Rotational symmetry breaking in superconducting nickelate Nd0.8Sr0.2NiO2 films

The infinite-layer nickelates, isostructural to the high-Tc cuprate superconductors, have emerged as a promising platform to host unconventional superconductivity and stimulated growing interest in the condensed matter community. Despite considerable attention, the superconducting pairing symmetry of the nickelate superconductors, the fundamental characteristic of a superconducting state, is still under debate. Moreover, the strong electronic correlation in the nickelates may give rise to a rich phase diagram, where the underlying interplay between the superconductivity and other emerging quantum states with broken symmetry is awaiting exploration. Here, we study the angular dependence of the transport properties of the infinite-layer nickelate Nd0.8Sr0.2NiO2 superconducting films with Corbino-disk configuration. The azimuthal angular dependence of the magnetoresistance (R(φ)) manifests the rotational symmetry breaking from isotropy to four-fold (C4) anisotropy with increasing magnetic field, revealing a symmetry-breaking phase transition. Approaching the low-temperature and large-magnetic-field regime, an additional two-fold (C2) symmetric component in the R(φ) curves and an anomalous upturn of the temperature-dependent critical field are observed simultaneously, suggesting the emergence of an exotic electronic phase. Our work uncovers the evolution of the quantum states with different rotational symmetries in nickelate superconductors and provides deep insight into their global phase diagram.

The conventional superconductivity with transition temperature (Tc) lower than 40 K was successfully explained by the Bardeen-Cooper-Schrieffer (BCS) theory, in which the electrons with anti-parallel spins and time-reversed momenta form Cooper pairs, and the superconducting order parameter is of isotropic s-wave symmetry 1,2 .However, the discovery of high-temperature superconductivity (Tc > 40 K) in cuprates is beyond the expectation of the BCS theory, and the superconducting order parameters of cuprates are believed to be of nodal d-wave symmetry 3,4 .Thereafter, the mechanism of unconventional high-Tc superconductivity has become one of the most important puzzles in physical sciences.Recently, the observation of superconductivity in infinitelayer nickelates with a maximal Tc of 15 K in Nd1-xSrxNiO2 has motivated extensive researches in this emerging new superconducting family [5][6][7][8][9] .Mimicking the d 9 electronic configuration and the layered structure including CuO2 planes of the cuprates, the isostructural infinite-layer nickelates are promising candidates for high-Tc unconventional superconductivity [5][6][7][8][9] .Discerning the similarities and the differences between the nickelates and the cuprates, especially in the symmetry of the superconducting order parameters, should be of great significance for understanding the mechanism of unconventional high-Tc superconductivity.
Theoretical calculations have suggested that the nickelates are likely to give rise to a d-wave superconducting pairing, analogous to the cuprate superconductors.However, the consensus has not been reached and there are several proposals, including dominant d x 2 -y 2 -wave 10,11 , multi-band d-wave 12,13 , and even a transition from s-wave to (d+is)wave and then to d-wave depending on the doping level and the electrons hoping amplitude 14 .Experimentally, through the single-particle tunneling spectroscopy, different spectroscopic features showing s-wave, d-wave, and even a mixture of them are observed on different locations of the nickelate film surface, which complicates the determination of the pairing symmetry in the nickelates 15 .The London penetration depth of the nickelates family are also measured, and the results on La-based and Prbased nickelate compounds support the existence of a d-wave component 16,17 .However, the Nd-based nickelate, Nd0.8Sr0.2NiO2,exhibits more complex behaviors that may be captured by a predominantly isotropic nodeless pairing 16,17 .The pairing symmetry of the superconducting order parameter in the nickelate superconductors, the fundamental characteristic of the superconducting state, is still an open question, thus further explorations with diverse experimental techniques are highly desired.
In addition to the mystery of the superconducting pairing symmetry, the strong electronic correlation in nickelates is another element that makes the nickelate systems intriguing.The strong correlation is theoretically believed to play an important role in the nickelates systems 8,9,18,19 and the strong antiferromagnetic (AFM) exchange interaction between Ni spins has been experimentally detected 20 .Generally, the strong correlated electronic systems are anticipated to host a rich phase diagram and multiple competing states including superconductivity, magnetic order, charge order, pair density wave (PDW), etc 21,22 .In the nickelate thin films, the charge order, a spatially periodic modulation of the electronic structure that breaks the translational symmetry, has been experimentally observed by the resonant inelastic X-ray scattering (RIXS) 23- 25 .However, the charge order is only observable in the lower doping regime where the nickelates are non-superconducting.The interplay between the superconductivity and the charge order as well as other underlying symmetry-broken states is still awaiting explorations.With these motivations, we investigate the polar (θ) and azimuthal (φ) angular dependence of the critical magnetic field and the magnetoresistance of the infinite-layer nickelate Nd0.8Sr0.2NiO2superconducting films.The perovskite precursor Nd0.8Sr0.2NiO3thin films are firstly deposited on the SrTiO3 (001) substrates by pulsed laser deposition (PLD).The apical oxygen is then removed by the soft-chemistry topotactic reduction method using CaH2 power.Through this procedure, the nickelate thin films undergo a topotactic transition from the perovskite phase to the infinite-layer phase, and thus the superconducting Nd0.8Sr0.2NiO2thin films are obtained 5 .Figure 1a presents the schematic crystal structure of Nd0.8Sr0.2NiO2.In agreement with the previous reports 5 , the temperature-dependence of the resistance R(T) exhibits metallic behavior from room temperature to low temperature followed by a superconducting transition beginning at T c onset of 14.7 K (Fig. 1b).Here, T c onset is determined at the point where R(T) deviates from the extrapolation of the normal state resistance (RN).
Note that the R(T) curve shows a considerably broad superconducting transition with a smooth tail, which can be described by the Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional (2D) superconductors [26][27][28][29] .As shown in the inset of Fig. 1b, the R(T) curve under 0 T can be reproduced by the BKT transition using the Halperin-Nelson equation 30 , ] (R0 and b are materialdependent parameters, and T c ' is the superconducting critical temperature), yielding the BKT transition temperature T BKT of 8.5 K.An apparent difference is also noted between the R(T) curves under in-plane and out-of-plane magnetic fields (inset of Fig. 1b), implying the anisotropy of the superconductivity.
To obtain more insight into the anisotropic superconductivity in Nd0.8Sr0.2NiO2thin films, the critical magnetic field and magnetoresistance under different magnetic field orientations are measured.Here, the Corbino-disk configuration is used to eliminate the influence of the current flow in angular dependent magnetoresistance measurements 31 , which cannot be completely avoided in standard four-probe measurements 32,33 .The schematic image and the optical photo of a Corbino-disk device are shown in Fig. 1c.
To start with, the temperature-dependence of the critical field Bc is measured under the magnetic field applied along the c-axis (denoted as ⊥), the a/b-axis (∥, 0⁰), and the ab diagonal direction (∥, 45⁰).Here, Bc is defined as the magnetic field required to reach 50% of the normal state resistance (RN = 98.9 Ω), and the Bc(T) curves are collected in Fig. 1d.The T-linear dependence of B c⊥ (T), and the (Tc-T) Ginzburg-Landau (G-L) formula 34 : where ϕ 0 is the flux quantum, ξG-L(0) is the zero-temperature G-L coherence length and dsc is the superconducting thickness.The consistency with the 2D G-L formula near Tc indicates the 2D nature of the superconductivity in the Nd0.8Sr0.2NiO2thin films.To further study the dimensionality of the superconductivity, we measure the polar angular dependence of the critical magnetic field Bc(θ) for Nd0.8Sr0.2NiO2thin film at T = 6 K.
Here, θ represents the angle between the magnetic field and the c-axis of the Nd0.8Sr0.2NiO2.As shown in Fig. 1e, the Bc(θ) curve exhibits a prominent angular dependence of the external magnetic field, and a cusp-like peak is clearly resolved around θ = 90⁰ (B⊥c-axis).The peak in the Bc(θ) around 90⁰ can be well reproduced by the 2D Tinkham model and cannot be captured by the 3D anisotropic mass model, which qualitatively demonstrates the behavior of 2D superconductivity 35 (inset of Fig. 1e).
To obtain a more comprehensive depiction of the anisotropy, the polar angular  and g, respectively), while four more subtle kinks at 90⁰ ± 20⁰ and 270⁰ ± 20⁰ can be seen under 12 T (marked by black dashed line Fig. 1g).Considering the crystal structure of the infinite-layer nickelate, the humps and kinks with relatively small variations may originate from the magnetic moment of the rare-earth Nd 3+ with 4f electrons, which slightly affects the superconductivity of adjacent NiO2 planes.and 315⁰ (45⁰ to the a/b-axis) from 8 K to 11 K, and becomes indistinguishable when the temperature is increased to 14 K (the top panel in Fig. 2c).
To confirm the correspondence between the minimum of the C4 R(φ) and the a/baxis of the Nd0.8Sr0.2NiO2thin films, control experiments have been carefully conducted.
Specifically, the Nd0.8Sr0.2NiO2Corbino-disk device is remounted and remeasured after rotating a finite in-plane degree Δφ.Through the comparison between the initial results R(φ) and the remeasured results after rotation R(φ+Δφ), the minimum (maximum) of the C4 symmetry are fixed with the a/b-axis (45⁰ to the a/b-axis), verifying the C4 symmetry of the R(φ) is an intrinsic property of the Nd0.8Sr0.2NiO2thin films (Supplementary Fig. 3).Note that the R(φ) curve at 14 K is already larger than the normal state resistance RN of 98.9 Ω, and is almost isotropic within the experimental resolution.Thus, the observed C4 symmetry below 11 K should be related to the superconducting characteristics of the Nd0.8Sr0.2NiO2thin films.Moreover, considering that the quasi-2D nature of the superconductivity in the Nd0.8Sr0.2NiO2and the large magnetoresistance amplitudes of the C4 anisotropy (ΔRC4/R) (approximately 10% under 8 K and 8 T in Fig. 2c; around 20% under 5.5 K and 12 T in the Supplementary Fig. 6), the C4 symmetry is not likely owing to the magnetic moment of the Nd 3+ between the NiO2 planes as discussed in Supplementary Fig. 8, but should be ascribed to the superconductivity in the NiO2 planes.Previously, the C4 symmetric R(φ) has also been reported in the cuprates.However, the C4 anisotropy is observed in the normal state with a magnitude of merely 0.05% and is attributed to the magnetic order 36 , representing a different mechanism from our observations.The in-plane critical field of a d-wave superconductor is theoretically predicted to exhibit a C4 symmetric anisotropy owing to the d-wave pairing symmetry 37 .The C4 anisotropic critical field as well as the C4 anisotropic magnetoresistance have been experimentally used to determine the dwave superconductivity in cuprate superconductors 38 and heavy fermion superconductors 39 , etc. Therefore, the C4 symmetry of our R(φ) curves is supposed to First, through the aforementioned remounted measurements after an in-plane rotation of Δφ, the C2 symmetry is confirmed to be invariant with respect to the sample mounting, since R(φ+Δφ) can nicely overlap with R(φ) after slight shifts (Supplementary Fig. 3).Second, the C2 features cannot be explained by the trivial misalignment of the magnetic field, because the φ angles of the C2 symmetry maxima are nearly corresponding to the minima of the misalignment-induced features (Supplementary Fig. 4).Third, the Corbino-disk configuration excludes the anisotropic vortex motion due to the unidirectional current flow, which has been reported in the previous works using the standard four-probe or Hall-bar electrode 32,33 .Fourth, the C2 features superimposed on the C4 symmetric R(φ) can be consistently observed in many other samples in the large magnetic field, demonstrating the strong reproducibility of the C2 and C4 anisotropy (Supplementary Fig. 13).
To quantitatively study the evolution of the C2 and C4 anisotropy of the R(φ), the C2 components (ΔRC2) and C4 components (ΔRC4) of each R(φ) curve at different temperatures and magnetic fields are extracted through trigonometric function fitting (Supplementary Fig. 7).Among them, the fitting curve of the R(φ) under 2 K and 16 T is shown in Fig. 2d and e.Here, the ratio between the average resistance of the R(φ) and the normal state resistance (Ravg/RN) are used as an independent variable for an intuitive comparison.Figure 2f shows ΔRC4 as a function of Ravg/RN under different magnetic fields, which exhibit similar parabolic behaviors with maxima approximately around 50% RN.Differently, the ΔRC2 are monotonically decreasing with increasing Ravg/RN as shown in Fig. 2g.With increasing magnetic field, the ΔRC4 show subtle decreasing tendency, while the ΔRC2 monotonically increase, exhibiting a magnetic field-mediated competition between the ΔRC4 and the ΔRC2.The parabolic Ravg/RNdependence of the ΔRC4 can be understood as the resistance anisotropy due to the superconductivity anisotropy becomes smaller when approaching the superconducting zero-resistance state or the normal state.However, the monotonic Ravg/ RN -dependence of the ΔRC2 cannot be explained by such a scenario, suggesting a different origin of the C2 symmetry.Generally, the observation of spontaneous rotational symmetry breaking in R(φ) curves would indicate the existence of nematicity 31,40 .However, in our measurements, the C2 component has a relatively small weight in the anisotropy of the R(φ) compared with the C4 symmetry (<12%), inconsistent with previous results of nematic superconductivity with a primary C2 feature 32 .To understand the origin of the C2 anisotropy, we recall the RIXS-detected charge order in the nickelates that is along the Ni-O bond direction and exhibits a competitive relationship with the superconductivity [23][24][25] .Considering the Ni-O bond direction of our C2 feature and the magnetic field-mediated competition between the ΔRC4 and the ΔRC2, our observation of the C2 anisotropy might result from the charge order in Nd0.8Sr0.2NiO2thin films.
The magnetic field suppresses the superconductivity and may alter the competition between the anisotropic superconductivity with C4 symmetry and the charge order fluctuations with C2 symmetry in our Nd0.8Sr0.2NiO2,leading to the monotonically magnetic field-dependent decrease of the ΔRC4 and increase of the ΔRC2 in our observations (Fig. 2g).As previously reported, the Sr doping dramatically lowers the onset temperature of the charge order in the nickelates 23 , which may explain the occurrence of the charge order in our Nd0.8Sr0.2NiO2only at low temperatures.In addition, since the C2 feature further breaks the rotational symmetry, our observation favors a stripe-like charge order in the nickelates, which is supported by theoretical proposals 18,19 , although further investigations are still needed.The phase diagram demonstrates an evolution of the superconducting states manifesting different rotational symmetries, depending on the external magnetic field.
Specifically, in the grey region labeled as ~isotropy (B < 5 T), the B c∥, 0⁰ 50% (T) and B c∥, 45⁰ 50% (T) curves overlap with each other and, consistently, the R(φ) curves are nearly isotropic within our measurement resolution, indicating the isotropic superconductivity (Fig. 3e).Under the magnetic field from 5 T to 11.5 T (the blue region labeled as C4), the R(φ) curves exhibit C4 rotational symmetric anisotropy (Fig. 3d), which should be ascribed to the superconductivity of the Nd0.8Sr0.2NiO2films since it disappears in the normal state (Fig. 2c).Simultaneously, B c∥, 0⁰ 50% (T) and B c∥, 45⁰ 50% (T) curves split in this region, according with the emergence of C4 anisotropy.When the magnetic field is increased above 11.5 T (the pink region labeled as C4+C2), an additional C2 anisotropy is observed in the R(φ) curves as a superimposed modulation on the predominant C4 anisotropy, which breaks the C4 symmetry (Fig. 3b   and c varying parameters 14 .Experimentally, the s-and d-wave mixture has been reported by the previous STM study 15 .The second transition with the rotational symmetry turning from C4 to C4+C2 takes place around 11.5 T. Also, the second transition is accompanied with an anomalous upturn of the in-plane critical field, unveiling the emergence of an electronic state unexplored before.As discussed above, we speculatively ascribe the second transition to the emergence of the charge order in the Nd0.8Sr0.2NiO2films.It is normally believed that the long-range charge order disfavors the superconductivity 18,22 . In our Nd0.8Sr0.2NiO2films, with increasing magnetic field, the superconductivity gets suppressed and the stripe charge order fluctuation with short-range correlation emerges, accounting for the relatively small C2 symmetric anisotropy in R(φ) curves.The coexistence between the ordered phases with different broken symmetries is relatively rare, and the intertwined orders would give rise to more unexpected quantum phenomena and more complex phase diagram.In our Nd0.8Sr0.2NiO2thin films, the short-range stripe order coupled to the superconducting condensation may induce a secondary PDW state, in which the superconducting order parameter is oscillatory in space 22,41,42 .Through pairing in the presence of the periodic potential of the charge order, the Cooper pairs gain finite center-of-mass momenta 42 , which is also a signature of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase that features an upturn of the critical magnetic field at the low temperatures 43,44 , resembling our observations.These phases were not expected previously in the Nd0.8Sr0.2NiO2systems.Our findings suggest that the nickelates would be a potential option to explore these exotic states.
Our experimental observations have important implications on current theoretical debates.The isotropic superconductivity requires a primary pairing mechanism that is not expected in optimally hole-doped superconducting cuprates.The successive phase transitions reveal a subtle balance between several competing interactions that are unique for infinite-layer nickelates and also cannot be explained as in cuprates.In the Mott-Kondo scenario, the phase transition may be attributed to the competition between the Kondo coupling and the AFM spin superexchange coupling 14 .The Kondo coupling can produce local spin fluctuations that support isotropic s-wave pairing 45 , while the AFM coupling favors d-wave pairing.In Nd1-xSrxNiO2, the superconductivity emerges around the border of the Kondo regime 46 .Therefore, applying magnetic field may suppress the local spin fluctuations, tilt the balance towards AFM correlation, and thus induce a secondary d-wave component, which explains the emergent C4 symmetry by either dor (d+is)-wave pairing 14 .The charge order also competes with the AFM correlation 19 as well as the superconductivity.Previous experiments have shown that the charge order phase boundary may even penetrate into the superconducting dome.
Further increasing magnetic field may reduce both the superconductivity and AFM correlation, thus promote the charge order, and lead to the observed C2 symmetry.The interplay of the Kondo effect, the AFM correlation, the superconductivity, and the charge order provides a potential playground for novel correlated phenomena, which is well beyond a simple scenario.Any candidate theory should be made in conformity with all these experimental observations.
In    confirming that the C2 anisotropic feature is not due to the trivial misalignment between the sample plane and the magnetic field.In the large magnetic field regime, the polar angular dependent anisotropy is weak, because the anisotropy ratio of the in-plane critical field to the out-of-plane critical field (Γ = Bc∥/Bc⊥) becomes smaller with increasing field, and is gradually approaching 1 (see Fig. 1d in the main text).Thus, the influence of the misaligned magnetic field is negligible in large magnetic field regime.The C4 symmetry with a smaller magnitude observed in the normal state of the Nd0.8Sr0.2NiO2can be ascribed to the magnetic moment of the rare-earth Nd 3+ .The magnetic order-induced C4 symmetric R(φ) curves have also been reported in the cuprates previously 36 , which is observed in the normal state with a magnitude of 0.05%, consistent with our Nd 3+ -resultant C4 anisotropy in the normal state.However, such a phenomenon cannot explain the C4 anisotropy measured in the superconducting transition region, which shows different orientations (C4 exhibits minima at 0⁰, 90⁰, 180⁰ and 270⁰ in the superconducting region, instead of 45⁰, 135⁰, 225⁰ and 315⁰ in the normal state) and much stronger magnitude (two orders of magnitude larger).
stimulated growing interests in the condensed matter community.Despite numerous researches, the superconducting pairing symmetry of the nickelate superconductors, the fundamental characteristic of a superconducting state, is still under debate.Moreover, the strong electronic correlation in the nickelates may give rise to a rich phase diagram, where the underlying interplay between the superconductivity and other emerging quantum states with broken symmetry is awaiting exploration.Here, we study the angular dependence of the transport properties on the infinite-layer nickelate Nd0.8Sr0.2NiO2superconducting films with Corbino-disk configuration.The azimuthal angular dependence of the magnetoresistance (R(φ)) manifests the rotational symmetry breaking from isotropy to four-fold (C4) anisotropy with increasing magnetic field, revealing a symmetry breaking phase transition.Approaching the low temperature and large magnetic field regime, an additional two-fold (C2) symmetric component in the R(φ) curves and an anomalous upturn of the temperature-dependent critical field are observed simultaneously, suggesting the emergence of an exotic electronic phase.Our work uncovers the evolution of the quantum states with different rotational symmetries and provides deep insight into the global phase diagram of the nickelate superconductors.

Fig. 1 |
Fig. 1 | Structure and the quasi-two-dimensional superconductivity in Nd0.8Sr0.2NiO2.a, Crystal structure of the infinite-layer nickelate Nd0.8Sr0.2NiO2.b, Temperature dependence of the resistance R(T) at zero magnetic field from 2 K to 300 K.The inset shows the R(T) curves below 20 K at 0 T (black circles), B∥ = 16 T (red circles), and B⊥ = 16 T (purple circles).Here, the B∥ is applied along the a/b-axis and B⊥ is along the c-axis.The blue solid line represents the BKT transition fitting using the Halperin-Nelson equation.c, Schematic image and optical photo (inset) of the Corbino-disk configuration for polar (θ) angular dependent magnetoresistance R(θ) measurements on the Nd0.8Sr0.2NiO2thin film.Here, θ represents the angle between the magnetic field and the c-axis of the Nd0.8Sr0.2NiO2.d, Temperature dependence of the critical magnetic field Bc(T) for the magnetic fields along the c-axis (denoted as ⊥),the a/b-axis (∥, 0⁰), and the ab diagonal direction (∥, 45⁰).Here, the Bc is defined as dependence of the magnetoresistance R(θ) at various temperatures and magnetic fields are measured, and two representative R(θ) curves at 5 K under 12 T and 8 T are shown in Figs.1f and g, respectively.The most notable features are two sharp dips at 90⁰ and 270⁰, corresponding to B⊥c-axis.The two sharp dips correspond to the cusp-like peak in the Bc(θ) curve, resulting from the quasi-2D anisotropy.With varying temperatures and magnetic fields, the two sharp dips are observed in all R(θ) curves measured in the superconducting region (Supplementary Fig.2), further confirming the quasi-2D nature of the superconducting Nd0.8Sr0.2NiO2thin films and suggesting that the layered superconducting NiO2 planes should mainly account for the superconducting properties in our transport measurements.Additionally, small humps at approximately 90⁰ ± 45⁰ and 270⁰ ± 45⁰ are observed under 8 T and 12 T (marked by black arrows in Figs.1f

Fig. 2 |
Fig. 2 | Azimuthal (φ) angular dependence of the magnetoresistance in Nd0.8Sr0.2NiO2.a, Schematic of the Corbino-disk device for azimuthal (φ) angular dependent magnetoresistance measurements.Here, φ represents the angle between the magnetic field and the a/b-axis of the Nd0.8Sr0.2NiO2.b, c, Azimuthal angle dependence of the magnetoresistance R(φ) at different temperatures under B = 8 T in the polar plot (b) and rectangular plot (c).d, e, Azimuthal angle dependence of the magnetoresistance R(φ) at T = 2 K under B = 16 T in the polar plot (d) and rectangular plot (e).Here, the logarithmic scale is used on the resistance-axis to specifically demonstrate the C2 symmetric feature.The blue solid lines are fits with the trigonometric function: R = Ravg + ΔRC4×sin(4φ) + ΔRC2×sin(2φ), where Ravg is the averaged magnetoresistance and ΔRC4 and ΔRC2 are the C4 and C2 components, respectively.The light blue area in (d) is a guide to the eye, representing the C2 anisotropy.f, g, Four-fold components ΔRC4 (f) and two-fold components ΔRC2 (g) versus the ratio between the averaged magnetoresistance and the normal state resistance (Ravg/ RN) under different magnetic fields.Here, the values of the C2 and C4 components are extracted by the trigonometric function fitting.
imply the C4 symmetric critical field of the predominant d-wave pairing in the Nd0.8Sr0.2NiO2thin films.The deduced d-wave pairing in the Nd0.8Sr0.2NiO2thin films cannot be definitely determined by our transport measurement results solely and requires further experimental investigations (e.g.phases-sensitive measurements).Remarkably, with further increasing magnetic field, additional two-fold (C2) symmetric signals are observed as small modulations superimposed on the primary C4 symmetry in the R(φ) curves.The representative R(φ) curve under 2 K and 16 T in the polar and rectangular plots are shown in Fig.2d and e, respectively.In addition to the predominant C4 symmetric R(φ), an additional C2 signal can be clearly discerned by R(φ = 0⁰ and 180⁰) being smaller than R(φ = 90⁰ and 270⁰), indicating the rotational symmetry breaking between a-axis and b-axis.In the following, elaborated experiments and analysis are discussed to exclude the possible extrinsic origin of the C2 feature.

Fig. 3 |
Fig. 3 | B versus T phase diagram for the Nd0.8Sr0.2NiO2.a, B versus T phase diagram for in-plane magnetic fields.The white region above B c∥, 0⁰ onset (T) (open circles) represents the normal state, and the dark blue region below B c∥, 0⁰ 1% (T) (open triangles) denotes the zero-resistance state (defined by R < 1% of RN).Between B c∥, 0⁰ onset andB c∥, 0⁰ 1% (T) is the superconducting transition region, which is separated into ~isotropy, C4, and C4+C2 regions from small magnetic field to large magnetic field regime, ).At the same time, the B c∥, 0⁰ 50% (T) curve shows an anomalous upturn at the low temperatures above 11.5 T, deviating from the saturating Bc at the low temperatures expected for a conventional superconducting state.The simultaneous occurrences of the rotational symmetry breaking in the R(φ) curves and the enhanced superconducting critical field behavior strongly indicate the emergence of an exotic state.The superconducting phase diagram may reveal two phase transitions characterized by spontaneous rotational symmetry breakings.The first transition occurs at approximately 5 T indicated by the change from isotropic superconductivity to C4 anisotropy.Considering that the R(φ) curves show the symmetry of the in-plane critical field and could reflect the superconducting pairing 37 , the first rotational symmetry breaking may suggest a transition from s-wave to d-wave superconductivity.The transition from s-wave to d-wave superconductivity is reminiscent of the theoretical phase diagram hosting s-, d-and (d+is)-wave superconductivity for nickelates with Fig. S1.R(T) and R(H) curves for the determination of Bc.R(T) under different magnetic field applieds along the a/b-axis (a), the ab diagonal direction (b), and the caxis (c), d, R(B) at different θ angles at 6 K.The B c∥, 0⁰ (T), B c∥, 45⁰ (T), B c⊥ (T) and

Fig. S2 .
Fig. S2.R(θ) curves showing the quasi-2D anisotropy.Polar angular dependence of magnetoresistance R(θ) at different temperatures under 2 T (a and b), 8 T (c and d), 12 T (e and f) and 16 T (g and h).Here, θ represents the angle between the magnetic field and the c-axis of the Nd0.8Sr0.2NiO2.The left panels show the rectangular plots and the right panels show the corresponding polar plots.

Fig. S3 .
Fig. S3.Remounted measurements after an in-plane rotation Δφ2.a, Schematic of the sample remounting after an in-plane rotation.b, c, The comparison between the initial results R(φ) and the remeasured results R(φ+Δφ1) after an in-plane rotation Δφ1 ~ 13⁰.The R(φ) and R(φ+Δφ1) curves under 16 T at 2 K (b) and 3 K (c) show the C4+C2 anisotropy.The R(φ+Δφ1) curves are shifted 13⁰ and 1.6 Ω for comparison, which can nicely overlap with initial R(φ) curves, confirming that the C4+C2 anisotropy is intrinsic.

Fig. S4 .
Fig. S4.Exclusion of the possibility that the C2 features are due to the misalignment.a, R(φ) under 16 T at 2 K, and the fit with the trigonometric function R = Ravg + ΔRC4×sin(4φ) + ΔRC2×sin(2φ), where Ravg is the averaged magnetoresistance and ΔRC4 and ΔRC2 are the C4 and C2 components, respectively (red solid line).b and e, R(φ) under 8 T at 2 K showing the misalignment-induced two-sharp-dips feature, and

Fig. S6 .
Fig. S6.R(φ) curves at different temperatures and magnetic fields.R(φ) curves in the polar plots at different temperatures and magnetic fields from 1 T to 16 T (a to p), showing the evolution of the rotational symmetry from isotropic to C4 symmetric and then to C4+C2 anisotropic.The temperatures and magnetic fields are labeled in the corresponding plots.The blue areas are guides to the eye, representing the C2 anisotropy.

Fig. S7 .
Fig. S7.Fit of the C4+C2 symmetric R(φ) curves at different temperatures and magnetic fields.R(φ) curves and the corresponding trigonometric function fits at different temperatures and magnetic fields from 12 T to 16 T (a to e).Each R(φ) curve is fitted by the trigonometric function R = Ravg + ΔRC4×sin(4φ) + ΔRC2×sin(2φ) to extract the C2 component (ΔRC2) and C4 component (ΔRC4).Here, the logarithmic scale is used on the resistance-axis in (c to e) to specifically demonstrate the C2 symmetric feature.

Fig. S8 .
Fig. S8.C4 anisotropy in the R(φ) curves of the superconducting state and the normal state.a, Rectangular plot of the R(φ) curve measured at 2 K and 16 T where the Nd0.8Sr0.2NiO2 is in the superconducting transition, showing C4+C2 anisotropy.b and e, Rectangular plot (b) and polar plot (e) of the R(φ) curves measured at 11 K and 16 T where the Nd0.8Sr0.2NiO2 is in the normal state, showing C4 and C2 anisotropy with different orientations and two orders of magnitude smaller compared with (a).c and f, Rectangular plot (c) and polar plot (f) of the R(φ) curves measured at 20 K and 16 T where the Nd0.8Sr0.2NiO2 is in the normal state, showing C4 anisotropy.d, R(T) curves under 0 T and 16 T.Three arrows indicate the corresponding temperature where (a), (b) and (c) are measured.The R(φ) curves shown here are measured with a current of 5 μA.

Fig. S10 .
Fig. S10.Reproducible critical field behaviors in sample S2.Temperature dependence of the critical magnetic field measured along a/b-axis (a), the ab diagonal direction (b), and c-axis (c), d, Comparison between the B c∥, 0⁰ 50% (T) and B c∥, 45⁰ 50% (T).e, Comparison between B c∥, 0⁰ 50% (T) and B c⊥ 50% (T).The blue and the orange solid lines are the 2D G-L fittings of the Bc(T) data near Tc.f, Bc at different θ angles at 6 K.The red solid line and blue solid line represent the theoretical fitting curves obtained by 2D Tinkham model and 3D anisotropic mass model, respectively.

Fig. S11 .
Fig. S11.Reproducible quasi-2D anisotropy in sample S2.Polar angular dependence of magnetoresistance R(θ) for sample S2 at different temperatures under 2 T (a and b), 8 T (c and d), 12 T (e and f), and 16 T (g and h).The left panels show the rectangular plots and the right panels show the corresponding polar plots.

Fig. S12 .
Fig. S12.Reproducible R(φ) behaviors in sample S2.Azimuthal angular dependence of magnetoresistance R(φ) for sample S2 at different temperatures under 4 T (a), 12 T (b), 16 T (c) in the polar plots, showing nearly isotropy, C4 anisotropy, and C4+C2 anisotropy, respectively.The R(φ) curve at 2 K and 16 T is further plotted in (d), where the C2 anisotropy can be better resolved.The blue area is a guide to the eye, representing the C2 anisotropy.

Fig. S13 .
Fig. S13.Reproducible C2 anisotropy superimposed on the C4 symmetry in more samples.R(φ) at different temperatures under 16 T from sample S2(a), S3(b), and S4(c), showing both the reproducible C4 and C2 symmetric features.The C4 anisotropy is manifested as four minima at 0⁰, 90⁰, 180⁰, and 270⁰ (maxima at 45⁰, 135⁰, 225⁰ and 315⁰) (a, b and c).The C2 anisotropy is manifested as R(0⁰) being larger than R(90⁰) (a) or R(0⁰) being smaller than R(90⁰) (b and c).The observations are consistent with the results of S1 shown in the main text.
1/2 -dependence of B c∥, 0⁰ and B c∥, 45⁰ near Tc show agreement with the phenomenological 2D