## Introduction

A universal question in strongly correlated electron materials is how the salient degrees of freedom are microscopically coupled to stabilize novel quantum states. Unconventional superconductivity in cuprate materials is a model example, where it is thought that the macroscopic coherent quantum state arises from intertwined charge (CDW) and spin-density wave (SDW) fluctuations1,2,3,4,5. A variety of experimental results alongside theoretical Hubbard model calculations show that the ground state energies of static CDW, SDW orders and unconventional superconductivity are nearly degenerate1,2,6,7,8,9. However, it remains unclear how charge and spin degrees of freedom are coupled to enable high-temperature superconductivity in distinct cuprate materials.

An effective way to tune the subtle balance among the different ground states is the inclusion of holes into the copper-oxygen (Cu-O) layers, which has notably established that the density-wave orders are most stable when the number of holes per Cu site p = 1/8. Interestingly, this is observed concomitant with a suppression of superconductivity. In addition, in YBa$${}_{2}$$Cu$${}_{3}$$O$${}_{7-y}$$, for instance, static CDW and SDW orders seem not to coexist10,11. This is in striking contrast with La-based cuprates where the static density-wave orders not only coexist but also show signatures of coupling1,2,9. Yet the two order parameters reveal very different dependencies on temperature and hole-doping. In La$${}_{2-x}$$Sr$${}_{x}$$CuO$${}_{4}$$ (LSCO) with x=p = 1/8 the SDW order is strongest revealing an onset temperature comparable to that of superconductivity, $${\rm {i}}.{\rm {e}}.$$ $${T}_{c}\approx$$ 27 K12, but short-range CDW order persists to temperatures above 100 K3. Similarly, while density-wave fluctuations exist over a broad hole-doping range in LSCO3,13, SDW and CDW orders coexist only for $$\sim$$0.1 $$<$$$$x <$$ $$\sim$$0.135 questioning whether they vanish in a common quantum critical point3,14,15,16,17,18. Summarizing, this raises the question how SDW and CDW orders are microscopically coupled, and how their fluctuations give rise to unconventional superconductivity.

To address this issue for La$${}_{1.88}$$Sr$${}_{0.12}$$CuO$${}_{4}$$, we leverage uniaxial pressure applied along the Cu-O bond direction as a newly-established extrinsic tuning parameter. Here recent x-ray experiments19 have demonstrated that one of the CDW domains can be suppressed through uniaxial strain. For the case of truly coupled CDW and SDW orders, we similarly expect to observe that uniaxial pressure generates a single-domain stripe SDW order (see Fig. 1). Using elastic neutron scattering with superior beam-focusing, we confirm this hypothesis and demonstrate that charge and spin orders respond to uniaxial pressure in the same manner. These results attest that adequate theories for high-temperature superconductivity must account for charge and spin degrees of freedom via uniaxial charge-spin fluctuations.

## Results

The outstanding experimental challenge of our study is that sufficiently high strains may only be achieved for tiny single crystals with well-defined shapes and small surface cross-sections perpendicular to the direction of the applied pressure. Such crystals of a few mg weight are typically not suitable for neutron experiments required to unambiguously probe weak SDW order. Crucially, the scattered neutron intensity scales linearly with the crystal mass while uniaxial pressure cells generally increase background scattering, resulting in an overall insufficient signal-to-noise ratio.

To overcome this challenge, we designed a neutron study that combines three major technical advances. Firstly, we exploit new developments in neutron-ray-tracing simulations, where the Union components of the McStas suite20 only since recently take into account background scattering processes resulting from advanced sample environments such as cryostats and pressure cells. This, in turn, allowed us to optimize the signal-to-background ratio (see Supplementary Information (SI) Note 2) and demonstrate that irrespective of geometrical and instrumental details, neutrons of longer wavelengths are advantageous in terms of signal-to-noise ratio. Secondly, armed with this knowledge, we leveraged the unique focusing capabilities of the adaptive virtual source and double focusing silicon (Si(111)) monochromator at the ThALES cold neutron triple-axis spectrometer at the Institut Laue-Langevin, Grenoble, France to probe the weak SDW order in a 55 mg LSCO $$x$$ = 0.12 single crystal21,22,23. Finally, this strongly reduced sample size (for comparison, a 1.06 g sample from the same batch was used for ambient pressure measurements. See Methods section), enabled the use of a scaled-up version of a uniaxial pressure cell, which we recently developed for x-ray studies of CDW order19,24, without compromising uniaxial pressure application (see Fig. 1b). We note that the exquisite focusing capability of ThALES not only strongly increased the flux on the tiny sample but equally importantly reduced background scattering, allowing for a vastly improved signal-to-noise ratio (further details are given in the Methods section and in the SI Note 3).

Figure 2 shows the peaks characteristic of the SDW order in La$${}_{1.88}$$Sr$${}_{0.12}$$CuO$${}_{4}$$ at ambient pressure and a temperature $$T$$ = 2 K (additional scans are plotted in SI Note 5) as probed by our experiment on the 1.06 g sample. The measured count rates are shown normalized to the sample mass to facilitate a later comparison with results obtained with the pressure cell. Within the high-temperature tetragonal unit cell notation SDW order in La-based cuprates manifests as peaks at incommensurate wavevectors $${\bf {Q}}_{{{{{\rm{SDW}}}}}}^{{{{{\rm{a}}}}}}$$ = (0.5 $$\pm$$ $${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0.5, 0) and $${\bf {Q}}_{{{{{\rm{SDW}}}}}}^{{{{{\rm{b}}}}}}$$ = (0.5, 0.5 $$\pm$$ $${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0) in reciprocal lattice units (rlu). The peaks may be interpreted as the signature of either two orthogonal magnetic domains in the well-known stripe model1,2,6,25,26,27,28,29 shown in Fig. 1a, or to two phase-related wavevectors describing a single-domain multi-Q magnetic order, which is consistent with the interpretation of earlier experimental results28,29,30,31,32,33,34,35,36,37,38,39. The incommensurability $${\delta }_{{{{{{\rm{SDW}}}}}}}$$ is doping dependent with $${\delta }_{{{{{{\rm{SDW}}}}}}}\approx 0.12$$ rlu around $$x$$ = 1/81,1,. Our measurements unambiguously show that magnetic intensity is present at both wavevectors $${\bf {Q}}_{{{{{\rm{SDW}}}}}}^{{{{{\rm{a}}}}}}$$ and $${\bf {Q}}_{{{{{\rm{SDW}}}}}}^{{{{{\rm{b}}}}}}$$. The data were fitted with a Gaussian line shape on a sloping background, revealing an average incommensuration $${\delta }_{{{{{{\rm{SDW}}}}}}}$$ = 0.118(4) rlu and an in-plane correlation length $${\xi }_{a}\,=\,(a/\pi )$$$${\rm {FWHM}}$$$${}^{-1}$$ = 67(9) Å, where FWHM is the full-width at half-maximum. These results are in agreement with earlier reports on La$${}_{1.88}$$Sr$${}_{0.12}$$CuO$${}_{4}$$12,40. We also observe a small perpendicular incommensuration (0.007(2) rlu shown in SI Note 5), which is often referred to as $$Y$$-shift and is hypothesised to arise through pinning to orthorhombic distortions40,41,42.

To probe the response of SDW order in La$${}_{1.88}$$Sr$${}_{0.12}$$CuO$${}_{4}$$ to uniaxial pressure, we applied compressive strain $${\varepsilon }_{a}$$ $$\approx$$ 0.02% (see Methods section) along the $$a$$-axis (Cu-O bond direction) of the 55 mg La$${}_{1.88}$$Sr$${}_{0.12}$$CuO$${}_{4}$$ sample (cf. Figure 1). In Fig. 3, we display reciprocal space scans across the two wavevectors at $$T$$ = 2 and 40 K. The high temperature data were measured above the SDW onset temperature of $${T}_{N}$$ $$\approx$$ 25 K (see Fig. 4a, b), and used as background calibration assuming a sloping behavior. The high counting statistics enabled by our setup, allowed us to observe a magnetic signal down to approximately 2% above the background (see SI Note 6). At this level of discrimination no magnetic Bragg peak is observed at $$T$$ = 2 K around (−0.5 + $${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0.5, 0) or other symmetry equivalent reflections of $${\bf {Q}}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{a}}}}}}}$$ (see SI Note 6). In strong contrast, a clear magnetic signal was found at $${\bf {Q}}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{b}}}}}}}$$ (see Figs. 3 and SI Note 6). We fitted the roughly 8% strong signal at (−0.5, 0.5 - $${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0) with a Gaussian lineshape over the fixed high-temperature background. The fit yields $${\delta }_{{{{{{\rm{SDW}}}}}}}$$ = 0.118(2) rlu and $${\xi }_{b}$$ = 66(13) Å, matching the ambient pressure results. We note that a slight 0.005(1) rlu Y-shift persists under strain (see SI Note 6) and that no evidence for a pressure-induced shift in $${T}_{N}$$ is detected (see Fig. 4a, b).

For a quantitative comparison of the data obtained at ambient conditions and under $$a$$-axis strain, the background contributions from the scans shown in Figs. 2 and 3 were substracted. The resulting background-corrected data are displayed in Fig. 4c–f. The peaks associated with Q$${\,}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{a}}}}}}}$$ and Q$${\,}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{b}}}}}}}$$ feature integrated intensities $${I}_{{{{{{\rm{int}}}}}}}$$ = 0.030(3) and 0.015(3) cts$$\cdot$$rlu/(s$$\cdot$$g) at ambient pressure, respectively. By contrast, under uniaxial strain conditions only Q$${\,}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{b}}}}}}}$$ carries a finite integrated intensity of $${I}_{{{{{{\rm{int}}}}}}}$$ = 0.07(1) cts$$\cdot$$rlu/(s$$\cdot$$g). Alongside our finding that strain has no effect on the SDW ordering vector, correlation length or onset temperature, this demonstrates that uniaxial pressure along the Cu-O bond direction leads to a dramatic redistribution of magnetic peak intensity. This observation allows us to gain insight into the magnetic structure. While extrinsic tuning parameters may modify the population of magnetic domains by breaking the symmetry on a macroscopic scale, they are not expected to differently affect the coexisting wavevectors in multi-Q structures43,44. Thus, our results provide direct experimental evidence for a single-$${\bf {Q}}$$ magnetic structure in LSCO, and that application of pressure along one of the Cu-O directions favors the domain characterized by a propagation vector along the perpendicular Cu-O direction. We note that this effectively rules out spin-vortex checkerboard structures and any other multi-Q single-domain structures proposed earlier29,30,31,32,33,35,37,39.

## Discussion

Our observation that the application of uniaxial strain causes domain repopulation reflected in the peak intensity without altering the other characteristics of the magnetic order, demonstrates that SDW and CDW order in La$${}_{1.88}$$Sr$${}_{0.12}$$CuO$${}_{4}$$ can be tuned by uniaxial pressure in an identical manner. Recent x-ray scattering results have shown that small uniaxial strain along the Cu-O bond direction also yields a suppression of the intensity of the CDW ordering vector along the direction of the applied pressure, and that its intensity is recovered upon strain release19. Together with our neutron scattering study these results strongly suggest an intertwined CDW and SDW order. Notably, this provides evidence for a charge-spin stripe arrangement (see Fig. 1a) as the fundamental density-wave state of LSCO. Theoretically, a direct coupling between static charge and spin order is supported by a microscopic mechanism in the strong coupling limit, where stripe order arises from local correlations1,7,8,45,46. In this picture holes are located at the antiphase SDW domain boundaries (see Fig. 1a) to minimize their kinetic energy, therefore establishing the relationship $${\delta }_{{{{{{\rm{CDW}}}}}}}$$ = 2$${\delta }_{{{{{{\rm{SDW}}}}}}}$$ between the incommensurations of the two orders. As the direct space picture naturally connects charge and spin order, uniaxial pressure must induce a single-domain state where populated charge and spin domains are oriented along the same direction. Our observations are harder to reconcile with a weak coupling picture where the electronic order arises from Fermi surface nesting instabilities47,48. In this case, uniaxial pressure affects the Fermi surface topology, which would need to act similarly at $${\bf {Q}}_{{{{{{\rm{SDW}}}}}}}$$ and $${\bf {Q}}_{{{{{{\rm{CDW}}}}}}}$$ to account for a unidirectional charge and spin domain states. To explain the observation that only one ordering vector survives the pressure application, significant and peculiar anisotropic Fermi-surface distortions would be required. In contrast to our observations, one would also expect subtle shifts of the ordering wavevectors in such a scenario.

Our observation of single-Q charge-spin stripe order in La-based cuprates also adds to the discussion on the interplay between density-wave orders and unconventional superconductivity. Several theories attest the intertwined coupling between superconductivity and density-wave order to the emergence of a spatially-modulated superconducting order parameter that makes up for a sizable fraction of the superconducting condensate1,2,3. This putative pair-density wave (PDW) phase is thought to change sign within the spin stripes to cope with magnetic long-range order. Because both the SDW order probed here, and the CDW order explored by our previous x-ray scattering study19 is not altered by uniaxial pressure along the Cu-O bond direction, this suggests that any PDW order will also involve uniaxial stripes. Notably, that the magnetic correlation length and SDW onset temperature, as well as the coupling between CDW order and superconductivity remain unchanged by uniaxial pressure application, suggests that in LSCO charge-spin stripe order and unconventional superconductivity are deeply intertwined. Interestingly, for La$${}_{2-x}$$Ba$${}_{x}$$CuO$${}_{4}$$ (LBCO) a different behavior has been observed when uniaxial pressure is applied along an in-plane axis close to the Cu-Cu bond direction, demonstrating direct competition between superconductivity and magnetic order49,50. Thus, further uniaxial pressure studies on LSCO and LBCO along different crystal directions will be crucial to gain new insight into the coupling between stripe order and unconventional superconductivity.

In addition, we note that despite these differences, short-ranged charge fluctuations often exist over a wider temperature and doping range than spin order in La-based cuprates3,14,15,16,17,18. This suggests that the charge degrees of freedom are responsible for the primary fluctuations that drive the coupled charge-spin state at low temperature. Our results add further constraints to the intertwined ground state of this strongly correlated material class. They attest that adequate theories for high-temperature superconductivity must account for charge and spin degrees of freedom via uniaxial charge-spin fluctuations. Future experiments directly assessing the spin fluctuations under uniaxial pressure may be a fruitful way forward to gain a deeper understanding of the pairing mechanism behind high-temperature superconductivity.

Beyond the cuprates, the exact nature of the intimate coupling among various electronic degrees of freedom is crucial to the understanding of all strongly-correlated electron materials51,52,53. Our study establishes that the latest-generation neutron instrumentation allows us to explore the effects of uniaxial pressure — a particularly clean and useful tuning parameter — on spin order with weak order parameters. This advancement will be key in improving our understanding of a large range of electronic ground states. Notably, the study of coupled PDW and SDW order in heavy-fermion superconductors54,55, coupling of ferroelectric and magnetic domains in multiferroics53, spin, charge and lattice coupling in electronic nematic states56,57,58, or the identification of topologically non-trivial multi-Q spin textures59 will benefit from this capability.

## Methods

### Sample preparation and characterization

A high quality LSCO $$x$$ = 0.12 crystal was grown by the travelling solvent floating zone technique. Standard magnetization measurements revealed a superconducting transition temperature $${T}_{c}$$ = 27 K (cf. SI Note 1). The magnetic response was checked using muon spin rotation (MuSR) at the GPS beamline of the S$$\mu$$S muon source, Paul Scherrer Institut, Switzerland (see SI Note 1). These measurements showed that the magnetically ordered state inside the superconducting phase is in line with the earlier studies on LSCO12,60.

After confirming the rod’s surface crystallinity by x-ray Laue diffraction, two samples of masses $$m$$ = 1.060 and 0.055 g were oriented and cut using a tungsten wire saw. The latter sample dedicated to fit into the uniaxial pressure cell had a cuboid shape of dimensions 3.17(5)x1.35(5)x1.83(5) mm$${}^{3}$$ along the axes of the high-temperature tetragonal unit cell. The tetragonal notation (see SI Note 1 for details) with $$a\,=\,b$$ = 3.759 Å and $$c$$ = 13.2 Å refined at $$T$$ = 2 K on ThALES is employed throughout the manuscript. Here the SDW wavevector $${\bf {Q}}_{{{{{{\rm{SDW}}}}}}}$$ is shifted by $${\bf {q}}_{{{{{{\rm{SDW}}}}}}}$$ = ($${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0, 0) and (0, $${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0) away from the antiferromagnetic wavevector $${\bf {Q}}_{{{{{{\rm{AF}}}}}}}$$ = (1/2, 1/2, 0). The charge order peaks at $${\bf {Q}}_{{{{{{\rm{CDW}}}}}}}$$ are offset by $${\bf {q}}_{{{{{{\rm{CDW}}}}}}}$$ = (2$${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0, 0) and (0, 2$${\delta }_{{{{{{\rm{SDW}}}}}}}$$, 0) with respect to structural Bragg peaks, yielding $${\delta }_{{{{{{\rm{CDW}}}}}}}$$ = 2$${\delta }_{{{{{{\rm{SDW}}}}}}}$$.

The bulk crystallinity of the two samples was checked using the triple-axis alignment station IN3, the neutron Laue diffractometer Orient Express and finally determined on the cold neutron spectrometer ThALES at the Institut Laue-Langevin, Grenoble, France. These data revealed that the small 55 mg sample was single crystalline within our resolution. The larger 1.06 g was composed of two crystallites shifted by 1.5 degrees with a relative intensity of 5:1 that was determined at $$T$$ = 2 K (see SI Note 4).

A strain cell similar to those used in recent x-ray diffraction studies19,24 was adapted to fit the cryostat dimensions of the ThALES instrument (see also Fig. 1b). We note that other neutron scattering experiments under uniaxial pressure have been performed in earlier pioneering studies61,62,63,64. The small ordered moment in La1.88Sr0.12CuO465 required us to construct a strain cell made of high purity aluminium with an open geometry to minimize background contributions. Uniaxial pressure was applied at room temperature along the $$a$$-axis using an M3 aluminium screw that was rotated by 60$${}^{\circ }$$. Following the calibration from ref. 19. this amounts to a compressive strain of $${\varepsilon }_{a}\,=\,\Delta a/a$$ $$\approx$$ 0.02%.

### Neutron scattering at ThALES

The neutron scattering experiment was performed at the high-flux cold neutron spectrometer ThALES at the Institut Laue-Langevin. ThALES is uniquely suited for the purpose of our experiment since its double-focusing silicon (Si(111)) monochromator allows aggressive focusing of the incident neutron at the sample position21,22,23. This was essential to minimize parasitic scattering from the strain cell while maximizing the signal from the sample. A sketch of the experimental setup is shown in Fig. 1b. We used $${k}_{i}={k}_{f}=1.55$$ Å$${}^{-1}$$ neutrons, which were cleaned from higher order contaminations via a velocity selector before the silicon monochromator and a cooled Be-filter before a double-focusing pyrolytic graphite (PG(002)) analyzer that was used to further reduce the background. The instrument was used in a $$W$$-configuration to minimize the extent of the instrumental resolution ellipsoid. The scattered neutrons were detected with a standard one-inch $${}^{3}$$He detector. The samples were aligned perpendicular to the (0, 0, 1) axis to access both SDW domains in the horizontal scattering plane. We improved the signal-to-noise ratio for the strain sample by optimizing the position and opening (8 mm) of the virtual slit, which is located between the velocity selector and the monochromator (see SI Note 3). Further improvements of the signal-to-noise ratio were gained through tight slit openings before (8 mm horizontal, 15 mm vertical) and after (10.5 mm horizontal, 13 mm vertical) the sample.

The temperature dependencies of the Q$${\,}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{a}}}}}}}$$ and Q$${\,}_{{{{{{\rm{SDW}}}}}}}^{{{{{{\rm{b}}}}}}}$$ peaks shown in Fig. 4a, b were measured on the optimized peak positions, i.e. at $$\bf Q$$ = (-0.387, 0.512, 0) and (-0.504, 0.382, 0), respectively. Additional reciprocal lattice scans on the 1.06 g sample were taken at intermediate temperatures to confirm the validity of the temperature dependent peak intensity measurement. Additional temperature dependent background data on the 55 mg sample were measured at $$\bf Q$$ = (-0.504, 0.347, 0) and (-0.504, 0.417, 0).