Introduction

In conventional superconductors, due to the fact that phase stiffness \(J\) significantly exceeds pairing energy \(\Delta\), fluctuations in the phase \(\theta\) of the superconducting order parameter \(\Delta {e}^{{{\rm{i}}}\theta }\) are energetically unfavored, rendering phase fluctuations a negligible role in determining the superconducting transition temperature \({T}_{{{\rm{c}}}}\)1,2. In contrast, in unconventional superconductors, including cuprates and iron-based materials, low charge carrier density results in small superfluid density, leading to superconductivity primarily governed by the phase fluctuations. This is demonstrated by the ubiquitous occurrence of pronounced phase fluctuations above the phase boundary of superconductivity3,4, while the quantitative relationship between \({T}_{{{\rm{c}}}}\) and phase stiffness \(J\) further underscores the central role of phase fluctuations in these materials5,6,7. However, the extensive regimes adjacent to superconductivity in the phase diagrams of unconventional superconductors are filled with various intertwined orders8,9. Their association with phase fluctuations complicates the understanding for a unified picture for the unconventional superconductivity.

Recently discovered superconducting nickelates provide a unique platform to address this issue. Superconductivity was first discovered in infinite-layer nickelates with \({T}_{{{\rm{c}}}}\) around 9–15 K10. Under applied pressure, \({T}_{{{\rm{c}}}}\) reached 30 K in trilayer nickelates11 and exceeded the boiling point of liquid nitrogen in bilayer nickelates12,13,14. More recently, compressive substrate strain enhanced superconductivity with \({T}_{{{\rm{c}}}}\) upto 40 K in bilayer nickelate thin films at ambient pressure15, analogous to that in cuprate thin films. These nickelates share similar physics with cuprates16, but notably lack translational symmetry-breaking orders at doping levels where superconductivity emerges. In particular, infinite-layer nickelates exhibit no long-range spin order17 and charge18,19,20 density-wave order near a doping level of Sr = 0.2. This distinctive feature offers an unprecedented opportunity to explore interrelation between the phase fluctuations and superconductivity, isolated from other intertwined orders.

Here, we show that the establishment of superconducting phase coherence in infinite-layer nickelate superconductors Nd0.8Sr0.2NiO2 and La0.8Sr0.2NiO2 favors a specific direction that deviates from the symmetry imposed by both electronic nematicity and underlying lattice. Electronic nematicity in normal state is evidenced by a two-fold symmetric angular resistance, where anisotropic in-plane resistance is probed by applying current along different directions. Interestingly, resistance anisotropy persists during the superconducting transition, but with the symmetric axes of angular resistance gradually shifting and ultimately stabilizing at a finite value \({\theta }_{{{\rm{SC}}}}\) as global phase coherence is achieved. Strikingly, the direction favored by electronic nematicity and superconductivity is sharply different. Further analysis of the anisotropy in non-linear current-voltage characteristics and vortex motion reveals that phase coherence is stiffer along \({\theta }_{{{\rm{SC}}}}\) than other directions. This anisotropy in phase stiffness \(J\) is fundamentally different from that of the gap energy \(\Delta\) that appears in the nematic superconductivity. Consistent observations in the cuprate superconductor La1.8Sr0.2CuO4 underscore that this unforeseen anisotropic phase stiffness may be ubiquitous among unconventional superconductors, providing crucial insights for developing a unified theoretical framework for unconventional superconductivity across various materials.

Results

Electronic nematicity in normal state

Nd0.8Sr0.2NiO2 films with a thickness of 15 nm were grown on tetragonal SrTiO3 substrates by pulsed laser deposition with successive topotactic reduction21 (see Methods). The residual resistance ratio is 4.3, and superconducting onset temperature \({T}_{{{\rm{c}}}}^{{{\rm{onset}}}}\) of the film is 15 K (temperature at which the \({R}_{{{\rm{L}}}}(T)\) deviates from the linear extrapolation of the normal state). Structural characterization indicates that the lattice symmetry of Nd0.8Sr0.2NiO2 is tetragonal without observable lattice distortions (see Methods). These observations are consistent with the literature16. We utilize the “vector current” technique (see Fig. 1a) to probe the angular resistance of tetragonal superconducting films22,23,24,25. Due to the vector nature of the electric field, the direction and amplitude of the net current \({I}_{0}\) can be continuously tuned by two orthogonal DC currents, \({I}_{{{\rm{X}}}}={I}_{0}\sin \theta\) and \({I}_{{{\rm{Y}}}}={I}_{0}\cos \theta\) (\(\theta\) is the angle between \({I}_{0}\) and crystal principal axis, see Fig. 1a). As a result, the corresponding longitudinal and transverse resistance along the \({I}_{0}(\theta )\) can be measured as \({R}_{{{\rm{L}}}}=({V}_{{{\rm{X}}}}\sin \theta+{V}_{{{\rm{Y}}}}\cos \theta )/{I}_{0}\) and \({R}_{{{\rm{T}}}}=({V}_{{{\rm{X}}}}\cos \theta -{V}_{{{\rm{Y}}}}\sin \theta )/{I}_{0}\), respectively. The current direction and amplitude can be arbitrarily tuned in situ in a single device, enabling the detection of possible anisotropy of the physical properties. The validity and reliability of the “vector current” method are confirmed by COMSOL simulations and control experiments, as shown in the Supplementary Information and Fig. S2.

Fig. 1: Angle-resolved measurements of in-plane resistance anisotropy in infinite-layer superconducting nickelate films.
Fig. 1: Angle-resolved measurements of in-plane resistance anisotropy in infinite-layer superconducting nickelate films.The alternative text for this image may have been generated using AI.
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a Illustration of the “vector current” technique. The net current \({I}_{0}\) is composed of two orthogonal injected DC current \({I}_{{{\rm{X}}}}\) (blue arrow) and \({I}_{{{\rm{Y}}}}\) (purple arrow), where the angle \(\theta\) between the net current \({I}_{0}\) and crystal principal axis as well as the amplitude of \({I}_{0}\) can be tuned continuously. The right inset shows the optical image of the patterned Nd0.8Sr0.2NiO2 film. b Temperature dependence of longitudinal resistance \({R}_{{{\rm{L}}}}\) of Nd0.8Sr0.2NiO2 at selected \(\theta\) ranging from 0° to 90°. c Angle-dependence of longitudinal resistance \({R}_{{{\rm{L}}}}(\theta )\) (blue line) and transverse resistance \({R}_{{{\rm{T}}}}(\theta )\) (green line) of Nd0.8Sr0.2NiO2 at \(T\) = 300 K, 100 K and 15 K. Black arrow marks the 45° phase shift between \({R}_{{{\rm{L}}}}(\theta )\) and \({R}_{{{\rm{T}}}}(\theta )\).

Figure 1b depicts the temperature dependence of the longitudinal resistance \({R}_{{{\rm{L}}}}(T)\) at representative directions for Nd0.8Sr0.2NiO2, showing a distinguishable resistance anisotropy between different directions. The angle dependence of the \({R}_{{{\rm{L}}}}(\theta )\) and \({R}_{{{\rm{T}}}}(\theta )\) at representative temperatures is illustrated in Fig. 1c, revealing a clear oscillation with a 180° period, observed from room temperature down to \({T}_{{{\rm{c}}}}^{{{\rm{onset}}}}\). Both \({R}_{{{\rm{L}}}}(\theta )\) and \({R}_{{{\rm{T}}}}(\theta )\) display the same periods and amplitudes, but are phase-shifted by 45°. Thus, the angular resistance can be described by Eqs. (1 and 2):

$${R}_{{{\rm{L}}}}\left(\theta \right)={R}_{{{\rm{L}}}0}-\Delta R\cos \left[2\left(\theta -{\theta }_{0}\right)\right]$$
(1)
$${R}_{{{\rm{T}}}}\left(\theta \right)=-\Delta R\sin \left[2\left(\theta -{\theta }_{0}\right)\right]$$
(2)

where \({R}_{{{\rm{L}}}0}\) is the isotropic background of the angular resistance, \(\Delta R\) is the anisotropic resistance amplitude, and \({\theta }_{0}\) represents the direction of the resistance minimum. The \(\sin (2\theta )\) relationship between resistance and current direction \(\theta\) is a natural result when the current is not aligned with the principal axes of the two-fold symmetric resistance26,27,28 (Supplementary Information Section 3). Control experiments, including angular resistance measurements on a “ring-shaped” device and on various materials (gold, anoxic SrTiO3, and NbN), were conducted to ensure the two-fold symmetric resistance is intrinsic to the Nd0.8Sr0.2NiO2 film (Fig. S2–S3). This two-fold symmetry in angular resistance indicates that the rotational symmetry of the tetragonal Nd0.8Sr0.2NiO2 thin film is broken. Since the translational symmetry in the infinite-layer superconducting nickelate is preserved due to the absence of the long-range spin17 and charge18,19,20 density-wave order, we attribute the two-fold symmetric angular resistance to electronic nematicity. Additionally, the direction of lowest resistance in the normal state appears at \({\theta }_{{{\rm{N}}}}\) = 125°, deviating from the crystal lattice’s principal axes (0° or 90°), suggesting the coexistence of \({{{\rm{B}}}}_{1{{\rm{g}}}}\) and \({{{\rm{B}}}}_{2{{\rm{g}}}}\) nematic orders. The persistence of resistance anisotropy up to room temperature (Fig. 1c) resembles that observed in La2–xSrxCuO426,28,29 and Sr2RuO430, awaiting further exploration.

Misaligned anisotropy in normal state and superconductivity

Further cooling below \({T}_{{{\rm{c}}}}^{{{\rm{onset}}}}\) reveals directional anisotropy in the sharpness of the superconducting transition (Fig. 2a). This anisotropy is quantified by defining \({T}_{{{\rm{mid}}}}\) at a given direction \(\theta\) as the temperature where resistance reaches 50% \({R}_{{{\rm{N}}}}\). Figure 2b shows that \({T}_{{{\rm{mid}}}}\) is clearly two-fold symmetric, with the maximum at \({\theta }_{{{\rm{SC}}}}\) = 160°. The \({\theta }_{{{\rm{SC}}}}\) consistently aligns at 160° in the analysis of the anisotropy of temperature when resistance reaches 30% and 90% \({R}_{{{\rm{N}}}}\) (Fig. S15). The deviation of \({\theta }_{{{\rm{SC}}}}\) from \({\theta }_{{{\rm{N}}}}\) suggests that the anisotropy of superconductivity is different from that in normal state.

Fig. 2: Shift of the symmetric axis θ0 during the superconducting transition.
Fig. 2: Shift of the symmetric axis θ0 during the superconducting transition.The alternative text for this image may have been generated using AI.
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a Superconducting transition of Nd0.8Sr0.2NiO2 at representative current direction \(\theta\), where \({R}_{{{\rm{N}}}}\) is defined as the resistance at \(T\)= 15 K. b Anisotropic superconducting transition of Nd0.8Sr0.2NiO2, where \({T}_{{{\rm{mid}}}}\) is defined as the temperature where \({R}_{{{\rm{L}}}}\) reaches the 50% of the \({R}_{{{\rm{N}}}}\). Arrow marks the maximum of \({T}_{{{\rm{mid}}}}\) the locates at \({\theta }_{{{\rm{SC}}}}\) = 160°. c Angle-dependent scaled \(\Delta {R}_{{{\rm{L}}}}(\theta )\) at a series of temperature \(T\) under zero field. d Angle-dependent scaled \(\Delta {R}_{{{\rm{L}}}}(\theta )\) measured under a series of perpendicular magnetic fields \({B}_{\perp }\) ranges from 0 T to 14 T, at fixed temperature \(T\)= 7.8 K. Curves in c, d are scaled and vertically shifted for comparison (original data are presented in Fig. S4). Triangles denote the location of the lowest longitudinal resistance as the \({\theta }_{0}\). e The symmetric axis \({\theta }_{0}\) of \(\Delta {R}_{{{\rm{L}}}}(\theta )\) as a function of temperature \(T\) and perpendicular magnetic field \({B}_{\perp }\), where the shaded area denotes the \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\). f Relevance between the zero-resistance temperature \({T}_{{{\rm{c}}}0}\) and temperature where \({\theta }_{0}\) reaches 160°, \({T}_{{\theta }_{0}={160}^{\circ }}\). The \({T}_{{{\rm{c}}}0}\) denotes the zero-resistance temperature when \({R}_{{{\rm{L}}}0}(T)\) reaches 0.5% \({\left.{R}_{{{\rm{L}}}0}\right|}_{T=15{{\rm{K}}}}\).

We investigate the evolution of the symmetric axes \({\theta }_{0}\) during the superconducting transition. Figure 2c shows the temperature dependence of the anisotropic part of the angular resistance \(\Delta {R}_{{{\rm{L}}}}\left(\theta \right)={R}_{{{\rm{L}}}}(\theta )-{R}_{{{\rm{L}}}0}\). At high temperatures, the \({\theta }_{0}\) (marked by triangles in Fig. 2c) for the normal state is fixed at \({\theta }_{{{\rm{N}}}}\) = 125°. However, a different picture appears below the \({T}_{{{\rm{c}}}}^{{{\rm{onset}}}}\): \({\theta }_{0}\) starts to shift and eventually saturate at \({\theta }_{{{\rm{SC}}}}\) = 160° near the zero-resistance temperature \({T}_{{{\rm{c}}}0}\). As the superconductivity is suppressed by the perpendicular magnetic field \({B}_{\perp }\), \({\theta }_{0}\) shifts back to the normal state value \({\theta }_{{{\rm{N}}}}\) (Fig. 2d). As shown in Fig. 2e, \({\theta }_{0}\) switches between the \({\theta }_{{{\rm{N}}}}\) for the normal state and \({\theta }_{{{\rm{SC}}}}\) for the superconducting state. This non-trivial dependence of \({\theta }_{0}\) on temperature and perpendicular magnetic field in Nd0.8Sr0.2NiO2 film provides strong evidence against the two-fold symmetric resistance caused by sample inhomogeneity or macroscopic lattice distortion. We interpret the distinct \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\) as evidence of the misaligned anisotropy in normal state and superconducting state.

Anisotropic superconducting phase stiffness

Figure 2f shows that temperature where \({\theta }_{0}\) reaches \({\theta }_{{{\rm{SC}}}}\) coincides with \({T}_{{{\rm{c}}}0}\) at each magnetic field, suggesting \({\theta }_{{{\rm{SC}}}}\) serves as an indicator for global superconducting phase coherence. To understand how the phase coherence develops, we performed current-voltage measurements at different current directions \(\theta\) to investigate superconducting phase fluctuations. Near the zero-resistance temperature, thermal fluctuations soften vortex-antivortex pairs, making them more susceptible to current-induced unbinding, leading to non-linear I-V characteristics, \(V \sim {I}^{\alpha \left(T\right)}\) (Fig. 3a). Figure 3b shows the extracted exponent \(\alpha (T)\) at representative directions \(\theta\)= 70° and 160°, where \(\alpha\) develops pronounced anisotropy below 12 K. The maximum \(\alpha (\theta )\) locates at \({\theta }_{{{\rm{PF}}}}\) = 160° (Fig. 3c), indicating the vortex-antivortex pairs are most resistant to current-unbinding at 160°. This anisotropy in the current unbinding process results in an anisotropic establishment of global phase coherence. By adopting the conventional notion of the Berezinskii-Kosterlitz-Thouless (BKT) transition31,32, the anisotropic establishment of global phase coherence manifests as the BKT transition temperature \({T}_{{{\rm{BKT}}}}\) (temperature when \(\alpha\)= 3), as shown in Fig. 3d. This is the evidence of anisotropic phase fluctuations, the intensity of which can be quantified by the normalized superconducting transition width, \(\eta=({T}_{{{\rm{mid}}}}-{T}_{{{\rm{BKT}}}})/{T}_{{{\rm{mid}}}}\). Figure 3e indicates that phase fluctuations are anisotropic, in which the phase fluctuations along 160° are weaker than other directions.

Fig. 3: Anisotropy in superconducting phase fluctuations and vortex motion.
Fig. 3: Anisotropy in superconducting phase fluctuations and vortex motion.The alternative text for this image may have been generated using AI.
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a Representative current-voltage characteristic measured at \(\theta \,\)= 160° for temperature ranges from 6.75 K to 12 K. Dashed black lines show the fit with the power-law relation given by \(V \sim {I}^{\alpha }\). b Extracted \(\alpha\) as a function of temperature for \(\theta\) = 70° and 160°. c Angle-dependent \(\alpha (\theta )\) at representative temperatures, arrows mark the maximum locates at \({\theta }_{{{\rm{PF}}}}\) = 160°. d Angle-dependent \({T}_{{{\rm{BKT}}}}(\theta )\) as obtained by \(\alpha (\theta )\) = 3, arrow marks the maximum locates at \({\theta }_{{{\rm{PF}}}}\) = 160°. e Angle-dependent \(\eta (\theta )\) as the intensity of phase fluctuations, arrow marks the minimum locates at \({\theta }_{{{\rm{PF}}}}\) = 160°. f Arrhenius plots of temperature-dependent resistance \({R}_{{{\rm{L}}}}(T)\) at \(\theta\) = 160° under a series of perpendicular magnetic field \({B}_{\perp }\). Dashed lines are fitted by the \({R}_{{{\rm{L}}}}(T)={R}_{0}\exp (-U(B)/{k}_{B}T)\). g Extracted activation energy \(U/{k}_{B}\) as a function of perpendicular magnetic field for \(\theta\) = 70° and 160°. Dashed lines indicate the \(U/{k}_{B}\) is proportional to \({\mathrm{ln}}({B}^{-1})\). h Angle-dependent \(U(\theta )/{k}_{B}\) at various perpendicular magnetic field \({B}_{\perp }\), arrows mark the maximum locates at \({\theta }_{{{\rm{VM}}}}\) = 160°.

The key to understanding anisotropic superconducting phase fluctuations lies in vortex dynamics. In type-II superconductors, vortex motion leads to phase decoherence33. Under a perpendicular magnetic field, vortices are generated and move with the activation of thermal fluctuations, described as “thermally-assisted flux-flow” (TAFF)33. As shown in Fig. 3f, the temperature-dependent resistance follows the TAFF behavior with \({R}_{{{\rm{L}}}}(T)={R}_{0}\exp (-U(B)/{k}_{B}T)\), where \({k}_{B}\) is Boltzmann’s constant and \(U(B)\) is the magnetic field-dependent activation energy for vortex34. The \(U\propto {\mathrm{ln}}(B)\) relation in Fig. 3g is consistent with the thermally assisted collective vortex-creep model34. Figure 3h shows that \(U\) exhibits two-fold symmetry under different magnetic fields up to 14 T. The largest vortex activation energy occurs at \({\theta }_{{{\rm{VM}}}}\) = 160°, suggesting vortex motion is more difficult to activate in this direction. Thus, superconducting phase coherence is stiffer along 160°.

We further investigate the anisotropy of superconducting state by extracting superfluid phase stiffness \(J\). Based on the model of thermally assisted collective vortex-creep34, the slope of \(U\propto {\mathrm{ln}}(B)\) in Fig. 3g relates to the penetration depth (See Methods), thus the phase stiffness \(J\) with the following ration

$$J=-16{{\rm{\cdot }}}\frac{a}{t}{{\rm{\cdot }}}\frac{d(U/{k}_{B})}{d{\mathrm{ln}}\left(B\right)}$$
(3)

where \(a\) is the separation length between adjacent superconducting planes for layered materials1,2, \(t\) is the sample thickness. We extract the phase stiffness along different directions. Fig. 4a shows the angle-dependent phase stiffness \(J(\theta )\) for Nd0.8Sr0.2NiO2 in a polar plot. \(J(\theta )\) exhibits clear two-fold symmetry with its maximum at \({\theta }_{{{\rm{SC}}}}\) = 160°, highlighting the anisotropy in the phase stiffness. Identical measurements and analysis were performed on La0.8Sr0.2NiO2 and La1.8Sr0.2CuO4 films to testify whether the anisotropic phase stiffness is unique to Nd0.8Sr0.2NiO2. Figure 4b, c present the angle-dependent phase stiffness \(J\left(\theta \right)\) for La0.8Sr0.2NiO2 and La1.8Sr0.2CuO4 in polar plots. In all three materials, \(J\left(\theta \right)\) displays two-fold symmetry, with its maximum located at \({\theta }_{{{\rm{SC}}}}\), aligning with the direction where \({T}_{{{\rm{mid}}}}\) is maximized. The phase stiffness to transition temperature ratio \(J/{T}_{{{\rm{c}}}}\) is approximately 1 in all three materials, considerably lower than in the conventional superconductors1,2, indicating the superconductivity is controlled by the phase stiffness. This conclusion is further supported by the alignment of \({\theta }_{{{\rm{SC}}}}\), \({\theta }_{{{\rm{PF}}}}\) and \({\theta }_{{{\rm{VM}}}}\) in all three materials (Fig. S7), demonstrating that the symmetry of superconducting properties is consistently controlled by the anisotropic phase stiffness. Since \({\theta }_{{{\rm{SC}}}}\) is decoupled from \({\theta }_{{{\rm{N}}}}\) imposed by electronic nematicity or 0°/90° of the crystal principal axes in all three materials, we infer that superconducting phase coherence is not preferentially favored along the symmetric direction imposed by electronic anisotropy. Although statistics show that exact values for \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\) fluctuate in different samples within a considerable range (Table. S2), the main conclusion—the misaligned anisotropy in normal state and superconducting state—remains highly reproducible.

Fig. 4: Anisotropic phase stiffness in three different unconventional superconductors.
Fig. 4: Anisotropic phase stiffness in three different unconventional superconductors.The alternative text for this image may have been generated using AI.
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a Linear-scale polar plot of the angle-dependent \(J(\theta )\) for Nd0.8Sr0.2NiO2. The isotropic background of the phase stiffness \({J}_{{{\rm{iso}}}}\) is defined as the minimum of \(J(\theta )\), shown as the dashed gray line. The difference between the maximum and minimum of the anisotropic \(J\left(\theta \right)\) denotes the \({J}_{{{\rm{aniso}}}}\), shown as the red arrow. The \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\) is defined as the location of \({R}_{{{\rm{L}}}}\left(\theta \right)\) minimum measured at \({T}_{{{\rm{c}}}}^{{{\rm{onset}}}}\) and \({T}_{{{\rm{c}}}0}\), respectively. The \({\theta }_{{{\rm{SC}}}}\) coincides with direction where the maximum of \(J(\theta )\) occurs in all three materials. The \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\) is marked by the gray and red lines, respectively. b, c Linear-scale polar plot of the angle-dependent \(J(\theta )\) for La0.8Sr0.2NiO2 and La1.8Sr0.2CuO4. Markers are defined as the same as that for Nd0.8Sr0.2NiO2. \(J/{T}_{{{\rm{c}}}}\) is calculated by using \({J}_{{{\rm{iso}}}}\) and \({T}_{{{\rm{c}}}0}\) to represent the phase stiffness and superconducting phase coherence temperature. In three materials, \(J/{T}_{{{\rm{c}}}}\) is found to be close to 1. This result is also consistent by choosing maximized phase stiffness \({J}_{{{\rm{aniso}}}}+{J}_{{{\rm{iso}}}}\) as \(J\).

Discussion

The present work uncovers an exotic superconductivity with anisotropic phase stiffness. While previous studies have primarily examined the role of nematic fluctuations in Cooper pair formation, our observations provide a unique insight into addressing their complexing interrelation by highlighting a connection between nematicity and the emergence of phase coherence, the critical importance of which is a defining feature of unconventional superconductors. The consistent observation of this misaligned anisotropy in both nickel- and copper-based superconductors suggests a fundamental but unforeseen mechanism governing the nematicity and superconductivity. Moreover, similar phenomena potentially exist in diverse quantum materials (Table. S3), implying that the misaligned anisotropy of normal and superconducting states is an essential and universal property. However, this misalignment has been largely overlooked in previous research, and its underlying origins remain poorly understood.

Building on our experimental data, we seek to discuss possible origins for the observations. Firstly, the shift of the symmetric axis of the anisotropic resistance is highly reproducible across different measurement setups—both in “vector current” and “ring-shaped” devices—and across various samples (See Methods). This consistency allows us to exclude macroscopic orthogonal distortions of the crystal, which would typically pin the symmetric axis along the crystal axes35. Secondly, the absence of the long-range spin or charge order in both superconducting (Nd, Sr)NiO217,18 and (La, Sr)NiO219 renders the magnetism36 and charge density-wave irrelevant. Thirdly, the consistency of observations between nickelates and cuprates (Fig. S7) suggests that features specific to nickelate thin films—such as topotactic reduction20,37, Nd3+ 4f magnetic moments38, Ruddlesden-Popper stacking faults and chemical inhomogeneity39—do not critically impact our findings. Moreover, the consistent behavior across all three materials points to an intrinsic anisotropy in phase stiffness, distinct from superconducting stripes typically induced by coupling with an external ferromagnetic overlayer in heterostructures36,40. It’s worth further exploring other possible origins from the properties of complex oxide thin films, such as orbital ordering driven by surface reconstructions41 or the Jahn-Teller effect42, or the dynamic orthorhombic distortion at atomic scale43.

Our observations of the anisotropic phase stiffness can hardly reconcile with the current understanding. While anisotropy in the electronic structure (arising from spin/charge order, electron nematicity, or other mechanisms) potentially modulates Cooper pair hopping strength along specific directions—thereby inducing anisotropic phase stiffness—our observations contradict this assumption. Specifically, the symmetry axis of the phase stiffness (\({\theta }_{{{\rm{SC}}}}\)) deviates sharply from both the axis imposed by electron nematicity (\({\theta }_{{{\rm{N}}}}\)) and crystalline axes (\(\theta\) = 0° and 90°). This misalignment suggests that the anisotropy in phase stiffness cannot be solely attributed to the normal-state electronic structure. By further comparing our observation with recent measurements of angle-resolved magnetoresistance (ADMR) on infinite-layer nickelates38,44, we demonstrate anisotropy of phase stiffness appears independent of the symmetry of the superconducting order parameter. Instead of characterizing phase stiffness, ADMR measurements reveal gap symmetry by applying an in-plane magnetic field \({B}_{\parallel }\) to break Cooper pairs. In this way, although the rotational symmetry is preserved in Nd0.8Sr0.2NiO2 near zero field44, that of the phase stiffness is found to be broken in our measurements (Fig. S7). This indicates that the superconducting state with anisotropic phase stiffness we uncover is essentially distinct from the nematic superconductivity manifested by the anisotropic order parameter45. Furthermore, the observed anisotropic phase stiffness does not arise from the coupling between superconductivity and unidirectional (charge or spin) density-wave order, which creates a secondary spatially-modulated pair-density-wave (PDW) order, naturally leading to anisotropic phase stiffness46. The lack of density-wave order in the normal state of infinite-layer nickelate superconductors rules out this scenario. The anisotropy of phase stiffness is consistent with the nature of a primary unidirectional PDW state. However, the crucial experimental evidence for PDW order, such as spectral evidence of finite-momentum pairing order or the observation of residual Fermi surface, is currently lacking for Nd0.8Sr0.2NiO2 and La0.8Sr0.2NiO2. Whether a primary PDW state underlies the anisotropy in phase stiffness remains an open and intriguing question for future study47.

Thus far, we suggest the observed superconductivity with anisotropic phase stiffness unifies two critical ingredients—phase fluctuations and nematicity—through an unforeseen and potentially universal mechanism. The misaligned anisotropy in electronic fluid and superfluid suggests that the relationship between nematicity and superconductivity extends beyond mere competition27 and cooperation48,49, providing new guidance for understanding them within a unified framework. Broadly, angle-resolved transport measurements provide valuable tools for tracking quantitative evolution of superconducting phase coherence along specific directions, stimulating the theoretical and numerical investigations of the quantum many-body models to further address the complex interrelation. Particularly, recent numerical calculation on the Hubbard model suggests the coexistence of superconductivity and partially filled stripe order in certain parameter regime50, indicating a possible microscopic origin for anisotropic phase stiffness. Further exploration of directional phase stiffness on the numerical level will promote fathoming the fundamental properties of the unconventional superconductivity and associated intertwined electronic orders. Our work motivates similar measurements on other unconventional superconductors, potentially paving the way towards the universal mechanism of unconventional superconductivity.

Methods

Film growth and structure characterization

Perovskite Nd0.8Sr0.2NiO3 films with a thickness of 15 nm were grown on single-crystal SrTiO3 (001) substrates using pulsed laser deposition (PLD, Demcon TSST) with a KrF excimer laser (\(\lambda\) = 248 nm). The SrTiO3 (001) substrates were pre-treated by HF etching and annealed at 1000 °C for 2 hours to obtain atomically flat TiO2-terminated surfaces. A uniform laser spot (1.2 × 2.8 mm) was used for ablation, generated by aperture imaging. The film growth temperature was set at 580 °C, with the oxygen partial pressure maintained at 200 mTorr. The topologically oriented reduction was conducted under the optimized conditions established in our previous study51.

High-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) was used to examine structural transformations following the reduction process (Fig. S1a). The Nd0.8Sr0.2NiO2 thin films primarily consist of the infinite planar layered structure with no significant blockiness or planar faults. Additionally, the Energy-dispersive X-ray spectroscopy (EDS) was also performed on superconducting samples to assess element distribution and composition. Fig. S1b shows sharp contrast between the film and substrate, indicating high crystal quality. EDS analysis confirmed the film composition closely matches the nominal Nd0.8Sr0.2NiO2 formula. Reciprocal space mapping (RSM) near the SrTiO3(103) reflection provided further structural details (Fig. S1c), showing that the film remained strained to the tetragonal SrTiO3 lattice, indicating the in-plane lattice symmetry of the film is consistent with that of SrTiO3. X-ray diffraction results (Fig. S1d) revealed a decrease in the c-axis lattice constant from 3.76 Å to 3.36 Å after reduction process. The X-ray reflectivity (XRR) analysis (inset of Fig. S1d) confirmed the post-reduction Nd0.8Sr0.2NiO2 film thickness to be 14.6 nm, in agreement with the nominal thickness of 15 nm film.

Atomic force microscopy (AFM) was performed on both SrTiO3 substrate (before the film deposition) and Nd0.8Sr0.2NiO2 film (after the growth) to characterize terrace orientations (Fig. S1e–f). It reveals that atomic terraces form at 75° relative to the [100] crystalline axes of the SrTiO3 surface, and this orientation is projected into the as-grown epitaxial Nd0.8Sr0.2NiO2 film. To investigate resistance anisotropy, the Nd0.8Sr0.2NiO2 film was patterned into an eight-terminal configuration for “vector current” measurements. Fig. S2g displays two-fold symmetric resistance, with minima occurring at \({\theta }_{{{\rm{N}}}}\) = 125° in the normal state, and \({\theta }_{{{\rm{SC}}}}\) = 146° near the zero-resistance temperature. These values align with prior observations in Table. S1. It’s important to note that both \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\) deviate from terrace orientation (\({\theta }_{{{\rm{AFM}}}}\) ~ 75°). Therefore, we conclude that the atomic terraces of the substrate are not related to the observed anisotropy in Nd0.8Sr0.2NiO2 film.

Angular resistance measurements

Electrodes (20 nm silver/20 nm gold) were evaporated onto the films, ensuring contact resistance below 1 Ω. For the “vector current” device, the films were patterned into an eight-terminal configuration (Fig. 1a) using standard photolithography. The central area of the device was 50 × 50 μm. DC currents were applied using Keithley Model 6221 current sources, and DC voltages were measured with a Keithley Model 2182 A nanovoltmeter. A “pulse-delta” method was employed, where the current source generated alternating positive (\(+I\)) and negative (\(-I\)) pulses, and the nanovoltmeter averaged the voltages as \([V\left(+I\right)-V\left(-I\right)]/2\). This method eliminates (1) the drift voltage of the nanovoltmeter and (2) Joule heating in the measurement of current-voltage characteristics. For the ring-shaped device (Fig. S3), angular resistance is characterized by measuring the transverse resistance with the standard lock-in technique. The diameter of the measured area is 1 mm. The temperature and magnetic field dependence of resistivity was measured using a commercial cryostat.

Validity of “vector current” method

To rule out artifacts or extrinsic factors that could contribute to the observed two-fold symmetric resistance, identical angular measurements were conducted on homogeneous and inhomogeneous gold films. In practice, thin films exhibit greater inhomogeneity (or disorder) near substrate edges compared to the more uniform central regions. Therefore, we choose devices located at the central substrate as the “homogenous gold film” and devices located near the substrate edge as the “inhomogeneous gold film”. Fig. S2a displays the angular resistance in homogeneous gold film, which lacks discernible maxima/minima and cannot be fitted to a \(\sin (2\theta )\) function. We note a residual resistance variation of ~0.005 Ω along different directions, which presumably relates to macroscopic-scale thickness variations. The background longitudinal resistance \({R}_{{{\rm{L}}}0}\) is 0.4 Ω, which is two orders of magnitude larger than the directional resistance variations. Combined with COMSOL simulations (Fig. S13), we confirm that the vector current method does not introduce spurious two-fold symmetric resistance.

However, devices fabricated near substrate edges exhibit a distorted two-fold symmetric resistance profile (Fig. S2b). To evaluate the role of structural inhomogeneity, we performed COMSOL simulations (Fig. S14), which demonstrate that macroscopic disorder (e.g., scratches, interfacial particulates) generates a weak two-fold transverse resistance superimposed on a background signal \({R}_{{{\rm{T}}}0}\) (Fig. S14). To experimentally validate this mechanism, we fabricated inhomogeneous films using distinct growth methods: magneto-sputtered NbN film, PLD-grown axonic SrTiO3 film, thermally-evaporated gold film. Fig. S2c shows the angular transverse resistance of these inhomogeneous systems, where \({R}_{{{\rm{T}}}0}\) is comparable or even larger than anisotropic amplitudes \(\Delta R\). In contrast, the \({R}_{{{\rm{T}}}0}\) remains negligible in La0.8Sr0.2NiO2, Nd0.8Sr0.2NiO2, and La1.8Sr0.2CuO4 (Fig. 1c, Fig. S3). These results exclude macroscopic structural disorder as a significant contributor to the resistance anisotropy in these nickelate and cuprate systems. We thus conclude that sample homogeneity is critical for reliable angular resistance measurements; otherwise, minor two-fold symmetric resistance artifacts may arise.

Reproducibility and statistics

The central observation of our work is the shift of symmetric axis during the superconducting transition, which indicates the misaligned anisotropy in normal state and superconducting state. Our key conclusion is that anisotropy of the superconductivity originates from anisotropic phase stiffness. To validate these findings, we conducted identical measurements on six devices to ensure (1) reproducibility of the central observation and (2) consistency of the key conclusion across different sample batches.

Table. S1 summarizes the symmetric axes of normal state (\({\theta }_{{{\rm{N}}}}\)) and superconducting state (\({\theta }_{{{\rm{SC}}}}\)) for six devices. We found that the “misaligned anisotropy” is highly reproducible, with \({\theta }_{{{\rm{N}}}}\) and \({\theta }_{{{\rm{SC}}}}\) varied within a narrow range. This variation may be attributed to either (1) discrepancies between \(\theta=0^{\circ}\) and the principal axis of the film and (2) variations in thin film quality across different sample batches (Fig. S10). We note that the \({\theta }_{{{\rm{N}}}}\) for La1.8Sr0.2CuO4 (p = 0.20) in our measurements is consistent with \({\theta }_{{{\rm{N}}}}\) for La1.8Sr0.2CuO4 (p = 0.21) measured previously by another group26, reinforcing the reproducibility of our measurements.

Identical measurements were also performed on NSNO-S3, LSNO-S1, and LSCO-S3. Figs. S5 and S7 demonstrate that the “misaligned anisotropy” is present across these three materials, exhibiting similar behaviors. Additionally, we measured non-linear I-V characteristics and thermally-assisted flux-flow, along with corresponding analyses for La0.8Sr0.2NiO2 and La1.8Sr0.2CuO4 (Supplementary Sections 8 and 9). All physical quantities related to superconductivity were consistently aligned among Nd0.8Sr0.2NiO2, La0.8Sr0.2NiO2, and La1.8Sr0.2CuO4 (Fig. S7), supporting the conclusion that “anisotropy in superconductivity originates from the anisotropic phase stiffness”.

Analysis of anisotropy of the superconducting phase stiffness

In this section, we present a detailed analysis of the anisotropic phase stiffness. The anisotropy of superconducting phase fluctuations (Fig. 3c) and vortex activation energy (Fig. 3h) suggests that the superconducting phase stiffness can be anisotropic. However, conventional characterizations of superconducting phase stiffness involve measuring penetration depth, which reflects an average of the in-plane properties \({\lambda }_{{ab}}\) and the possible difference between the \({\lambda }_{a}\) and \({\lambda }_{b}\) is hidden. Besides, there’s limited theory focusing on the in-plane anisotropy of Cooper-pair mass/penetration depth/phase stiffness29. Hence, the anisotropy of phase stiffness is hard to reveal due to limited tools. Our strategy at current stage is to use different theoretical models to extract phase stiffness, and evaluate (1) whether the value of the extracted phase stiffness is consistent with that measured in experiments, and (2) whether different models yield consistency of anisotropic phase stiffness.

We firstly analyze based on the Feigelman-Geshkenbein-Larkin (FGL) model34, which is applicable for various layered materials with large anisotropy. Figure 3g depicts that the activation energy of vortex motion exhibits a linear dependence on perpendicular magnetic fields in a semilogarithmic plot, \(U\left(B\right)\propto {\mathrm{ln}}(B)\), a behavior observed in many 2D superconductors52. This is also consistent with the two-dimensional nature of superconductivity in Nd0.8Sr0.2NiO244 and La0.8Sr0.2NiO253 thin films. According to the FGL model, the slope of \(U\left(B\right)-{\mathrm{ln}}(B)\) scales with the penetration depth \({\lambda }^{-2}\)

$$U\left(B\right)=\frac{{\varPhi }_{0}^{2}t}{64{\mu }_{0}{\pi }^{2}{\lambda }^{2}}{{\mathrm{ln}}}\left(\frac{{B}_{0}}{B}\right)$$
(4)

where \(t\) is the sample thickness, \({\varPhi }_{0}\) is the magnetic flux quantum, \({\mu }_{0}\) is the vacuum permittivity, \(\lambda\) is the penetration depth, \({B}_{0}\) is the numeric parameter, \({k}_{B}\) is the Boltzmann constant, \(e\) is the element of charge. The phase stiffness \(J\), which is proportional to the superfluid density \({n}_{s0}\), is defined as \(J={\hslash }^{2}a{n}_{s0}/4{m}^{*}\). Substituting the relation \({n}_{s0}={m}^{*}/{\mu }_{0}{e}^{2}{\lambda }^{2}\) into this expression yields \(J={\hslash }^{2}a/4{\mu }_{0}{e}^{2}{\lambda }^{2}\). This allows us to relate the slope of \(U\propto {\mathrm{ln}}\left(B\right)\) directly to the phase stiffness \(J\) by:

$$J=-\frac{a}{t}\cdot \frac{16{\pi }^{2}{{{\hslash }}}^{2}}{{e}^{2}{\varPhi }_{0}^{2}}\cdot \frac{dU/{k}_{B}}{d{{\mathrm{ln}}}\left(B\right)}$$
(5)

where \(\hslash\) is the reduced Planck constant. Equation (3) shown in the main text is obtained by inserting the definition of flux quantum \({\varPhi }_{0}=\pi \hslash /e\) into Eq. 5. The length scale \(a\) is determined by the following: it corresponds to the separation length between adjacent superconducting plane \(l\) when \({\xi }_{\perp } < l\) (e.g., in cuprates), the film thickness \(d\) when (\({\xi }_{\perp } > d\)) (e.g., in superconducting thin films), or the coherence length \({\xi }_{\perp }\) when \({\xi }_{\perp } < d\) (e.g., in three-dimensional superconductors)1,2. By considering there are two superconducting planes in one unit-cell, the \(a\) should be half of the c-axis constant, measured as 0.168 nm, 0.173 nm, and 0.66 nm for Nd0.8Sr0.2NiO2, La0.8Sr0.2NiO2, and La1.8Sr0.2CuO4, respectively. As shown in Table. S2, an excellent consistency is found between the extracted \(J\) and the measured zero-temperature phase stiffness \({J}_{0}\) in La1.8Sr0.2CuO46. Although the TGL-model describes the properties at finite temperature, it provides reasonable value of \(J\) that is close to the zero-temperature phase stiffness \({J}_{0}\), as revealed by previous study54. The accurate value of \({\lambda }_{0}\) for As for the superconducting infinite-layer nickelates has not been experimentally confirmed yet16, awaiting further experimental confirmation. We further use this analysis to extract \(J\) in other materials, we find the consistency between the extracted \(J\) and experimentally measured zero-temperature \({J}_{0}\) (Table. S2). This demonstrates that the value of the extracted phase stiffness in our experiments is reasonable.

Below we discuss the anisotropy of the phase stiffness analyzed by different strategies. Firstly, two separated theoretical models connect the phase stiffness to the vortex activation energy34,55. In both models, the slope of \(U\propto {{\mathrm{ln}}}\left(B\right)\) is proportional to \(J\) through the relation of \(d(U/{k}_{B})/d{{\mathrm{ln}}}\left(B\right)\propto {\lambda }^{-2}\propto J\). Therefore, we can measure the \(d(U/{k}_{B})/d{{\mathrm{ln}}}\left(B\right)\) along different \(\theta\) to extract anisotropic phase stiffness \(J\). Besides, the exponent \(\alpha\) in the non-linear I-V curve \(V \sim {I}^{\alpha }\) is proportional to the phase stiffness through the relation of \(\alpha=1+\pi J/T\). Furthermore, by adopting the conventional notion of BKT transition, the BKT transition temperature is directly proportional to the phase stiffness (Fig. S16) \({T}_{{{\rm{BKT}}}}=\pi J/2{k}_{{{\rm{B}}}}\). To eliminate the influence of theoretical details on the analysis, we choose the normalized phase stiffness \(\Delta J/{J}_{{{\rm{iso}}}}\) for estimating anisotropy. Fig. S8 shows the phase stiffness \(J\) extracted from (a) vortex activation energy, (b) non-linear I-V curve, and (c) BKT transition exhibits a clear two-fold symmetry, with the symmetric axis consistently aligned with \({\theta }_{{{\rm{SC}}}}\). This consistency strongly supports the existence of anisotropic superconducting phase stiffness in Nd0.8Sr0.2NiO2, La0.8Sr0.2NiO2, and La1.8Sr0.2CuO4.

Hints of the misaligned anisotropy in other materials

In our study, we define evidence for the “misaligned anisotropy” as the distinct symmetry axes in the normal and superconducting states, manifesting as a shift in the symmetric axis of angular resistance, \({R}_{{{\rm{L}}}}(\theta )\), during the superconducting transition (Fig. 2c–d). If we extend this concept to include the shift of the symmetric axis of a physical quantity related to superconductivity with external parameters (e.g., temperature, magnetic field, or doping level), we find hints of the “misaligned anisotropy” in a wide range of quantum materials. Table. S3 lists the phase-shift (\(\Delta \theta\)) of various anisotropic physical quantities in materials, including transition metal dichalcogenides (2H-NbSe256, 4Hb-TaS257), Kagome superconductors58, Moire superlattice27, and doped topological insulator (SrxBi2Se359, CuxBi2Se360) and cuprate superconductors La2-xSrxCuO4 with varying doping level29.