Introduction

Since the milestone discovery of high-temperature superconductivity in La2-xBaxCuO41, whether high-temperature superconductivity could be achieved in its nickelate analog has become an outstanding question for the community2,3,4,5,6,7,8,9,10. In 2019, the discovery of superconductivity in infinite-layer nickelate Nd1-xSrxNiO2 thin film, which shares a similar crystalline structure as the infinite cuprate superconductor CaCuO211 and shows a superconducting transition temperature (Tc) ranging from 9 to 15 K12, opens a fresh avenue for exploring the high-Tc puzzle. Very recently, nickelate superconductor family has been successfully extended to the Ruddlesden-Popper (RP) phases Lan+1NinO3n+1 with n = 213 and n = 314,15,16 and the maximum of Tc determined by transport measurement on La3Ni2O7 is beyond the liquid nitrogen temperature. However, different from that in infinite-layer nickelates Nd1-xSrxNiO2 thin film, the realization of superconductivity in the RP phase Lan+1NinO3n+1 requires a high-pressure (P) environment13, which greatly hinders the spectroscopic and thermodynamic studies of the superconducting state and thus the pairing symmetry. Up to now, the experimental study of the high-pressure superconductivity in RP phase Lan+1NinO3n+1 is still rather limited and the superconducting mechanism is under hot debate17. The exploration of superconductivity at ambient pressure in the RP phase Lan+1NinO3n+1 is one of the most important directions to solve above dilemma.

In both La3Ni2O7 and La4Ni3O10, the NiO6 octahedron is tilted from the c axis at ambient pressure13,14,15,16, leading to an orthogonal or monoclinic distortion of tetragonal structure (Fig. 1a–d). The bond angle of Ni-O-Ni between adjacent NiO6 octahedrons is 168° (167°) in La3Ni2O7 (La4Ni3O10) which is thought to be critical for the interlayer coupling between NiO planes13. These distorted structures host spin and/or charge orders as implied by density-wave-like transitions in resistivity18,19,20,21,22,23,24. In La4Ni3O10, neutron scattering and X-ray measurements indicate that the density-wave transition at ~ 136 K involves a simultaneous development of spin-density-wave (SDW) order and charge-density-wave (CDW) order25. In La3Ni2O7, although a similar SDW order has been confirmed by recent spectroscopy measurements below ~150 K26,27,28,29, the existence of CDW order is still elusive17. With increasing pressure, the volume of unit cell shrinks remarkably, making the titled NiO6 octahedron unstable13,14,15,16. In orthogonal La3Ni2O7, a pressure-induced structure transition from Amam to Fmmm has been observed by recent X-ray diffraction experiment under high pressure30. Moreover, more X-ray diffraction experiments under high pressure further demonstrate that, when the superconductivity appears with increasing pressure, a first-order structural transition into the tetragonal phase (I4/mmm) occurs almost concurrently in both pressurized La3Ni2O7 and La4Ni3O1015,16,30,31 (see Fig. 1e). Moreover, the density-wave transition completely disappears in the high-pressure tetragonal structure13,14,15,16,32,33. As such, the tetragonal structure (I4/mmm) is conjectured to be a prerequisite for the emergence of superconductivity17, whereas the relation between the density-wave order and superconductivity remains unclear29,30,31. Here, with high-oxidative-environment growth technology, we successfully synthesize tetragonal La4Ni3O10 microcrystals without octahedral tilting at ambient pressure. Both density-wave and superconducting transitions are absent up to 160 GPa in the tetragonal La4Ni3O10 phase. Such unexpected absence of electronic orders in tetragonal RP phase nickelates put constraints on the theories modeling the interactions therein and provide important insights into the mechanism of pressure-induced superconductivity.

Fig. 1: Crystal structure, phase diagram, resistance curves, and magnetic torque curves of La4Ni3O10.
figure 1

a, b The crystal structure of orthogonal La3Ni2O7 at ambient pressure and tetragonal La3Ni2O7 under pressure. There is a small tilting angle between the adjacent NiO6 octahedron in the orthogonal La3Ni2O7, which disappears in the tetragonal La3Ni2O7. c, d The crystal structure of monoclinic La4Ni3O10 at ambient pressure and tetragonal La4Ni3O10 under pressure. A small tilting angle between the adjacent NiO6 octahedron also exists in the monoclinic La4Ni3O10 and disappears in the tetragonal La4Ni3O10. e The Temperature-Pressure (T-P) phase diagram of La4Ni3O10 determined by the data in Fig. 3a, b. The information on structure under different pressure comes from ref. 15,16. The circular, triangle, and square data dots represent the Tc (onset) for sample S1, the Tc (onset) for sample S2 and density wave transition temperature (TDW) for sample S1, respectively. These data are obtained from Figs. 3a, b and Supplementary Fig. 7. There is a phase boundary between the density wave state and superconductivity, accompanied with a structure transition from monoclinic phase to tetragonal phase. f The temperature-dependent resistance curve of monoclinic (black curve, Sample S1) and tetragonal (red curve, Sample S3) La4Ni3O10 at near-ambient pressure (below 1 GPa). There is a density-wave transition evidenced by an up-turn in the resistivity at around 135 K for the monoclinic La4Ni3O10, which is absent in the tetragonal La4Ni3O10. g The temperature-dependent magnetic torque (τ(T)) data of monoclinic and tetragonal La4Ni3O10, respectively. More magnetic torque data is shown in Supplementary Fig. 9. There is a kink in τ(T) at 135 K for monoclinic La4Ni3O10, which is consistent with the anomaly in resistance due to density-wave transition in Fig. 1f. Similar evidence for the density-wave transition in monoclinic La4Ni3O10 has been also observed by specific heats and magnetic susceptibility measurements16. In contrast, both resistivity and magnetic torque results indicate that such a density-wave transition is absent in the tetragonal La4Ni3O10. a–d The green, blue, and red circles represent the atoms of La, Ni, and O, respectively.

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Results

Tetragonal La4Ni3O10 at ambient pressure

Figure 1c, d show the structures of monoclinic La4Ni3O10 (ambient pressure) and tetragonal superconducting La4Ni3O10 (high pressure phase)15,16, respectively. Both structures consist of a (LaNiO3)3 sub-structure which contains three-layer perovskites inter-grown with a LaO rock-salt sub-structure, forming the RP phase. For the ambient-pressure monoclinic phase, the bond angle of O-Ni-O in the NiO6 octahedron is about 177° (Fig. 2a). Furthermore, the adjacent NiO6 octahedron tilts towards each other. The bond angle of Ni-O-Ni between the adjacent NiO6 is about 167° (Fig. 2a). Such a tilt introduces a buckling Ni-O plane in the perovskite layer. However, in the high-pressure tetragonal phase, the bond angles of O-Ni-O in the NiO6 octahedron are all 180° (along a, b and c axis Fig. 2b). The adjacent NiO6 octahedrons arrange parallel to each other, and the bond angles of Ni-O-Ni between the adjacent NiO6 octahedrons are also 180° (Fig. 2b). The absence of tilting between the adjacent NiO6 octahedron leads to a flat plane of the perovskite sub-structure. The temperature-pressure phase diagram of La4Ni3O10 (Fig. 1e) implies that the phase boundary between the density wave state and superconductivity is close to the pressure-induced monoclinic to tetragonal structural transition. This suggests a substantial impact of the structural transition on the electronic band structure, and a tetragonal phase seems to be more favorable to the appearance of superconductivity. More importantly, such correlation between structural transition and the emergence of superconductivity has been revealed in both La3Ni2O7 and La4Ni3O1013,14,15,16. As a result, this engenders a fundamental question: can the tetragonal phase, supposedly hosting superconducting ground state, be stabilized at ambient pressure? Indeed, a metastable orthorhombic phase (Bmab) has been reported to be stabilized by postgrowth rapid cooling under 20 bar oxygen pressure34. This might give a clue that the growth of La4Ni3O10 with higher structural symmetry could be favored in a strong oxidative atmosphere. We have adopted an oxidative molten salt method under oxygen pressure to synthesize La4Ni3O10 crystal (details see Methods). The major products are the ambient-pressure tetragonal La4Ni3O10 microcrystals.

Fig. 2: The crystal structure, bond length, and bond angle comparison between different La4Ni3O10.
figure 2

a–c The connection between the NiO6 octahedron for monoclinic phase at ambient pressure (P21/a-AP), tetragonal phase at 19.5 GPa (I4/mmm–HP) and tetragonal phase at ambient pressure (I4/mmm-AP), respectively. d The atom labels of Ni atoms and O atoms. The O5 site only exists in the monoclinic phase, which becomes indistinguishable from O2 in the tetragonal phases. e The bond length and bond angle list between Ni atoms and O atoms. f The powder XRD pattern for tetragonal La4Ni3O10 under the pressure range from ambient pressure to 27.7 GPa. g An enlarged view for (f) with 2θ between 10 − 12 degree. a–d The blue and red circles represent the atoms of Ni and O, respectively.

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The structure of the as-grown La4Ni3O10 microcrystal grown under flowing oxygen environment is solved by the refinement of single crystal X-ray diffraction (SC-XRD) data. The reciprocal lattice data along a*, b* and c* axis is shown in the Supplementary Fig. 1 which satisfies a I4/mmm space group and thus corroborates the tetragonal symmetry. The details of crystal structure solved from the SC-XRD at 300 K is shown in Table 1. Furthermore, the selected area electron diffraction (SAED) data along the [100] and [001] axis shown in the Supplementary Fig. 2 also confirms the tetragonal structure solved by the SC-XRD data. For comparison, the reciprocal lattice data for the monoclinic La4Ni3O10 single crystal is shown in Supplementary Fig. 3 and the detail structure information is also shown in Table 1. Figure 2a–c summarizes the connection between the NiO6 octahedron for monoclinic phase at ambient pressure (P21/a-AP), tetragonal phase under high pressure (I4/mmm-HP)15,16 and tetragonal phase at ambient pressure (I4/mmm-AP), respectively. A general description of the connection between the NiO6 octahedron in these three structures is shown in Fig. 2d. We note that site O5 only exists in the monoclinic phase, which becomes equivalent to site O2 in the tetragonal phases due to higher symmetry. The detailed bond lengths and bond angles are also provided in Fig. 2e. Considering these results, we note that the I4/mmm-AP phase and I4/mmm-HP phase is almost identical with only slightly different specific bond lengths. The bond angle in the NiO6 octahedron (Ni3-O2-Ni4 and O4-Ni1-O4) and between the adjacent NiO6 (Ni2-O4-Ni1) are all 180° in these two tetragonal phases. Furthermore, the powder XRD pattern for tetragonal La4Ni3O10 is also collected by grinding several pieces of tetragonal single crystals under the pressure range from ambient pressure to 27.7 GPa. As shown in Fig. 2f, all diffraction peaks can be well indexed by the I4/mmm tetragonal structure. For example, to enlarge the diffraction peaks between 10 and 11.5 degrees as shown in Fig. 2g, the two diffraction peaks can be well indexed by (107) and (110) for a tetragonal structural phase and no additional splitting due to the monoclinic distortion is observed in this range of 2θ. These results indicate that the tetragonal structure is kept with pressure at least up to 27.7 GPa. The specific lattice parameters and cell volume of the tetragonal La4Ni3O10 under different pressures are shown in Supplementary Fig. 4. In addition, it should be noted that the tetragonal structure is also kept down to 110 K under ambient pressure (see Supplementary Table 1).

Table 1 Crystal data and structure refinement for La4Ni3O10
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In Fig. 1f, we compare the temperature dependent resistance curves R(T) for monoclinic La4Ni3O10 (sample S1) and ambient-pressure tetragonal La4Ni3O10 (sample S3). There is a density wave transition manifested by a huge hump in the resistivity at around 135 K for the monoclinic phase, which is consistent with the previous report35. On the other hand, the R(T) of the tetragonal phase shows a metallic behavior without notable anomalies, hinting at the absence of the density wave order. Moreover, no superconductivity is observed in our tetragonal La4Ni3O10 down to 0.4 K. Such metallic behavior without density wave transition has been repeated on three pieces of tetragonal samples (the Supplementary Fig. 6). To confirm the absence of density wave transition in our tetragonal sample, we have also performed magnetic torque measurement on both monoclinic and tetragonal La4Ni3O10 samples (see Methods for details). As shown in Fig. 1g a prominent feature around 135 K is observed in the monoclinic La4Ni3O10 sample, implying the impact of the density-wave transition on magnetization, while it is absent in the tetragonal La4Ni3O10 sample.

Pressure effect in monoclinic and tetragonal La4Ni3O10

To further investigate the physical properties of La4Ni3O10 with different structures, we performed high-pressure resistance measurements using liquid pressure medium (see details in method). For the monoclinic La4Ni3O10 (sample S1), the resistance gradually decreases with increasing pressure, and the anomaly associated with the density wave transition can be gradually suppressed as shown in Fig. 3a. Above 20 GPa, the density wave transition is completely suppressed; a weak resistive drop occurs at ~3 K at 25.3 GPa, signifying a superconducting transition. With further increasing pressure, the transition becomes more pronounced and its temperature Tc gradually increases; zero resistance is achieved at around 4 K at P = 72 GPa (Fig. 3b). We have also measured another monoclinic La4Ni3O10 (sample S2) using a solid pressure medium (see details in Methods), the data are presented in Supplementary Fig. 7. Superconductivity is observed above 24 GPa, with a maximum Tc of up to 24 K, similar to that in sample S1. The T-P phase diagram constructed based on these data (Fig. 1e) is consistent with previous findings14,15,16.

Fig. 3: Pressure effect for La4Ni3O10 with different structures.
figure 3

a Temperature dependence of resistance for monoclinic La4Ni3O10 (sample S1) under various pressure. The resistance at room temperature gradually decreases with increasing pressure. The anomaly corresponding to the possible density-wave transition is gradually suppressed, and superconductivity emerges above 20 GPa. With further increasing pressure, the superconducting transition becomes more pronounced, and the Tc continuously increases. The summarized phase diagram for density-wave and superconducting transition is show in Fig. 1e. b Enlarged area for the resistance at low temperature which shows superconductivity. Zero resistance can be achieved above 60 GPa. c Temperature dependence of resistance for tetragonal La4Ni3O10 (sample S3) up to 20.2 GPa. The resistance gradually decreases with increasing pressure below 14 GPa, then starts to increase. No anomaly related to the density-wave transition is observed. d Temperature dependence of resistance for tetragonal La4Ni3O10 (sample S3) above 20.2 GPa. The resistance at room temperature decreases with increasing pressure and shows a weak upturn at low temperatures above 20 GPa. e, f Pressure dependence of resistance at high and low temperatures for monoclinic sample S1 (e) and tetragonal sample S3 (f).

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In contrast, for the tetragonal La4Ni3O10 (sample S3), no anomaly related to the density wave transition is observed at ambient pressure. The general metallic behavior of the T-dependent resistance persists up to above 60 GPa without indications of superconductivity (Fig. 3c, d). The resistance gradually decreases with increasing pressure below 14 GPa, then starts to increase as shown in Fig. 3c. Above 20 GPa, the resistance decreases again with increasing pressure and shows a weak upturn at low temperatures as shown in Fig. 3d; the nonmonotonic P-dependence is illustrated in Fig. 3f. We have also conducted high-pressure resistance measurements on another tetragonal La4Ni3O10 (sample S4) up to 158.9 GPa, as shown in Supplementary Fig. 8. It exhibits similar behaviors as those in sample S3, and no superconductivity is observed up to the highest pressure. From our high-pressure resistance measurements, we conclude that the as-grown tetragonal La4Ni3O10 exhibits strikingly different high-pressure behavior comparing to that in the monoclinic La4Ni3O10, although the monoclinic structure can also be tuned to tetragonal phase at a certain pressure where the superconductivity sets in (Fig. 1e).

DFT calculations of monoclinic and tetragonal La4Ni3O10

The band structures and Fermi surfaces (FSs) of three different La4Ni3O10 crystal structures have been investigated by performing first-principles calculations based on the density functional theory (DFT). For the P21/a-AP phase and the I4/mmm-HP phase, the band structures and Fermi surfaces are shown in Fig. 4a, b, d, e, respectively. In the case of the P21/a phase, the tilted NiO6 octahedrons fold the Brillouin zone (BZ) into the dashed \(\sqrt{2}\times \sqrt{2}\) BZ displayed in Fig. 4d, whereas the tetragonal phases are characterized by the typical square BZ shown in Fig. 4d. Considering the Ni 3 d orbitals, there are 12 and 6 eg bands for the P21/a-AP (Fig. 4a) and I4/mmm-HP (Fig. 4b) phases, respectively. The FSs for both phases comprise preponderant dx2-y2 orbital component, which occasionally mixes with the dz2 orbitals (Fig. 4d, e); The calculated FSs of the P21/a-AP phase (Fig. 4d) matches well with the recent ARPES results36. For the newly-synthesized I4/mmm-AP phase introduced in this work, the band structure and FSs are presented in Fig. 4c, f for a comparison. We can see that the most salient feature for this phase is the appearance of two additional FS pockets \(\gamma,\varepsilon\) centered at the Γ point and M point, respectively. According to the orbital projection plot (Fig. 4f), these additional FSs are mainly attributed to the dz2 orbital. Their emergence is due to the elongated c axis in the I4/mmm-AP structure which reduces the dz2 interlayer coupling. On the contrary, the I4/mmm-HP phase suffers from the compressed c axis and thus the \(\gamma,\varepsilon\) pockets disappear (Fig. 4e).

Fig. 4: DFT calculation on La4Ni3O10.
figure 4

a The band structure of ambient pressure P21/a phase. b The band structure of high pressure (40 GPa) I4/mmm phase. c The band structure of ambient pressure I4/mmm phase. d The Fermi surface of ambient pressure P21/a phase. The Brillouin zone (dashed lines) is folded due to tilted octahedrons. e The Fermi surface of high pressure (40 GPa) I4/mmm phase. f The Fermi surface of ambient pressure I4/mmm phase. Notice that there are two additional Fermi pockets around Brillouin zone center and corner.

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Discussion

Now we turn to discuss the possible implications of the present work on exploring the superconducting mechanism in RP-phase nickelate superconductors. In previous studies, it is broadly speculated that a key impact of high pressure on introducing superconductivity is to stabilize the tetragonal structure without octahedron tilting in both La3Ni2O7 and La4Ni3O10. Such a scenario has immediately revealed the importance of synthesizing tetragonal La3Ni2O7 and La4Ni3O10 crystals at ambient pressure. By substantially reducing the oxygen content in both compounds, the tetragonal structure is indeed achieved at ambient pressure in La3Ni2O6 and La4Ni3O8, respectively24,37,38. Previous transmission electron microscope (TEM) results indicate that the removal of oxygen predominantly happens in the apical oxygen sites39. In these oxygen-deficient tetragonal phases, an insulating transport behavior always prevails at low temperatures, which inevitably hampers the development of superconductivity24,40. For our I4/mmm-AP La4Ni3O10 samples grown in high oxygen atmosphere, although precise determination of the oxygen content is difficult due to the small sample size, the XRD refinement of crystal structure implies an oxygen content very close to the stoichiometry (see Methods). By this way, the oxygen content of the I4/mmm-AP La4Ni3O10 is found to be slightly larger than that of the P21/a-AP La4Ni3O10 by ~ 0.04. Furthermore, we also used integrated differential phase contrast (iDPC) imaging techniques to accurately visualize the oxygen atoms on the tetragonal La4Ni3O10. As shown in Supplementary Fig. 5, this result shows no obvious oxygen vacancy or defects. In this sense, the absence of superconducting state in the I4/mmm-AP La4Ni3O10 samples at both ambient and high pressure can hardly be attributed to oxygen deficiency (this is also demonstrated by the metallic resistance behavior shown in Fig. 3c), but it places a direct challenge on the idea that the high-pressure superconductivity ubiquitously emerges in RP-phase nickelates once a tetragonal structure develops.

A tempting explanation is the existence of robust magnetic order in these ambient-pressure tetragonal samples. In monoclinic La4Ni3O10, SDW order has already been revealed by the neutron scattering experiment25. Very recently, presence of a similar SDW order has been confirmed in La3Ni2O7 by various spectroscopy probes including resonant inelastic X-ray scattering (RIXS), nuclear magnetic resonance (NMR) and muon spin rotation (μSR) experiments26,27,28,29. Our DFT calculation indicates that the magnetic ground state is still more favorable in the I4/mmm-AP phase at least in the case without applying pressure (see Methods and Supplementary Table 2 and 3). However, our transport and magnetic torque measurement does not show any signature for the SDW transition up to 160 GPa. One possible reason is that multiple competing (and nearly-degenerate) magnetic states coexist in the I4/mmm structure give rise to a frustrated paramagnetic ground state. Further spectroscopy measurements (such as NMR and μSR) are urgently needed to clarify the ground state and examine whether the absence of SDW transition and superconductivity is internally connected. In addition, the role of dz2 orbital on the high-pressure superconductivity in La3Ni2O7 and La4Ni3O10 is under hot debate17. In the tetragonal La4Ni3O10 at ambient pressure, our DFT calculations indicate that the dz2 orbital contributes two additional FS pockets located at the Γ point and M point, respectively (Fig. 4f). With increasing pressure, these two pockets (\(\gamma,\varepsilon\)) disappear from the Fermi level, suggesting a Lifshitz transition. By analyzing the pressure-dependent resistance, a nonmonotonic behavior is observed around 20 GPa (see Fig. 3f), which further supports the occurrence of a Lifshitz transition. The superconducting state is rigorously absent in the I4/mmm-AP samples, as mentioned above, both at ambient pressure and under applied high pressure up to 160 GPa, i.e., irrespective of the presence or absence of the dz2-derived FS pockets. Therefore, our results do not support a critical role of dz2 orbital on high-pressure superconductivity. As a final note, spectroscopy measurements on the new I4/mmm-AP La4Ni3O10 are needed in the near future, which would be undoubtedly decisive for understanding the key ingredients in the electronic states that determining the fate of superconductivity in RP-phase nickelates.

Methods

Sample growth

The La4Ni3O10 crystals were synthesized using a previous reported molten salt flux method35. Specifically, the La2O3 was baked at 1000°C for 10 h and then quickly transferred into the glove box with O2 and H2O content less than 1 ppm. The La2O3, NiO and K2CO3 were mixed with the mole ratio of 2:3:35 and loaded into a corundum crucible. The temperature was also controlled as previous reported. The temperature was increased to 1050°C in 400 min, kept for 72 h and then slowly cooled down to 900 °C in 7 days. For monoclinic La4Ni3O10, the crystals were synthesized at air. For tetragonal La4Ni3O10, the crystals were synthesized under a flow oxygen. Crystals with size 70–120 μm were selected from the product after dissolving the flux using water and the crystal structure were checked with the four-circle X-ray diffractometer.

Structural characterization

The single crystal X-ray diffraction (SC-XRD) was conducted on a four-circle diffractometer (Rigaku, XtaLAB PRO 007HF) with Cu-Kα radiation in Core Facility Center for Life Sciences, USTC. The data collected at 300 K (or 110 K) were reduced and finalized using CrysAlisPro software and the structure were solved and refined using Olex-2 with ShelXT and ShelXL packages41,42,43,44. We note that crystals grown under oxygen were firstly solved and refined with the space group P21/a as previous reported and results with a moderate R1 and wR2 were obtained. After careful check with PLATON software and online checkCIF website, a higher space group I4/mmm was suggested and adopted to solve and refine the crystal structure. A significant improvement in both R1 and wR2 confirm the I4/mmm space group. Based on the refinement from the SC-XRD data, we note that the site occupation of all the oxygen sites, especially the apical oxygen site, shows negligible vacancy which indicates the oxygen content should be very close to the ideal chemical stoichiometry. In situ high-pressure X-ray diffraction (XRD) experiments were performed using a MetalJet E1 + 160 kV source equipped with an In-Ga target, generating an X-ray wavelength of 0.5124 Å. The X-ray beam was precisely focused to an area of approximately 100×100 μm². Two-dimensional diffraction images were recorded by a PILATUS R CdTe detector and subsequently processed into one-dimensional XRD patterns using the Dioptas software. Helium gas was employed as the pressure-transmitting medium to ensure optimal hydrostatic pressure conditions, with pressure calibrations verified through the shift in the fluorescence peak of a ruby indicator. The cross-sectional transmission electron microscope (TEM) sample for selected area electron diffraction (SAED) and integrated differential phase contrast (iDPC) measurement is prepared using a scanning electron microscope equipped with a focused ion beam (Carl Zeiss Crossbeam 550 L FIB-SEM). The iDPC data collected on the tetragonal La4Ni3O10 sample shows no obvious oxygen vacancies or defects. The chemical composition of single crystals was characterized using a Hitachi SU8220 field emission scanning electron microscope (FE-SEM) equipped with an energy-dispersive X-ray spectroscopy (EDX, Oxford Instrument X-MaxN 150).

Magnetic torque characterization

Torque magnetometry was performed using an SCL piezoresistive cantilever. The sample was attached to the tip of cantilever, which was mounted to the horizontal rotator of Physical Properties Measurement System (PPMS, Quantum Design Inc., DynaCool-14 T). We first rotated the sample in a range of θ (the angle between magnetic field vector H and the c axis of La4Ni3O10 crystal) from 0° to 90° under isothermal condition and determined that the largest signal occurs at ~θ = 45°. The data shown in Fig. 1g were collected at θ ≈ 45° with uncertainty of a few degrees from misalignment. We note that the measured data for this technique is the apparent resistance of the piezoresistive sensor; it indeed represents the output voltage of the Wheatstone bridge circuit inside the sensor that consists of two components, one is proportional to the magnetic torque of the sample and the other is a sample-independent background. While the latter may vary with temperature, the salient feature as shown in Fig. 1g unambiguously reflects a magnetic transition of the sample.

High pressure transport

Diamond anvils with various culets (100 to 300 μm) were used for high-pressure transport measurements. NaCl or Daphne oil 7373 were used as a pressure transmitting medium and the pressure was calibrated by using the shift of ruby florescence and diamond anvil Raman at room temperature. During transport measurements, the pressure was applied at room temperature using the miniature diamond anvil cell. The transport measurements were carried out in a refrigerator system (HelioxVT, Oxford Instruments). Four pieces of La4Ni3O10 microcrystals are used for high-pressure transport measurement in this study. S1 and S2 are monoclinic La4Ni3O10 samples. S3 and S4 are tetragonal La4Ni3O10 samples.

DFT calculations

Our DFT calculations employ the Vienna ab-initio simulation package (VASP) code45 with the projector augmented wave (PAW) method46. The meta-GGA functionals that consider the Laplacian of the electron density47 are used here. Typically, this can be more accurate compared to the GGA method. The cutoff energy for expanding the wave functions into a plane-wave basis is set to be 500 eV. The energy convergence criterion is 10−8 eV. All calculations are conducted using the primitive cell to save time. The Γ-centered 9×9×4, 11×11×15 and 11×11×15 k-meshes are used for the ambient pressure P21/a phase, ambient pressure I4/mmm phase and high pressure I4/mmm phase respectively. In order to determine the magnetic state, we set three different magnetic states which are usually encountered in this type of materials: ferromagnetic (FM), G-type antiferromagnetic (G-AFM) ordering at (π, π) and double stripe ordering at (π/2, π/2). We compare them with the non-magnetic (NM) state. At all U values shown here, we find the energies of FM state are relatively lower than other two competing magnetic states. We also calculate the magnetic moment of three magnetic states. In two AFM states (G-AFM and double stripe), the middle layer has zero magnetic moment, and the sign of magnetic moment is opposite in top and bottom layer. We calculate the spin orders using the simplified rotation invariant approach to the DFT + U, introduced by Dudarev et al. 48.