee's engagement with this subtle point.\n3. The hysteresis and re-entrant superconductivity is explained by the formation of superconducting domains for varying magnetic field. The current heating and coupling measurements support this scenario for hysteresis (Fig.2h at 230 mK). I am curious whether similar measurements can be done at the temperature where re-entrant superconductivity appears, such as at 450 mK. This measurement will further strength the proposed scenario and conclusions of the paper.\nResponse: We agree with the referee that performing such measurements near the re-entrant superconducting regime will help confirm the scenario, and we have conducted exactly the proposed experiment as shown in the figure below.\nThe red curve shows the original field-dependent resistance exhibiting re-entrant superconductivity. We then applied a current heating and cooling protocol at 0.1 T (purple curve). After the current heating and cooling process, the resistance drops from the peak into the superconducting state, and the system remains superconducting up to a higher positive critical field (purple) than in the original curve (red).\nThis behavior is consistent with what we observe at lower temperatures (e.g. in Fig. 2h), and further supports the proposed scenario and conclusions of the paper.\nReviewer #3 (Remarks to the Author):\nThe authors have addressed the majority of my concerns and questions with their revisions, which have clarified their arguments for their interpretation of the data. While the results presented are not proof of p-wave superconductivity, they are certainly suggestive; the new title of the article is more appropriate to this level of evidence.\nOne minor correction from the first review seems to have been missed.\nOriginal comment: The caption of Fig. S6 indicates that it is showing data for a variety of magnetic field directions, presumably with the current direction fixed. However, the text (lines 166-168) says, “In RbV3Sb5, the re-entrant superconductivity appears in all current directions with respect to a fixed in-plane magnetic field direction (as shown in Fig. S6 of Supplementary Information).” Which of these is accurate?\nResponse from authors: We apologize for the earlier misinformation and appreciate your attention to this detail. To clarify, we used a 3D vector magnet to rotate the in-plane magnetic field while keeping the current direction fixed. We have now corrected this in the manuscript.\nLines 188-190 in the new version still read, “in RbV3Sb5, the re-entrant superconductivity appears in all current directions with respect to a fixed in-plane magnetic field direction.”\nPlease correct this to indicate that the current direction is fixed and the magnetic field direction is changing.\nIn addition, please note the temperatures of measurements shown in Figs. S3, S4, S6, S8, S11, and S12 (either in figures themselves or in captions).\nWith the above changes, I would recommend the manuscript for publication.\nResponse: We thank the referee’s detailed and helpful suggestions. We apologize for the misinformation provided in the track-changes manuscript. We have modified and added the corresponding temperature information in the supplementary figures.\nReviewer #4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have satisfactorily addressed my concerns, and the paper has been improved by moderating the direct claims about spin-triplet pairing. In my opinion, the observed unconventional hysteresis and superconductivity re-entrace are quite intriguing, which significantly contributes to the understanding of intriguing pairing of kagome superconductor. Therefore, I recommend this paper for publication in Nature communications."}, "author_response": ""}]

Decision letter