Introduction

Strongly correlated materials, particularly those hosting unconventional superconductivity, exhibit complex and intricate phase diagrams, where multiple broken-symmetry phases often emerge in response to external parameters such as doping, pressure, stress, impurities and magnetic fields1,2,3,4,5. Among them, La2−xBaxCuO4 (LBCO) stands out as a compelling example harboring a rich phase diagram of intertwined electronic orders2. Near a hole doping level of x = 0.125, LBCO undergoes a dramatic suppression of the bulk three-dimensional (3D) superconducting (SC) transition temperature (Tc,3D)6, coinciding with the emergence of static charge7,8,9 and spin-stripe order10,11,12,13. This suppression is accompanied by a structural transition from the low-temperature orthorhombic (LTO) to the low-temperature tetragonal (LTT) phase, which introduces a lattice distortion, discussed as crucial for stabilizing the stripe order by anchoring it to the lattice potential8,14,15. Within each of the CuO2 layers, stripes align preferentially along a specific Cu-O bond direction, undergoing a 90∘ rotation between adjacent layers, as illustrated in Fig. 1a.

Fig. 1: Experimental schematic and anisotropic response to uniaxial stress.
figure 1

a Schematic representation of stripe order stacking along the c-axis in LBCO, showing charge and spin stripe modulation within CuO2 layers and their alternating 90∘ rotation between adjacent layers. The blue and green arrows indicate the in-plane (at an angle of 30∘ to the Cu-O bond direction, denoted as a-axis) and c-axis compressive stress, respectively. b The uniaxial stress sample holder used for the μSR experiments. Backside view of the sample showing the AC susceptibility coil and the Cernox temperature sensors (left panel). View from the direction of the incoming muon beam (right panel). Hematite pieces masking the holder frame exposed to the muon beam. c–e c-axis (hollow symbols) and in-plane (solid symbols) uniaxial stress dependence of 3D superconducting transition temperatures Tc,3D (c) derived from AC susceptibility measurements, spin-stripe ordering temperatures (Tso) (d) derived from weak transverse field μSR experiments and the magnetically ordered fraction Vm at base-T ≃ 0.7 K (e) derived from zero-field μSR experiments for La1.885Ba0.115CuO4. The results for the in-plane strain are taken from our previous work ref. 4. Error bars represent the uncertainty from temperature resolution in Tc,3D and Tso, and fitting uncertainty in Vm. The small difference in zero-stress Tc,3D and Tso between the in-plane and c-axis datasets arises from measurements on two separate LBCO crystals with nominally the same doping (x = 0.115), but likely differing slightly in actual Ba content – a known sensitivity near the 1/8-anomaly.

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The strong suppression of 3D superconductivity at x = 0.125 suggests an inherent incompatibility between stripe order and long-range SC coherence. However, experiments reveal a more nuanced picture, showing the coexistence of an unconventional 2D SC state with spin-stripe order16,17. This phenomenon is interpreted in terms of a pair-density-wave (PDW) state, where SC pairing occurs along quasi-1D charge stripes with the phase of the pair wave function shifting by π from one charge stripe to the next18,19,20. In adjacent CuO2 layers, PDW states on orthogonal charge stripes frustrate the interlayer Josephson coupling, suppressing uniform 3D SC while preserving 2D SC correlations18,21. This highlights the intricate electronic interactions in cuprates.

The major factors governing the interplay between spin-stripe order, PDW, and bulk superconductivity remain a subject of debate, with significant implications for understanding the mechanisms behind high-Tc superconductivity22,23,24,25,26,27,28,29,30. Because these competing orders are delicately balanced, even small external perturbations can act as effective tuning parameters, selectively stabilizing different emergent phases. Among various tuning methods, hydrostatic pressure has been widely explored in LBCO. However, despite exceeding the critical pressure threshold where the long-range structural anisotropy (LTT phase - believed necessary for pinning charge stripes) is eliminated, the effect on Tc remains surprisingly modest31,32. In contrast, uniaxial stress has proven to be a far more effective tool in manipulating the competing orders4,33,34,35,36,37,38,39,40,41,42. Specifically, in-plane compression along the SC CuO2 planes in LBCO (x = 0.115, 0.135) has been shown to dramatically enhance Tc while simultaneously suppressing the magnetic volume fraction (Vm), which is inversely correlated with 3D SC coherence4,33. Additionally, in-plane (at an angle of 30∘ to the Cu-O bond direction, denoted as a-axis) uniaxial stress of  ~0.1 GPa destabilizes the long-range LTT phase – thought to reinforce charge stripe order–while still preserving the onset temperature of spin-stripe order (Tso), which intriguingly coincides with Tc.

While the effects of in-plane stress have been studied extensively, the influence of uniaxial stress along the c-axis on spin-stripe order and superconductivity remains unexplored. Understanding this effect could provide crucial insights into the role of crystallographic anisotropy in governing electronic interactions in cuprates. In this study, we investigate the response of an LBCO-0.115 crystal43 to compressive stress applied along the c-axis, perpendicular to the SC CuO2 layers, using an in situ piezoelectrically controlled stress device (Fig. 1b)44,45,46. To probe the microscopic response of spin-stripe order, we employ muon spin rotation (μSR) spectroscopy, while AC susceptibility measurements track the SC transitions. The summary of our results is shown in Fig. 1c–e, while the details of the AC susceptibility and μSR results are provided in Figs. 2 and 3 below, respectively. Unlike in-plane stress, which strongly enhances Tc, compressive stress along the c-axis results in a modest suppression of Tc (Fig. 1c). Additionally, while in-plane stress leads to a rapid decrease in the onset of spin stripe order, Tso and the magnetic volume fraction (Vm), both remain nearly constant up to the highest applied c-axis compression (~0.2 GPa) (Fig. 1d, e). Our findings therefore reveal a pronounced anisotropy in how stripe-order and SC properties respond to c-axis versus in-plane uniaxial compression (Fig. 1c–e).

Fig. 2: Superconducting response under c-axis stress.
figure 2

a The temperature dependence of the AC diamagnetic susceptibility (\({\chi }_{{{{\rm{AC}}}}}^{{\prime} }\)) for La1.885Ba0. 115CuO4 under varying c-axis compressive stress. The arrow indicates the bulk 3D superconducting transition temperature (Tc,3D), defined as the point where \({\chi }_{{{{\rm{AC}}}}}^{{\prime} }=-0.5\). b Schematic temperature-doping (x) phase diagram of LBCO near x = 0.125, illustrating the contrasting effects of uniaxial stress on superconductivity. The suppression of Tc,3D under c-axis compression is shown for LBCO-0.115, while the pronounced enhancement of Tc,3D under in-plane stress is depicted for both LBCO-0.115 and LBCO-0.135. The dashed line represents the anticipated superconducting phase boundary under in-plane stress, shown as a qualitative guide, based on the experimental data on x = 0.115, 0.1354,33.

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To investigate the impact of c-axis compressive stress on superconductivity in LBCO-0.115, in situ AC susceptibility measurements were performed either immediately before or after each μSR experiment at each applied stress. The excitation field was applied primarily perpendicular to the c-axis, to ensure that the shielding currents flowed perpendicular to the CuO2 layers, which makes the measurement sensitive to the onset of three-dimensional (3D) superconductivity16. The results are presented in Fig. 2a. At zero applied stress, the diamagnetic response corresponds to measurements taken under the zero-force condition of the stress apparatus. To quantify the changes in Tc, we define the midpoint of the transition as an indicator of 3D SC order (Tc,3D), as marked in Fig. 2. The strongest diamagnetic signal is taken to represent the full SC volume fraction. Notably, as the compressive stress increases, a modest suppression of Tc,3D is observed, decreasing from 12.4 K to 9.9 K, with a linear suppression rate of 11.9 K/GPa. As shown in Fig. 2b, this suppression occurs from an already reduced Tc at ambient pressure, further decreasing under the maximum applied c-axis compressive stress (\({\sigma }_{\max }\approx 0.2\) GPa). In contrast, for both LBCO-0.115 and LBCO-0.135, an applied in-plane stress of \({\sigma }_{\max }\approx 0.1\) GPa (only half the maximally applied c-axis stress) is sufficient to rapidly enhance Tc,3D to its optimal SC temperature observed in LBCO4,33. This striking contrast underscores the strong anisotropic stress response of Tc,3D.

Results

To uncover the evolution of spin-stripe order under compressive stress, we performed a combination of weak transverse-field (wTF) and zero-field (ZF) μSR measurements. The μSR technique provides a unique capability to probe the local magnetic order parameter and the magnetically ordered volume fraction independently, making it an invaluable tool for distinguishing between phase separation and homogeneous magnetic order. This is particularly crucial in cuprates, where competing electronic phases often coexist, leading to microscopic inhomogeneity4,33. In a typical μSR experiment, spin-polarized muons are implanted into the sample, where their spins precess around the local internal magnetic field. The muons subsequently decay into positrons, which are emitted preferentially along the initial muon spin direction. By monitoring the time evolution of positron decay, one can extract information about the muon spin polarization P(t). (Further details on the methodology can be found in the methods section.)

Figure 3 a presents a selection of muon-spin polarization PTF(t) curves measured under a weak transverse field (wTF) of 5 mT, as a function of increasing c-axis compressive stress. In wTF-μSR, muons implanted in a nonmagnetic environment exhibit long-lived precessional oscillations with maximum amplitude. The initial polarization PTF(0) in the paramagnetic state thus serves as a direct measure of the paramagnetic volume fraction. In contrast, when magnetic order is present, the implanted muons experience a broad distribution of internal fields, leading to rapid dephasing and a reduction in the oscillation amplitude. This allows for a precise determination of the magnetically ordered volume fraction, which can be estimated using the relation Vm = 1 − PTF(0). At 40 K and zero applied stress, PTF(0) remains close to 1, confirming the absence of magnetic order. However, at 0.7 K, the value decreases significantly to PTF(0) ≈ 0.1 and remains unchanged up to the maximum applied stress, indicating the development of spin-stripe order in the majority of the sample volume. The temperature dependence of the extracted magnetic volume fraction (Vm) under varying c-axis compressive stress is shown in Fig. 3b. The onset of spin stripe ordering Tso and Vm remains almost unchanged, independent of the applied pressure up to  ~0.21 GPa. These results suggest that both Tso and Vm are largely insensitive to c-axis stress, highlighting the robustness of stripe order in LBCO-0.115 despite c-axis uniaxial perturbations. Notably, at the highest applied stress, Tc,3D remains almost three times smaller than Tso.

Fig. 3: Spin-stripe order under c-axis stress.
figure 3

a Weak transverse field μSR spectra of La1.885Ba0.115CuO4 at 40 K under zero stress and at base-T ≃ 0.7 K under varying c-axis compressive stress. The error bars are the standard error of the mean (s.e.m.) in about 106 events. b Temperature dependence of the magnetically ordered fraction (Vm) under different stresses, extracted from weak transverse field μSR data in (a). The error bars represent the standard deviation of the fit parameters. c Zero-field μSR spectra at 50 K under zero stress and T ≃ 0.7 K under different c-axis stresses. The error bars are the standard error of the mean (s.e.m.) in about 106 events. d Temperature dependence of internal fields under various stresses, derived from zero-field μSR data. Solid and empty symbols indicate the values extracted for forward-backward and left-right detector sets (see the Supplementary Note 1 and the Supplementary Fig. S1), respectively.

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Figure 3c illustrates the evolution of zero-field (ZF) muon spin polarization PZF(t) in LBCO-0.115 under increasing c-axis compressive stress. In ZF-μSR measurements, the implanted muon spins precess solely in response to the internal magnetic field inside a sample. The observed signal reflects the collective behavior of muon spins, averaging over their spatial distribution relative to local variations in the internal field. At high temperature (50 K) and zero applied stress, PZF(t) exhibits a weakly decaying Gaussian-like behavior, characteristic of the paramagnetic regime. In this state, muon depolarization is primarily governed by static nuclear dipole moments, as the electronic moments fluctuate too rapidly to significantly influence the implanted muons. In contrast, at T ≃ 0.7 K (T ≪ Tso), PZF(t) develops rapidly damped oscillations at early times (t < 1 μs), a hallmark of long-range spin-stripe order30,47. As shown in Fig. 3c, these oscillations persist even as the compressive stress increases, with their amplitude remaining largely unchanged up to the highest applied stress. In single-crystal ZF-μSR measurements, the oscillation amplitude is influenced by both the magnetically ordered volume fraction (Vm) and the angle between the muon spin and the internal magnetic field (μ0Hint). Assuming that the stress does not alter the direction of the internal field at the muon stopping site, the observed insensitivity of the oscillation amplitude to applied stress is likely due to the unaltered behavior of Vm, consistent with findings from wTF-μSR experiments. The characteristic internal field (μ0Hint) at the muon stopping site, extracted from the oscillation frequency (as detailed in methods section), also remains mostly unchanged under increasing stress as shown in Fig. 3d. This indicates that the spin-stripe order maintains its long-range nature even at the highest applied c-axis compressive stress, further underscoring its robustness in LBCO-0.115 against c-axis uniaxial stress.

Discussion

Two important aspects can be highlighted from our results. First, they demonstrate a significant difference between the effects of c-axis and in-plane stress. Additionally, c-axis stress leads to a modest reduction in Tc,3D, while the spin-stripe order remains completely unaffected. The stability of spin-stripe order is supported by our observation of a perfectly linear stress (c-axis)-strain (force-displacement) response (as shown in Fig. 4), providing indirect evidence that the LBCO-0.115 crystal structure retains its LTT symmetry. In contrast, a structural transition would typically manifest as a change in the slope of the stress-strain dependence4. While pressure-induced doping has previously been discussed in Y-based cuprates, in the present case, uniaxial stress is highly unlikely to alter the doping level for several reasons: (a) Even a minor change in hole concentration would be expected to produce correlated shifts in both the stripe ordering temperature and the superconducting transition temperature, neither of which is observed. (b) In our prior work, we studied the effect of in-plane uniaxial stress on samples with doping levels x = 0.115 and x = 0.135, both of which showed a substantial increase in Tc. Since these dopings lie on opposite sides of the 1/8 anomaly, a doping-induced effect would result in opposing trends in Tc; instead, we observe a similar enhancement in both cases, ruling out a change in doping. (c) Finally, the maximum uniaxial stress applied in our study is just 0.2 GPa–considerably lower than the hydrostatic pressures reported48,49. Such low stress is unlikely to cause change in the doping level. One possible explanation for the absence of a c-axis stress effect on stripe order is the following: as shown recently using X-ray diffraction7 on LBCO-0.125, the screening of the stripes is correlated between the layers through the displacement of La ions and the apical oxygen atoms. These correlated displacements play a crucial role in stabilizing the stripe order across adjacent layers. Since c-axis strain does not break the existing symmetry of the crystal structure, it is unlikely to disrupt this interlayer coupling mechanism. In contrast to in-plane strain, which can directly affect the charge and spin stripe periodicity by modifying the CuO2 plane geometry, c-axis stress primarily alters interlayer spacing without significantly perturbing the essential electronic interactions that underpin stripe order. As a result, the stripe order remains robust despite the applied stress along the c-axis. This interpretation aligns with the observed resilience of stripe order under c-axis stress and highlights the role of interlayer interactions in governing the stability of charge and spin stripes in LBCO.

Fig. 4: Stress-strain curve for LBCO-0.115.
figure 4

The force sensor output is proportional to stress, while the displacement sensor reading is proportional to strain. The dashed lines represent linear fits to the data across the entire applied stress region. The inset shows the distorted CuO6 octahedron in the low-temperature tetragonal structure.

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An important question arises: if the stripe order remains unchanged, why does Tc,3D decrease under stress? To gain insight, we recall that the strong suppression of 3D superconductivity (as shown in Fig. 2b) in LBCO near x = 0.125 is typically explained1,2,18 by the fact that the anti-phase 2D SC states (PDW state) residing within the orthogonal stripes in adjacent layers disrupt interlayer phase coherence. This, in turn, frustrates the interlayer Josephson coupling and naturally explains the suppression of 3D SC order at ambient pressure. The gradual suppression of Tc,3D up to highest applied pressure  ~0.21 GPa indicates that c-axis stress further disrupts 3D SC coherence. One plausible explanation is that c-axis stress enhances frustration of the inter-layer Josephson coupling. This can be understood by considering the coexistence of the PDW and 3D uniform d-wave SC regions. In adjacent layers, the 2D d-wave SC patches, whose projections (along c) overlap, mediate finite inter-layer coupling. However, if such patches remain sparse, PDW order dominates intra-layer physics, leading to frustrated inter-layer couplings. In-plane uniaxial stress has been shown to suppress stripe order and enhance bulk 3D superconductivity until σcr ≃ 0.05 GPa, thereafter Tso and Tc,3D level off and coincide with each other4,33. It has been well understood that in-plane stress can suppress stripe order by inducing significant distortions in the crystal structure, ultimately breaking the low-temperature tetragonal (LTT) symmetry at σcr ≃ 0.05 GPa. Subsequently, this structural modification facilitates the formation of uniform SC patches in adjacent layers, enhancing inter-layer Josephson coupling and driving the system toward bulk 3D superconductivity. Conversely, the steady suppression of Tc,3D under c-axis stress cannot be attributed to an enhancement of spin-stripe order as Vm remains nearly unchanged. Moreover, c-axis stress does not alter the crystal symmetry (as shown in Fig. 4). However, stress likely induces subtle local lattice distortions that modify the spatial distribution of d-wave SC. Expanding on this, the charge density within the two-dimensional d-wave patches differs from both the layer-averaged charge density and that of the stripy regions. Consequently, adjacent patches likely experience Coulomb repulsion. Stress applied along the c-axis enhances this effect by reducing the interlayer distance, which could, in turn, favor an anticorrelated arrangement of patches in neighboring layers, thereby decreasing their overlap. This would further enhance frustration in the interlayer Josephson coupling.

Although our measurements do not directly capture the onset of two-dimensional (2D) SC order (Tc,2D), it can be reasonably approximated based on the well-established fact that, under ambient pressure, Tc,2D ≈ Tso16,17,47. Consequently, the significant energy scale separation between Tc,3D and Tso under c-axis stress strongly suggests that 2D SC order (PDW state) persists despite the suppression of 3D superconductivity.

Conclusions

In conclusion, we investigated the evolution of spin-stripe order and superconductivity in LBCO-0.115 under c-axis compressive stress using muon spin rotation and AC susceptibility measurements. Our results reveal that while stress induces a modest suppression of 3D superconductivity, spin-stripe order remains largely unaffected. The persistence of a significant energy scale separation between Tc,3D and Tso throughout the applied c-axis stress regime suggests that 2D SC correlations continue to coexist with the spin-stripe order in this system. The contrasting response of stripe and SC correlations to in-plane and c-axis stress underscores the profound role of crystallographic anisotropy in controlling phase competition in LBCO. Our findings emphasize the need for a comprehensive theoretical framework to understand the stress-driven interplay between 3D superconductivity and spin order in the stripe phase of cuprates, offering potential insights into high-Tc superconductivity. They also pose critical questions regarding the precise impact of c-axis stress on crystal structure and charge-stripe order, emphasizing the need for further experimental exploration.

As an outlook and future direction, an intriguing avenue for exploration is the behavior of interlayer plasmons50 and their evolution in response to a potential reorganization of d-wave superconducting patches. Given the interplay between charge density variations and stress-induced changes in interlayer spacing, the spatial arrangement of these patches could significantly impact plasmonic excitations. In particular, an anticorrelated positioning of patches in neighboring layers may disrupt the coherence of interlayer Josephson coupling, leading to distinct plasmonic signatures. Understanding this evolution could offer valuable insights into the role of electronic inhomogeneity in shaping collective charge dynamics in layered superconductors.

Methods

Principles of the μSR technique

In a μSR experiment (see the Supplementary Note 1 and the Supplementary Fig. S1), a high-intensity beam of 100% spin-polarized positive muons is directed into the sample, where the muons thermalize at interstitial lattice sites and act as highly sensitive local magnetic probes. The muons, carrying a momentum of pμ = 29 MeV/c, experience the local magnetic field Bμ, causing their spins to precess at the Larmor frequency ωμ = 2πνμ = γμBμ, where γμ/(2π) = 135.5 MHz/T is the muon gyromagnetic ratio.

The muon-spin relaxation (μSR) experiments under zero field (ZF) and weak transverse field (wTF), the latter being applied parpenicular to the initial muon-spin polarization, were performed at the Dolly spectrometer (πE1 beamline) at the Paul Scherrer Institute, Villigen, Switzerland. In this technique, the implanted muons decay with a mean lifetime of τμ = 2.2 μs, emitting positrons preferentially along the muon spin direction. A set of detectors surrounding the sample records the arrival of muons and the subsequent emission of positrons. The detection process begins when a muon enters the sample, initiating an electronic clock. This clock stops when the decay positron is detected in one of the positron detectors, and the time interval is stored in a histogram. This process is repeated for millions of muon decay events, creating a time-resolved positron count histogram.

In the Dolly instrument, the sample is surrounded by four positron detectors–Forward, Backward, Left, and Right–positioned relative to the muon beam direction. The recorded positron counts in these detectors, Nα(t) (where α = F, B, L, R), follow an exponential decay due to the muon’s finite lifetime. This count distribution also includes a time-dependent polarization function P(t), which encodes the information about the local magnetic environment. The equation governing this decay is given by:

$${N}_{\alpha }(t)={N}_{0}{e}^{-t/{\tau }_{\mu }}\left[1+{A}_{0}P(t){n}_{\alpha }\right]+{N}_{{{{\rm{bg}}}}}$$
(1)

where N0 is proportional to the number of recorded events, A0 is the initial asymmetry factor dependent on detector geometry and positron scattering, and Nbg represents the background contribution from uncorrelated events. The initial asymmetry A0 typically ranges between 0.2 and 0.3.

Since positrons are preferentially emitted along the muon spin direction, the signals recorded by the Forward and Backward detectors exhibit oscillations corresponding to the muon precession frequency. To remove the exponential decay component associated with the muon’s finite lifetime, a reduced asymmetry function A(t) is introduced:

$$A(t)=\frac{{N}_{F,L}(t)-{N}_{B,R}(t)}{{N}_{F,L}(t)+{N}_{B,R}(t)}={A}_{0}P(t)$$
(2)

where NF,L(t) and NB,R(t) denote the positron counts recorded in the Forward/Left and Backward/Right detectors, respectively. The asymmetry function A(t) and the polarization function P(t) provide direct insight into the static and dynamic properties of the local magnetic environment at the muon site. These functions serve as a powerful tool for investigating the spatial distribution of magnetic fields and their fluctuations in complex materials. For clarity, all ZF and TF μSR asymmetry spectra shown in the main text are plotted after normalization by the initial asymmetry A0 obtained in the high-temperature paramagnetic state.

Analysis of ZF-μSR data

The ZF-μSR signals shown in were analyzed over the entire temperature range by decomposing the asymmetry signal into contributions from both magnetic and nonmagnetic regions. The zero-field μSR polarization function is expressed as (see also the Supplementary Note 2):

$${P}_{{{{\rm{ZF}}}}}(t) = \, {V}_{{{{\rm{m}}}}}\left[{f}_{\alpha }{e}^{-{\lambda }_{T}t}{J}_{0}({\gamma }_{\mu }{B}_{{{{\rm{int}}}}}t)+(1-{f}_{\alpha }){e}^{-{\lambda }_{L}t}\right]\\ + (1-{V}_{{{{\rm{m}}}}}){e}^{-{\lambda }_{{{{\rm{nm}}}}}t}.$$
(3)

Here, Vm represents the magnetically ordered volume fraction, while Bint denotes the maximum internal field associated with the Overhauser distribution. The depolarization rates λT and λL characterize the transverse and longitudinal relaxation of the magnetic regions, respectively. The parameters fα and (1 − fα) correspond to the fractions of the oscillating and non-oscillating components of the magnetic μSR signal. The function J0 is the zeroth-order Bessel function of the first kind, which describes the incommensurate nature of the spin-density wave and accounts for broad internal field distributions spanning from zero to a maximum value. This behavior is commonly observed in cuprates exhibiting static spin-stripe order4,30,33,47. λnm represents the relaxation rate of the nonmagnetic volume fraction, where spin-stripe order is absent. The complete analysis of μSR time spectra, including both zero-field (ZF) and transverse-field (TF) measurements, was performed using the open-source μSR data fitting software musrfit51.

Analysis of weak-TF μSR data

The weak-transverse-field (μSR) asymmetry spectra were analyzed using a model describing the time evolution of the muon-spin polarization (see also the Supplementary Note 3):

$${P}_{{{{\rm{TF}}}}}(t)={P}_{{{{\rm{TF}}}}}(0){e}^{-\lambda t}\cos (\omega t+\phi ),$$
(4)

where PTF(t) represents the muon-spin polarization at time t after implantation, PTF(0) denotes the initial polarization amplitude linked to the paramagnetic volume fraction, and λ is the depolarization rate associated with paramagnetic spin fluctuations and nuclear dipolar interactions. The frequency ω corresponds to the Larmor precession, which depends on the applied transverse magnetic field, while ϕ accounts for the phase offset.

For magnetically ordered samples, baseline asymmetry shifts were corrected by allowing PTF(t) to be adjusted for each temperature. The magnetically ordered volume fraction at temperature T was determined using the relation:

$${V}_{{{{\rm{m}}}}}(T)=1-{P}_{{{{\rm{TF}}}}}(0,T).$$
(5)

In the high-temperature paramagnetic phase (T > Tso), where no long-range magnetic order exists, the initial polarization reaches its maximum value, PTF(0, T > Tso) = 1.

In weak-TF μSR measurements, the signal typically consists of long-lived oscillations from muons precessing in the applied external field, alongside strongly damped oscillations from muons in magnetically ordered regions, where they experience a broad internal field distribution due to the combined influence of applied and internal magnetic fields. ZF-μSR measurements reveal rapid depolarization on a timescale of ~0.5 μs. However, in TF-μSR data, variations in the angle between the applied and internal fields create a highly inhomogeneous field distribution in magnetically ordered regions, leading to significant muon-spin dephasing. This, combined with data binning, results in the damped signals from magnetic patches being effectively suppressed in the weak-TF μSR spectra.

Estimation of applied c-axis stress

A detailed description of the uniaxial stress apparatus and the calibration of the force sensor is provided in refs. 4,44. Based on this calibration, we estimated the c-axis compressive force applied to the LBCO-0.115 crystal. As shown in Fig. 4, the change in the force sensor signal from the zero-force baseline to the maximum applied stress corresponds to 16.6 μV. We find that a change of 1 μV in the force sensor reading corresponds to an applied force of ≃30.5 N, resulting in a maximum applied force of ≃506 N. The LBCO-0.115 sample used in this study had dimensions of 7 mm × 4 mm × 0.6 mm (length × width × thickness), with uniaxial stress applied along the c-axis (i.e., across the 0.6 mm thickness). Given the cross-sectional area of 2.4 mm2, this yields a maximum applied stress of ~0.21 GPa.