## Nuclear magnetic resonance

Figure 1(a) shows the representative 35Cl NMR spectra for *µ*0*H*=10.5 T//*b*. We observe six Lorentzian lines at high temperatures. As the temperature is lowered below 14 K, the central lines (P3 and P4) are merged into a single Gaussian line. The observed peak splitting can not be explained by simple cylindrical electric field gradient (EFG) (see Supplementary Fig. 1). Rather, it should be understood in terms of asymmetric internal fields at the 35Cl sites, in conjunction with spatial electric field gradient EFG distortion due to the low symmetry orthorhombic structure.

To determine the thermal and field evolution of spin dynamics, we measured the spin-lattice relaxation 1/*T*1 for *µ*0*H*//*c* and *µ*0*H*//*b*, as plotted in Fig. 1(b). With decreasing temperature, 1/*T*1||*b* shows a broad maximum at around *T*max = 50 K (comparable to the energy scale of *J*) and decreases rapidly down to 20 K, forming a dip. On further cooling, 1/*T*1||*b* shows a power-law increase down to 4 K. On the other hand, 1/*T*1||*c* exhibits a weaker power-law-like decrease below *T*max and then flattens out below 20 K. The small increment below 5 K alludes to the development of 3D short-range magnetic correlations as *T*→*T*N=1.13 K. Such a disparate thermal evolution of 1/*T*1 between the intra- and interchain directions implies the spatial anisotropy of quasiparticle excitations and spin dynamics inherent in the ATL above *T*N.

Figure 2 exhibits the low-*T* dependence of 1/*T*1 in selected magnetic fields 3.2, 6, 8.25, and 10.5 T. 1/*T*1(*T*, *µ*0*H*) is well described by the power-law behavior 1/*T*1∝*T**α* below 20 K (Supplementary Fig. 2). This together with the *T*-linear heat capacity and gapless spinon excitation31–33 corroborates the stabilization of the TLL regime11,34,35. In case the transverse (longitudinal) spin fluctuations are dominant in the TLL phase, the spin-lattice relaxation rate follows 1/*T*1∝*T*1/2*K*−1 (1/*T*1∝*T*2*K*′−1)34–37. Here, *K* and *K*′ are the TLL parameters that characterize the sign and strength of the interactions between the spinless fermions, for the cases of predominant transverse and longitudinal fluctuations, respectively.

In consideration of the Ca3ReO5Cl2 crystal structure, the transverse (longitudinal) fluctuations couple to 35Cl through the isotropic (anisotropic) terms of the hyperfine interaction tensor. The Knight shift and the full width at half maximum (FWHM) of the NMR spectrum are related to the magnetic response of the 35Cl nuclei via the hyperfine coupling tensor A. Our 35Cl NMR results reveal that the Knight shift (~ 10 mT) is about twice larger than the FWHM (~ 5 mT). Therefore, the transverse fluctuations predominantly contribute to 1/*T*1, leading to the predicted power-law relation 1/*T*1∝*T*1/2*K*−1. The field dependence of the TLL parameter *K* is displayed in Fig. 2(e). We find that *K* is in the range of 0.6 < *K* < 0.85 and increases towards 1 as *µ*0*H*→0 T. As such, Ca3ReO5Cl2 realizes repulsive TLLs (*K* < 1) as reported in a majority of quasi-1D AFM compounds, including a strong-rung ladder and a spin chain34,35. Our observations motivate future theoretical calculations that illuminate whether the attractive and repulsive TLL regimes can be tuned by the magnetic parameter in ATLs. Taken together, our 35Cl NMR data advocate that Ca3ReO5Cl2 hosts repulsive spinon excitations and is in the vicinity of non-interacting (*K* = 1) or 1D quantum critical regime in zero field.

## Muon spin relaxation

Complementary to 35Cl NMR, we carried out the LF-*µ*SR experiments to shed light on anisotropic and quantum critical spin dynamics. The LF dependence of the slow muon spin relaxation rate λs(*H*LF) is well described by the Redfield formula38, while the fast λf(*H*LF) negates its description within 1D diffusion, ballistic, or Redfield formula (Supplementary Fig. 4). This suggests that λf(*H*LF) is substantially influenced by anisotropic spin diffusion39.

In the presence of anisotropic electronic spin diffusion, the LF dependence of the muon spin relaxation rate follows40,

$${{\lambda }}_{\text{f}}\left({H}_{\text{L}\text{F}}\right)=\frac{1}{20}\left[3{D}^{2}f\left({\omega }_{\mu }\right)+\left(5{A}^{2}+7{D}^{2}\right)f\left({\omega }_{e}\right)\right]$$

1

,

where *ω**µ* = *γ**µ**H*LF and *ω**e* = *γ**e**H*LF, *A* and *D* are the contact and dipolar hyperfine coupling constant, respectively, and *f*(*ω**µ*,*e*) is the spectral density. Because *A* and *D* cannot be determined by the *µ*SR experiments, we assume *D*≫*A* as expected for magnetic insulators and deduce the value of *D* = 46.21 mT/*µ*B from the 35Cl NMR results (Supplementary Fig. 2). In addition, *f*(*ω**e*)≫*f*(*ω**µ*) since there is no evidence for muon diffusion. Based on these assumptions, the latter term in Eq. (1) is predominant and the anisotropic spin diffusion is given by41,

$${{\lambda }}_{\text{f}}\left({H}_{\text{L}\text{F}}\right)={{\lambda }}_{\text{f}}\left(0\right){\left(\frac{1+\sqrt{1+{\left({H}_{\text{L}\text{F}}/2{H}_{\text{c}}\right)}^{2}}}{2\left(1+{\left({H}_{\text{L}\text{F}}/2{H}_{\text{c}}\right)}^{2}\right)}\right)}^{n}$$

2

.

Here, \({{\lambda }}_{\text{f}}\left(0\right)=7{D}^{2}/20\sqrt{2{D}_{\parallel }{D}_{\perp }}\) is the constant value for *H*LF<*H*c with *H*c taken as the cutoff magnetic field. For 1D QHAFs, *D*|| and *D*⊥ correspond to the intra- and interchain spin diffusion rates, respectively. In the case of *n* = 0.5, Eq. (2) is tantamount to the 1D spin diffusion *H*LF−1/2. Note that the exponent *n* is related to the spin-spin correlations or diffusive behavior. For the exponential spin correlation function, \(\mathcal{S}\left(t\right)=⟨{S}\left(t\right)\cdot {S}\left(0\right)⟩\propto {e}^{-\nu t}\), the spectral density follows a quadratic field dependence \(\mathcal{S}{\left(\omega \right)}^{-1}\propto 1/{{\lambda }}_{\text{f}}\propto {\left({\gamma }_{\mu }{H}_{\text{L}\text{F}}\right)}^{2}\). On the other hand, the power-law decaying spin correlation function, \(\mathcal{S}\left(t\right)\propto {t}^{-\left(1-n\right)}={t}^{-1/z}\), the spectral density is described by \(\mathcal{S}\left(\omega \right)={\omega }^{-n}\). Recent theoretical studies of the 1D *S*=1/2 QHAF proposed a thermal crossover from low-T ballistic (*z*=1) or diffusive (*z*=2) TLL to high-*T* superdiffusive (*z*=3/2) hydrodynamics42. Here, the dynamical exponent *z* depends on the microscopic model of the system.

As displayed in Figs. 3(a)-(d), λf(*H*LF) is well reproduced by the anisotropic spin diffusion model. With increasing temperature, *H*c also increases gradually. Notably, n decreases from 1.0(1) to 0.70(5) as the temperature is lowered (see the inset of Fig. 3(e)), reflecting the dimensional reduction from 2D ATL to 1D spin chain. This observation is well consistent with the previous results31–33. From the fittings using Eq. (2), we extract the anisotropic spin diffusion rates *D*|| and *D*⊥ as a function of temperature in Figs. 3(e) and 3(f). *D*⊥ linearly decreases upon cooling, whereas *D*|| is nearly temperature-independent *D*||avg=1.86×1010 s− 1. The reduction of *D*⊥ with decreasing temperature indicates the slowing down of spin fluctuations along the interchain direction (*c*-axis). The distinct thermal evolution of *D*|| and *D*⊥ along with the *T* dependence of the exponent *n* supports the notion that one-dimensionalization induced by geometrical frustration is reflected in the disparate spin diffusion.

## Scaling behaviors

Having established the occurrence of TLL by dimensional reduction and the proximate non-interacting and/or 1D QC in zero field, we naturally expect the universal scalings of spin correlation functions.

Figure 4 manifests the scaling behaviors of 35Cl 1/*T*1 and the muon spin polarization *P*z(*t*). We observe that 1/*T*1 at different fields collapses onto a constant value below 20 K in the 1/*T*1*T*− 0.5 vs. *T* plot. In addition, the muon spin polarization shows the time-field scaling with the exponent of *γ* = 0.5. The scaling exponent reflects that within the *µ*SR time window, the spin-spin correlations have a power-law decay as a function of time. Notably, both the non-interacting TLL (*z* = 1) and 1D QC (*z* = 2) can lead to the predicted scaling argument 1/*T*1∝*T*− 0.5 17,43. From our LF-*µ*SR data, we infer the dynamical exponent *z* = 3.33 (*n* = 0.70), pointing out the proximity to the 1D QC (*z* = 2). Therefore, the concomitant scaling behaviors in 1/*T*1 and *P**z*(*t*) should be taken as evidence for the quantum critical TLL phase.

In summary, a local probe study of Ca3ReO5Cl2 demonstrates the stabilization of TLL by dimensional crossover from the 2D ATL to the 1D spin chain. The thermal evolution of the anisotropic spin diffusion provides evidence for one-dimensionalization. In the TLL phase, the spinons are nearly non-interacting (or weakly coupled via repulsive interactions). The scaling behavior observed in two different experiments corroborates the 1D QC in the weakly interacting (or non-interacting) TLL phase. Our findings call for in-depth theoretical and experimental investigations of TLL physics in an ATL.